APPLYING DATA MINING TO JOB-SHOP SCHEDULING USING
REGRESSION ANALYSIS
A thesis presented to
the faculty of
the Fritz J. and Dolores H. Russ
College of Engineering and Technology of Ohio University
In partial fulfillment
of the requirements for the degree
Master of Science
Alok D. Innani
August 2004
This thesis entitled
APPLYING DATA MINING TO JOB-SHOP SCHEDULING USING
REGRESSION ANALYSIS
BY
ALOK D. INNANI
has been approved for
the Department of Industrial and Manufacturing Systems Engineering
and the Russ College of Engineering and Technology by
David A. Koonce
Associate Professor of Industrial & Manufacturing Systems Engineering
R. Dennis Irwin
Dean, Fritz J. and Dolores H. Russ
College of Engineering and Technology
INNANI, ALOK D. M.S. August 2004. Industrial and Manufacturing Systems
Engineering
Applying Data Mining to Job-Shop Scheduling using Regression Analysis (188 pp.)
Director of Thesis: David Koonce
Regression analysis is an established statistical analysis technique that has been
employed to learn more about the relationship between several predictor variables and a
dependent variable. The purpose of using this technique for scheduling job-shop
problems is to obtain better schedules.
Earlier research has shown that attribute-oriented induction can be used to learn
from job-shop scenarios to develop rules. The approach followed in this research was to
use test problems to create a regression equation. The solutions of the genetic algorithms
(GA) were extended to include other attributes. The solutions had the following
attributes: job number, operation number, processing time, remaining processing time
and machine load. The solutions were used to develop the regression equation. Using this
equation, new scenarios were scored and the results were sorted in an ascending priority
to develop new schedules. These schedules were developed incorporating a left-shift
technique. This was followed by analysis of the problems with the shortest processing
time (SPT) heuristic. Attribute-oriented induction rules were then applied to the same
problems. The makespans for the obtained sequences were developed by the same
procedure of plotting by Gantt chart.
The results from all the four techniques being obtained, the next step was to
perform statistical analysis. An ANOVA on the makespans showed that the best
schedules were obtained by the genetic algorithms. Regression analysis provided
makespans that were identical to slightly higher than those obtained by the GA. The
results were also subject to multiple comparison technique to determine which two
techniques provided closer solutions. Dunnett’s test showed that GA and regression
analysis solutions were much closer than those provided by the SPT heuristic and the
attribute-oriented induction rules.
Approved:
David Koonce
Associate Professor of Industrial and Manufacturing Systems Engineering
5
TABLE OF CONTENTS
ABSTRACT ........................................................................................................................3
LIST OF TABLES.............................................................................................................. 7
LIST OF FIGURES ............................................................................................................ 8
CHAPTER 1.
INTRODUCTION .................................................................................. 9
1.1
Scheduling Problems .............................................................................. 9
1.2
Description of a job-shop...................................................................... 10
1.3
Statement of Purpose ............................................................................ 11
CHAPTER 2.
BACKGROUND .................................................................................. 14
2.1
Classical Job-shop Scheduling Problems (JSSP).................................. 14
2.2
Approaches to solve JSSP..................................................................... 20
2.3
Genetic Algorithms (GA) ..................................................................... 34
2.3.1 Description of Genetic Algorithm (GA) ............................................... 35
2.3.2 Working of Genetic Algorithm............................................................. 37
2.3.3 Application of genetic algorithms to job-shop scheduling problems ... 43
2.4
Knowledge Discovery in Databases (KDD) and Data Mining............. 45
2.4.1 The process of Knowledge Discovery in Databases............................. 46
2.4.2 An Overview of Data Mining ............................................................... 48
2.4.3 Attribute-Oriented Induction in Data Mining ....................................... 49
2.4.4 Application of Data mining to job-shop scheduling problems ............. 56
2.5
CHAPTER 3.
3.1
Regression Analysis.............................................................................. 57
METHODOLOGY ............................................................................... 62
Approach............................................................................................... 62
6
3.2
Attribute-Oriented Rule Sets................................................................. 63
3.3
Procedure .............................................................................................. 70
CHAPTER 4.
4.1
CHAPTER 5.
RESULTS ............................................................................................. 73
Statistical Analysis................................................................................ 73
CONCLUSIONS AND FUTURE RESEARCH .................................. 80
5.1
Conclusions........................................................................................... 80
5.2
Future Research .................................................................................... 81
REFERENCES ................................................................................................................. 84
APPENDIX A TEST PROBLEMS ................................................................................. 88
APPENDIX B GA SOLUTIONS AND MAKESPANS (IN LAST ROW) .................... 90
APPENDIX C REGRESSION ANALYSIS.................................................................. 173
APPENDIX D SCHEDULES GENERATED BY SPT PROCEDURE........................ 180
APPENDIX E NORMAL DISTRIBUTION PLOTS OF MAKESPANS .................... 185
7
LIST OF TABLES
Table 2.1 A 4x3 job-shop problem ................................................................................... 15
Table 2.2 A 3x3 job-shop problem ................................................................................... 18
Table 3.1 Generic rules for the rule-set ............................................................................ 67
Table 3.2 Rule set of 47 rules learned from 10 cases ....................................................... 69
Table 4.1 Makespans obtained by GA, Regression, SPT and Induction rules techniques 77
8
LIST OF FIGURES
Figure 2.1 A Gantt chart representation of a solution to the 4 x 3 job-shop problem in
Table 2.1 ................................................................................................................... 16
Figure 2.2 A Gantt chart representations of Semi-active and Active schedules............... 19
Figure 2.3 A Venn diagram illustrating the relation between active, semi-active and nondelay schedules ......................................................................................................... 20
Figure 2.4 A comparison of conventional and genetic approaches .................................. 37
Figure 2.5. A binary representation of chromosomes....................................................... 39
Figure 2.6 An overview of the steps comprising the KDD process.................................. 47
Figure 2.7 Examples of concept hierarchy representation........................................... 54-55
Figure 3.1 A Rule Network tree........................................................................................ 66
Figure 4.1 ANOVA statistics for the design database ...................................................... 74
Figure 4.2 Blocked ANOVA showing difference in the techniques ................................ 78
Figure 4.3 Blocked ANOVA showing difference in the techniques without SPT ........... 79
9
CHAPTER 1. INTRODUCTION
1.1 Scheduling Problems
Scheduling is the allocation of resources over time to perform a collection of tasks.
Pinedo [58] describes it as a decision-making process with the goal of optimizing one or
more objectives. In a job-shop it is concerned with the simultaneous and synchronized
ordering of operations on several machines. In manufacturing, the scheduling function
interacts with other decision making functions such as Materials Requirement Planning
(MRP) within the plant. The four primary stages of scheduling decisions are formulation,
analysis, synthesis and evaluation [5]. Scheduling theory is concerned primarily with
mathematical models which relate to the scheduling function. The development of useful
models is the continuing interface between theory and practice.
The theoretical or quantitative approach attempts to capture the problem in concise
mathematical form, in which the decision making goals are translated into explicit
objective functions and the decision making restrictions into explicit constraints. A
solution to a scheduling problem is any feasible resolution of the two types of constraints
namely limits on the capacity of available resources and technological restrictions on the
order in which tasks can be performed. In a manufacturing organization production
scheduling is a unifying problem which relates such diverse elements of the organization
as sales, cost control, purchasing, capital budgeting and many others. The complexity of
10
scheduling increases with the increasing number of jobs, resources and constraints.
Essentially each scheduling problem is an optimization problem defined on the finite set
of so-called active schedules. The large cardinality of this set makes the complete
enumeration of all elements unfeasible. A common feature of many of these NP-hard
problems is that no efficient algorithm is known yet for solving it to optimality in
polynomial time. Dynamic problems are those in which jobs arrive randomly over a
period of time while those in which the processing times are uncertain are called
stochastic. French [28] provides a detailed discussion for solving various types of
machine scheduling problems using approaches such as branch and bound, integer
programming, algorithms and heuristics.
1.2 Description of a job-shop
Job-shop scheduling is an activity that comprises of a set of jobs to be processed on
a set of machines. The job-shop scheduling problem can be defined as the allocation of
machines over time to perform a collection of jobs to minimize/maximize a specific
performance measure while satisfying the operation precedence constraints, machine
capacity constraints, processing time and ready time requirements. The resources thus are
the machines and the basic task modules are called jobs. Each job may be comprised of
several elementary tasks called operations, which are interrelated by precedence
constraints. The processing of an operation requires the use of a particular machine for an
uninterrupted duration, called the processing time of the operation. Each machine can
process only one operation at a time. The routing, processing times and precedence
11
constraints are specified by a process plan. The main distinction between the classic flowshop and a job-shop is that, in the former case each job passes the machines in the same
order whereas in the latter case the machine order may vary per job. Since workflow in a
job-shop is not unidirectional, scheduling becomes quite tedious. For a particular jobshop process plan, several feasible schedules can be generated. Given a cost function by
which the cost of each possible solution can be measured, the processing order on each
machine that minimizes the corresponding cost is desired. The optimality of a schedule is
thus decided by the objectives such as minimization of process cost, makespan and
flowtime or maximization of throughput, system/ resource utilization and production rate.
1.3 Statement of Purpose
An optimal solution for a general job-shop scheduling problem is known to be hard
to achieve, particularly when the size of the problem is large. While mathematical models
such as integer and linear programming have been developed to solve similar constraint
based problems, the computational cost associated with solving the job-shop scheduling
problem increases with increasing numbers of jobs, resources and constraints. The
computation and analysis of these problems is time-consuming. Heuristics such as
dispatching rules, search algorithms and Artificial Intelligence (AI) approaches have been
used to solve the job-shop problem. Stochastic search techniques such as Genetic
Algorithms (GA), based on the evolutionary concepts of evolution and survival of the
strongest genes, provide solutions that are optimal or near optimal. GA’s however, are
considered unreliable, as they do not guarantee an optimal or even good schedule for
12
every run. Most of the heuristic job-shop scheduling procedures are based on priority
dispatching rules such as Shortest Processing Time (SPT), First Come First Serve
(FCFS), Most Work Remaining (MWKR), etc. A dispatching rule is used to select the
next job to be processed from a set of jobs requiring processing. The subset of operations
is designed to produce a schedule such that no operation can be started earlier without
delaying some other operation and no machine is idle at a time when it could begin
executing some operation. Heuristic procedures of the greedy type are fast and usually
find adequate solutions. However with the rapid increase in the speed of computing and
the growing needs of efficiency in scheduling, it becomes increasingly important to
explore ways of obtaining better schedules at minimal computational cost, short of going
all the way of finding a guaranteed optimal solution by explicit enumeration.
This thesis presents an approach for incorporating multivariate analysis into the
data mining process, and shows what proportion of the variability of the dependent
variable, in this case the objective function, is uniquely explained by each individual
predictor. Regression analysis was used to learn from good or optimal schedules for a 6 x
6 job-shop. In these schedules, quality was based on the objective to minimize the
makespan, i.e. the time needed for processing all the jobs. These good schedules were
generated by a genetic algorithm, an approach which has shown in prior research [44] to
produce good solutions to problems of this size.
The schedules and makespans obtained by the developed regression model are
applied to 19 test problems and compared with schedules obtained by the GA scheduler,
13
as well as solutions obtained by using the SPT heuristic. Finally the problems were
subject to the rule sets developed by Kantak [44] using the data mining approach of
attribute-oriented induction.
14
CHAPTER 2. BACKGROUND
2.1 Classical Job-shop Scheduling Problems (JSSP)
The classic job-shop scheduling problem is a well known machine scheduling
problem and among the hardest combinatorial optimization problems [29]. Due to the
inherent intractability of the problem, heuristic procedures are often employed. The
classic JSSP consists of n jobs to be scheduled on m different machines with explicit
processing times is denoted as an n x m problem. Each job is composed of a sequence of
operations and the operation order on the machines is pre-specified. Each operation is
characterized by the required machine and the fixed processing time. The process plan
specifies the routing, processing times and precedence constraints among operations of
each job. There are several constraints on jobs and machines:
•
a job does not visit the same machine twice
•
there are no precedence constraints among operations of different jobs
•
an operation cannot be preempted
•
each machine can process only one job at a time
•
neither release times nor due dates are specified
•
there are no machine breakdowns throughout the scheduling process and
•
the job-shop is static and deterministic in nature i.e., there is no randomness
as the jobs, machines, processing times and all other parameters necessary
for defining the job-shop problem are known and fixed.
15
Table 2.1 is an example of a 4 x 3 job-shop problem. The data is in the format
(Machine, Processing Time) and shows the machine assignments for all jobs, i.e. routing.
Table 2.1 A 4x3 job-shop problem
Job
1
2
3
4
1
(1 , 4)
(2 , 1)
(3 , 3)
(2 , 3)
Operation
2
(2 , 3)
(1 , 4)
(2 , 2)
(3 , 3)
3
(3 , 2)
(3 , 4)
(1 , 3)
(1 , 1)
In the example, the routing of job 1 is through machine 1 for 4 time units, machine
2 for 3 time units and finally through machine 3 for two time units. The precedence
constraints are thus imposed on the operations for each job. The simplest and most
widely used model to aid in decision making for a variety of scheduling problems is a
Gantt chart, as shown in Figure 2.1, which is a graphical representation of resource
allocation over time, thus conveying basic information about system status for scheduling
purposes.
16
2,2
Machine 1
1,1
4,1
Machine 2 2 , 1
3,2
3,1
Machine 3
1
4,3
1,2
4,2
2
3
4
5
3,3
2,3
6
7
8
9
1,3
10
11
12
13
14
TIME
Figure 2.1 A Gantt chart representation of a solution to the 4 x 3 job-shop problem in
Table 2.1
For any job-shop problem an infinite number of feasible schedules can be
generated as arbitrary amounts of idle times can be inserted at any machine between
adjacent pairs of operations [54]. It can be stated that superfluous idle time exists in a
schedule if some operation can be started earlier in time without varying the operation
sequences on any machine. Since, it is desirable that the operations be processed as
compactly as possible, superfluous idle time should be eliminated. A schedule should be
a feasible resolution of the logical constraints such that all operations of each given job
can be placed on one time axis in precedence order and without overlap. Each feasible
schedule can be classified as semi-active, active and non-delay schedule.
Semi-active schedules: An improvement in the objective of the set of all feasible
schedules can be brought about by moving operations towards the left in a Gantt chart.
17
This results in reduced idle times while preserving the operation sequences. This type of
adjustment which results in a compressed schedule on a local machine without modifying
the sequence of operations is known as local left-shift. The set of all schedules in which
no local-left-shift can be made is called the set of semi-active schedules. It is the most
compact feasible schedule with no surplus idle time.
Active schedules: An improvement can be brought about in the semi-active schedules by
shifting some operations ahead of some operations in the schedule without delaying their
processing. This involves shifting an operation into a slot earlier in the sequence such that
the feasibility of the schedule is preserved. This type of shift is knows as a global left
shift. The set of all schedules in which no global left shift can be made is called the set of
active schedules and are a subset of the semi-active schedules.
Non-delay schedules: A dispatch or non delay schedule is one in which no machine is
kept idle at a time when it could begin processing some operation. All non delay
schedules are active schedules since no left-shifting is possible.
Table 2.2 is an example of a 3 x 3 job-shop problem. The data is in the same
format as the previous example, i.e., (Machine, Processing Time) and the routing of jobs
is shown.
18
Table 2.2 A 3x3 job-shop problem
Job
1
2
3
1
(1 , 3)
(1 , 1)
(2 , 3)
Operation
2
(2 , 3)
(3 , 5)
(1 , 2)
3
(3 , 2)
(2 , 3)
(3 , 3)
Active schedules are generally the smallest dominant set in the job-shop problem
[5]. The non-delay or dispatch rules are smaller in number but are not dominant. The best
non-delay schedule may often be a very good solution, if not optimal. A discussion on the
characteristics and classification of schedules is provided by Conway et al. [13]. Figure
2.2 (a) shows a semi-active schedule and Figure 2.2 (b) shows an active schedule
generated by left-shifting the semi active schedule.
19
Machine 1
1,1
Machine 2
3,1
2,1
3,2
1,2
2,3
Machine 3
2,2
1
2
3
4
5
1,3
6
7
TIME
8
9
10
3,3
11
12
13
14
11
12
13
14
(a). Semi-active schedule with no local left-shift
Machine 1 2 , 1
Machine 2
3,2
1,1
3,1
1,2
2,2
Machine 3
1
2
3
2,3
1,3
3,3
4
5
6
7
TIME
8
9
10
(b). Active schedule with no global left-shift
Figure 2.2 A Gantt chart representations of Semi-active and Active schedules
A better way to look at the role of semi-active and active schedules is with the use
of a Venn diagram. The large rectangle represents the set of all schedules. The point
represents some optimal schedule which lies in the active subset. Figure 2.3, adapted
from Morton and Pentico [54], represents a Venn diagram demonstrating the relationship
between various schedules.
20
All Schedules
Semi-Active
y
Active
Non-Delay
Figure 2.3 A Venn diagram illustrating the relation between active, semi-active and nondelay schedules
2.2 Approaches to solve JSSP
A great deal of research has been focused on solving the job-shop problem,
resulting in a wide variety of approaches. The JSSP is among the hardest combinatorial
optimization problems and due to its intractability it is considered to belong to the class
of decision problems which are NP [29]. It is not only NP-hard but is one of the worst
members in the class. An indication of this is given by the fact that one 10 x 10 problem
formulated by Muth and Thompson [55] remained unsolved for over 20 years [2]. As
described by Baker [5] the number of feasible semi-active schedules for a JSSP is shown
to be bounded by (n!)m. This means that Muth and Thompson’s 10 x 10 problem had at
most 3.9594 x 1065 feasible semi-active solutions; clearly exhaustive enumeration is not
currently possible for this class of problems. Coffman [14] provides a detailed discussion
on the topic of the complexity of sequencing problems. Kan [43] develops optimizing
algorithms for the problem that may be either polynomial-bounded or of the implicit
21
enumeration type and suggests that the use of heuristics is unavoidable for some
problems. This section discusses the various approaches considered by researchers to deal
with the complex job-shop machine scheduling problem.
In one of the earliest papers in this field, by Giffler and Thompson [31], algorithms
using priority dispatching heuristics were incorporated to solve production scheduling
problems. The algorithms were designed to generate any one or all schedules of a
particular subset of all possible schedules called the active schedules. They demonstrated
that every optimal schedule is the same as an active optimal schedule. The objective
function was to minimize the length of production schedules. While their approach
helped to define the job-shop solution process, the integer programs did not lead to
practical methods of solutions. They also did not consider other optimality criteria’s such
as due dates, operation priorities and routing problems in their model.
Brooks and White [9] developed a procedure for finding optimal solutions for the
production scheduling problem incorporating a wide variety of possible criteria and
constraints, using the algorithm developed by Giffler and Thompson [31]. They
developed the proposal of using the lower bound as the decision rule for developing a
single near-optimal solution. The assumptions of negligible setup and transportation
times do not make the algorithm practical. Even for moderate size problems, the
computation time and the procedure followed are slow and thereby not cost effective.
22
The computational cost of solving these NP-hard problems increases exponentially
with the problem size. This has initiated the use of heuristic approaches based on priority
dispatching rules which produce schedules that are semi-active. Dispatching rules such as
Shortest Processing Time (SPT), Most Work Remaining (MWKR), and First Come First
Serve (FCFS) are most commonly used because of their speed and reasonable results.
Priority rules assign a value to each waiting job according to some criteria or priority and
then select the job with the minimum or maximum value. Heuristics can be simply
described as following some rule of thumb whereas scheduling rules comprise of a
combination of one or more priority rules and /or one or more heuristics.
Panwalker and Iskander [57] have carried out the most well known and
comprehensive survey of scheduling heuristics. A total of 113 priority dispatching rules
are presented, reviewed and classified. They have categorized these rules as per the
specific area of application.
The first category includes simple priority rules that are based on some information
related to a job such as setup times, due date, processing time, costs, slack, arrival times
etc. Rules based on random selection which are not based on specific job attributes are
also included in this section.
The second section comprises of some combination of simple priority rules which
work by dividing a queue into two or more priority groups. Different groups are then
subject to application of different rules. In many instances such as FIFO/ SI or Cost/
Time, two rules apply to the same queue under different circumstances.
23
The third section comprises rules that are a combination of the previous two
sections or their combination with different weights applied.
The fourth section comprises heuristic scheduling rules which involve a more
intricate consideration such as alternate routing effects, alternate operation scheduling or
expected machine loading. They are used in combination with the above three set of rules
and in some cases can involve aspects such as inserting a job in an idle time slot by visual
inspection of a schedule.
The last section comprises rules designed for a specific application or combination
of priority indexes based on some mathematical functions of job attributes or rules not
categorized in the above four sections.
Besides the various definitions and classifications of scheduling rules, Panwalker
and Iskander provide, for each rule, a reference number of all surveyed articles that use
the particular rule. They also describe the type of problem and measures of performance
used in each of the surveyed articles. Due to the form in which this exhaustive survey of
scheduling rules is presented, it can be readily used by both practitioners and researchers.
Blackstone et al. [7] provide a comparison of several dispatching rules. Their work
illustrates several measurement criteria’s used to evaluate dispatching rules and identifies
several rules that exhibit good overall performance. They present a formal definition of
34 representative dispatching rules, selected from articles or actual industrial usage.
They discuss the state of the art in the study of dispatching rules that includes
analytical approaches, simulation techniques, bias of estimated produced by simulation,
sample size and evaluation criterion. The dispatching rules have been classified and
24
discussed as per four classes namely those involving processing time, due-dates, simple
rules and a combination of the first three classes. The non-cost measurement criteria’s
used most often to study dispatching rules include flowtime, lateness and tardiness
besides cost effectiveness. Analytical methods such as queuing theory techniques have
been widely used in the study of job-shops. They also compare several of the listed
dispatching rules using the results of published studies and suggest those exhibiting good
performance. They present a discussion of secondary dispatching heuristics which are
procedures that can be used to identify circumstances under which the job indicated by
the dispatching rule should not be processed first. Such heuristics have significantly
increased the performance of some dispatching rules. They have not discussed job
sequencing procedures which is another approach to the problem of job selection.
Jones and Rabelo [42] provide a description of how different approaches, revolving
around a series of technological advances, have been applied to solving scheduling
problems. They discuss a few of the most important contributions and recent trends in
areas such as Mathematical techniques, Dispatching rules, Artificial Intelligence, Neural
Networks, Neighborhood search methods and Fuzzy logic.
Job-shop scheduling problems have been solved extensively by applying
mathematical programming techniques such as integer, mixed-integer and dynamic
programming. Due to the NP-complete nature of these problems, the use of these
applications was limited, initiating research to break down into sub-problems and
overcome these deficiencies. The authors detail two decomposition strategies proposed
by researchers. They also discuss enumerative techniques for solving the integer-
25
programming problems like branch-and bound which conceptualizes the problem as a
decision tree and Lagrangian relaxation that solves the problems by omitting specific
integer-valued constraints and adding the corresponding costs to the objective function.
They discuss various papers based on the study and classification of dispatching rules.
They provide detailed study of Artificial Intelligence approaches such as expert and
knowledge based systems and distributed scheduling systems. Artificial neural network
techniques such as supervised learning neural network, relaxation models, competitionbased unsupervised neural networks and temporal reinforcement learning have also been
discussed. They consider neighborhood search methods such as tabu search, simulated
annealing and genetic algorithms. They provide an explanation of concepts such as Fuzzy
logic, Reactive scheduling and Learning in scheduling and its beneficial incorporation in
solving machine scheduling problems.
Cheng et al. [17] provide a tutorial survey of recent works on solving classical
JSSP using genetic algorithms. They give a tutorial survey of recent works on solving the
classic JSSP using genetic algorithms. They address the key issue of encoding a GA for
solving the JSSP and present a detailed analysis of nine representation schemes for the
job-shop scheduling problem. They classify these representations into two basic encoding
approaches: direct and indirect. A direct approach encodes the solution of JSSP into a
chromosome and uses GA’s to evolve these chromosomes to obtain a better schedule.
The indirect approach varies considerably as it encodes a sequence of dispatching rules
for job assignment into the chromosome instead of a schedule and uses the GA to find out
a better sequence of dispatching rules. They also present a relative discussion on the
26
following features: Lamarkian property of chromosome, complexity of decoder, property
of encoding space and mapping and memory requirements.
In part II of this paper [18] a discussion of the various hybrid approaches of genetic
algorithms and conventional heuristics is provided. The underlying logic behind hybrid
genetic algorithms being to use the genetic algorithms to evolve an appropriate
permutation of operations, subject to constraints, and subsequently use a local search
technique in the GA’s loop of recombination and selection, to construct the solution. The
aim is to enhance the performance of genetic search by incorporating traditional heuristic
methods. They classify the hybridization methods for JSSP into the following categories:
adapted genetic operators, heuristic-featured genetic operators and hybrid genetic
algorithms. Adapted genetic operators revise or invent the genetic operators to meet the
characteristics of a particular encoding representation. The heuristic-featured approach
creates new genetic operators incorporating techniques from conventional heuristics. The
hybrid GA’s involves a combination of local search techniques with the main loop of
GA’s which provides a dual search possibility i.e., the GA’s perform global exploration
in the population while the heuristics perform local exploitation around chromosomes.
They have provided three combination techniques for hybrid GA’s. Since the authors
have not tested these encoding techniques with benchmark problems, a judgment on their
performance cannot be made.
An extension of the hybrid genetic algorithm is offered by Cheung and Zhou [19]
by incorporating sequence dependent setup times in the model. They provide a
27
framework for integrating heuristic rules with genetic algorithms to solve the scheduling
problem and also present a summary of existing researches on the topic. The hybrid
algorithm considerably reduces the size of the solution space thereby improving the
search efficiency. The hybridization does not guarantee an optimal solution as there is the
possibility of the optimal solution getting lost during the reduction of the search space.
However, a proper design and implementation of the heuristic rules should help retain the
high performance solutions. The authors demonstrate the robustness of the algorithm by
comparison with an effective heuristic method and incorporate flexibility to it by
including a small simulator, which calculates the objective function, to implement
heuristic rules. The drawback of the proposed approach is the computational time
consumed in developing the solutions.
The Shifting Bottle neck Procedure (SBP) is an approach developed to bridge the
gap between the time consuming priority dispatching rules and combinatorial
optimization methods. The SBP strategy involves sequentially sequencing the machines
one by one. Each time the bottle neck machine among those not yet sequenced is
selected. The SBP is designed to solve the problem iteratively, each time, as a single
machine scheduling sub-problem using the approach of Carlier [10] and treating the jobs
as independent. The solutions are compared and the machines are ranked on the basis of
their solution. The bottle neck machine is the unsequenced machine with the largest
solution value. The procedure sequences the bottle neck machine based on already
scheduled machines and ignores the remaining unsequenced machines. The method
selects the bottle neck machine earlier as scheduling it at a later stage would deteriorate
28
the makespan further. The selection and scheduling of a bottle neck machine restarts the
loop. All the previously scheduled machines that are liable to improvement are locally
reoptimised by solving the one machine problem again. The benefit of the SB procedure
over randomized local search methods such as tabu search and simulated annealing is that
they produce good solutions for the JSSP in relatively short computation time.
Adams et al. [1] propose some modifications to the shifting bottle neck procedure,
which uses a primitive form of iterated local search to produce substantially better
schedules than were previously computed. They refer to this technique as SBI procedure
and have found it to be better than the solutions produced by priority dispatching rules.
The results produced by the enumerative version of SB procedure (SBII) were
substantially better than those produced by the straight version. Out of the 40 problems
solved by SBI, SBII and priority dispatching rules, SBII provided better or similar results
than the dispatching rules in all the cases. The typical improvement was between 4 and
10 percent.
Many extensions to the SB procedure have been suggested making it adaptable to
job-shops with practical features such as precedence constraints between jobs,
simultaneous resource requirements and setup times.
Schutten [59] extends the SB procedure of Adams [1] for the classic job-shop to
deal with attributes such as transportation times and convergent job routings. They
discuss possible extensions and corresponding modeling of the SB procedure for
29
conditions such as: release and due dates, setup times, parallel machines, transportation
times, unequal transfer and production batches, multiple resources, down times,
convergent and divergent job routings and open shops. The authors state that the SB
procedure with extensions has performed well in companies and is currently part of a
commercial shop floor control system called “JOBPLANNER”.
Dauzere-Peres and Lasserre [20] propose a modified SB procedure which is more
general modification of Carlier’s algorithm and allows some precedence constraints
between jobs. They report several drawbacks to the strategies proposed by Adams et al.
[1]. Precedence constraints can be created between pairs of jobs on an unsequenced
machine by SBI procedures when they develop the sequence on a machine. These
constraints are known as delayed precedence constraints. They arise as sequencing a
given machine can impose conditions on the sequence on some other machine. The
processing of another job can thereby be delayed by a specified time lapse. Due to this
job dependency, it is possible that the real bottle neck machine is not selected and the
best sequence is not computed. The final solution of the SBI can thus be infeasible as
reoptimization does not guarantee a monotonic decrease of makespan [41].
Applegate and Cook [2] design and implement a new heuristic cutting- plane
scheduling method for obtaining lower bounds and a combinatorial branch and bound
algorithm. They construct an initial procedure known as “Bottle-k” (k is chosen as 4, 5
and 6). The algorithm branches by selecting each remaining machine in turn, for the last
‘k’ unscheduled machines They formulate an algorithm called “Shuffle” with Edge-finder
30
as it core. The initial solution is constructed by Bottle-k. This is then followed by Shuffle
fixing the processing order of one or a small number of heuristically selected machines.
The remainder of machines is optimally solved by Edge-finder. They combine the
heuristic and the algorithm to solve the well-known 10 x 10 problem of Muth and
Thompson [55] within a reasonable time.
Kuroda and Wang [49] introduce the fuzziness of scheduling information into the
classic job-shop scheduling problem. They define dynamic problems as types in which
jobs arrive randomly over a scheduling period and the scheduler has no information on
the jobs prior to scheduling them. They introduce the fuzziness on due-dates and
processing times into the model and detail the solution approach using the branch and
bound method. The selected performance criterion is completion time. The results
indicate the fact that the sensitivity of fuzzy information may control the performance of
the produced schedules. The methods can thus help show us which fuzzy information
should be considered and which one can be neglected under the given conditions.
Baptiste et al. [6] present both constraint based optimization algorithm and a
constrained approximation algorithm. They show that the performance of optimization
algorithms can be enhanced by using the approximation algorithms. They use two
important ideas from Applegate and Cook [2] and Nuijten [56] to develop their own
approximation algorithm. Applegate and Cook use the shuffle procedure to improve
solutions to the JSSP while Nuijten uses a randomized procedure to solve a wide range of
scheduling problems. The optimization algorithm is launched when the approximation
31
algorithm terminates. The empirical performance analysis performed shows that both the
algorithms perform well.
Binato et al. [4] apply the concept of greedy randomized adaptive search
procedures (GRASP) to the job-shop scheduling problem. GRASP is a metaheuristic
procedure for combinatorial optimization problems and is a competitive algorithm for
finding approximate solutions for JSSP. The iterative GRASP process comprises two
phases: construction and local search. The intensification-enhanced construction phase
builds a feasible solution whose neighborhood is explored by local search and the best
solution over all iterations is returned as the result. The construction phase makes use of
the Proximate Optimality Principle (POP) to correct imperfections made in this phase.
The shortcoming of the standard GRASP framework is that it does not learn from the
history of solutions found in previous iterations as it discards a solution that does not
show improvement. They have developed a specialized JSSP intensification scheme
based on the approach of Fleurent and Glover [27]. The POP principle (POP) [32] states
that good solutions of partial schedules with n operations are close to good solutions of
partial schedules with (n + 1) operations. The authors implement POP for the JSSP using
a slightly modified disjunctive graph based local search. The algorithm produced optimal
or near-optimal solutions on all instances for the 66 evaluation problems. They suggest
that the solution can achieve linear speed-up with parallel implementation and an
improved intensification scheme to search the solution space around the elite solutions
can help find better solutions on some very difficult problems.
32
Chryssolouris and Subramaniam [12] present a scheduling method that uses genetic
algorithms on a dynamic job-shop problem with multiple job routes. They demonstrate
the effectiveness of this approach by simulating a dynamic job-shop using mean job
tardiness and mean job cost as the performance measures. Their scheduling method is
based on GA’s and they replace the “generational replacement” policy of the crossover
operator with an “elitist recombination” policy thereby enhancing the performance of the
proposed method. The results indicate that the performance of the proposed method is
considerably better than common dispatching rules with respect to the obtained schedules
as well as the computational time.
Steinhofel et al. [61] present two simulated annealing-based algorithms for the jobshop scheduling problem with a makespan minimization objective. Simulated annealing
is a stochastic local search technique that was introduced as an analogy from statistical
physics of the computer simulation of the annealing process of a hot metal until its
minimum energy state is reached and is based on the independent proposals of
Kirkpatrick et al. [47] and Cerny [11] who adapt the work of Metropolis et al. [53] to
constraint optimization problems [41]. They use the model of disjunctive graphs to
represent the problem. The algorithms implemented and evaluated on several small and
large benchmark problems obtain good approximations of optimum values within a
relatively short time.
Steinhofel et al. [62] present an extended version of their earlier paper that deals
with parallelized versions of simulated annealing-based heuristics for the classic job-shop
33
scheduling problem. The same approach as that followed in the previous paper is used to
represent the problem i.e., disjunctive graph model and the objective is to minimize the
length of the longest paths. They carry out a complexity analysis of three simulated
annealing-based algorithms designed for their implementations on parallel random access
machines. The algorithms use a neighborhood relation which was introduced by
Laarhoven et al. [50] and involves transitions that always guarantee a feasibility of
schedules. They analyze potential upper bounds for famous benchmark problems of size
6 x 6 till 50 x 10.
Tagawa [64] proposes a new model of job-shop scheduling system which consists
of five decision systems and can be applied to actual shops. They outline the need for a
scheduling system that examines the accuracy of information for the entire period of time
from the order being placed by the customer to the product being completed. They
provide an outline of their new concept of job-shop scheduling system with hierarchical
decision model comprising of: order entry system, master scheduling system, job
scheduling system, operation scheduling system and dispatching system.
Blazewicz et al. [8] provide an overview of the solution techniques available for
solving the job-shop problem. They provide a detail of the literatures available on
machine scheduling and job-shops and the variety of scheduling rules that have been
considered for job-shops by various researchers. The authors provide a history of the
various approaches considered by researchers for solving the job-shop scheduling
problem starting from the 10 x 10 benchmark problem of Fisher and Thompson [26] to
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Baptiste et al., 1995 [6]. They concentrate on branch and bound algorithms, exploring
specific knowledge about the critical path of the JSSP, branching strategies and
approximation algorithms belonging to the class of opportunistic and local search
scheduling. Their analysis finds local search methods as the most powerful tools for
scheduling job-shops. They also assign importance to other methods that incorporate
problem specific knowledge either by problem decomposition, special purpose heuristics,
constraints and propagation of variables’ domain modification, neighborhood structures,
etc. or any combination of such methods.
Koonce and Tsai [48] show that data mining can be used for the extraction of
knowledge from a set of job-shop schedules produced by genetic algorithms. They apply
the data mining methodology of attribute oriented induction to explore the patterns in
solutions and generate rules that can be used to construct adequate solutions. The most
effective optimization techniques to date have been branch and bound algorithms based
on the disjunctive graph model for job-shop scheduling problem.
2.3 Genetic Algorithms (GA)
The simulation of the natural evolutionary process of life results in stochastic
optimization techniques called evolutionary algorithms, which can often outperform
conventional optimization methods when applied to real-world problems. The three main
avenues of this research are: genetic algorithms, evolutionary programming and evolution
strategies. Among them, genetic algorithms are the most widely known type of
35
evolutionary algorithms today. GA’s are inspired by Darwin’s theory of evolution which
states that evolutionary process is governed by the rules of natural selection. Problems are
solved by evolutionary concepts of survival of the fittest individuals. These surviving
individuals then produce the offspring, thereby passing the good genes along. This cycle
continues for generations producing a better fitting generation over time. GA’s are a part
of evolutionary computing, which is a rapidly growing area of artificial intelligence
2.3.1 Description of Genetic Algorithm (GA)
Genetic Algorithms are powerful and broadly applicable stochastic search
techniques based on the mechanisms and principles of natural selection and natural
genetics. The space of all feasible solutions is called search space. Each point in the
search space represents one possible solution. Each possible solution can be marked by
its value or fitness for the problem. The aim of the GA is to look for the best solution
among a number of possible solutions – represented by one point in search space. The
GA works on the population of points using the fitness function as their information
source for guiding the search. GA’s are a combination of blind search and heuristic
search algorithms as they randomly investigate the problem space, improvising their
search by using the information gained from the last experience. Their intelligence is
indicated from the fact that they not only exploit the best solution but also explore the
vast search space for better solutions [30].
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GA’s differ from conventional search techniques as they start with an initial set of
random solution set called population. Each individual in the population is called a
chromosome and represents a solution of the problem to be solved. The chromosome is a
string of symbols called genes and is usually represented by a binary bit string. The
chromosomes evolve through successive iterations called generations. During each
generation the chromosomes are evaluated using some measure of fitness. The objective
or fitness function is analogous to the environment that evaluates the performance of the
individuals in adapting to their surroundings. The next generation is created by forming
new chromosomes called offspring either by merging two chromosomes from current
generation using a crossover operator or modifying a chromosome using a mutation
operator. The fitter chromosomes have higher probabilities of being selected for creating
the new population while unfit chromosomes are discarded, to keep the population size
constant. Mutation is used during the mixing process to avoid creation of chromosomes
that are too similar to their parents. The new generation of solutions produced is better
than the preceding generation. The algorithm’s solutions converge to the best
chromosome after several generations, which should represent the optimal or nearoptimal solution to the problem.
Figure 2.4, adapted from Gen and Cheng [30], provides a comparison of
conventional and genetic approaches that are followed while solving a problem.
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Conventional
Method
Genetic
Algorithm
Initial single point
Initial population
Improvement
(problem-specific)
Improvement
(problem-independent)
Terminal ?
Terminal ?
Yes
End
Yes
End
Figure 2.4 A comparison of conventional and genetic approaches
2.3.2
Working of Genetic Algorithm
The functionality of the conventional GA can be explained as follows:
a) [Start] Generate random population of n chromosomes (suitable solutions for the
problem)
b) [Evaluate] Evaluate the fitness f(x) for each chromosome x in the population
c) [Repopulate] Create a new population by repeating following steps until the new
population is complete
d) [Select] Select two parent chromosomes from a population according to their
fitness (Higher fitness means higher chance of being selected)
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e) [Crossover] With a crossover probability combine the parents to form a new
offspring. If no crossover was performed, the offspring is exact copy of the
parents.
f) [Mutate] With a mutation probability, mutate new offspring at a locus (position in
chromosome)
g) [Accept] Place new offspring in the new population
h) [Replace] Use new generated population for a further run of the algorithm
i) [Test] If the end condition is satisfied, stop, and return the best solution in current
population
j) [Loop] Go to step 2
The outline of the basic GA is very general and there are many parameters and settings
that can be implemented differently in various problems. A detailed discussion about the
various features of the GA process follows.
Encoding
Genetic Algorithms operate on an initial set of random solutions which essentially
is comprised of a population of chromosomes. The procedure of the GA is aimed at
obtaining a chromosome (solution) which is hopefully optimal. It is essential that the
chromosome should, in some way, contain information about the solution that it
represents. The question of how to encode a solution of the problem into a chromosome
is therefore a key issue for GA’s. The most used way of encoding is to use a binary
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string. A chromosome could then look like Figure 2.5, where each bit in the string
represents some characteristics of the solution.
Chromosome 1
Chromosome 2
1101100100110110
1101111000011110
Figure 2.5. A binary representation of chromosomes
Each gene in the chromosome can be encoded to a finite length using binary or
non-binary strings. Various non-string encoding techniques have been created for
problems like real number coding for constrained optimization problems and integer
coding for combinatorial optimization problems [30].
Evaluation
The fitness of each chromosome in the population is evaluated using a measure of
performance to split them into groups. The objective function f(x), where x is the value of
the decoded chromosome, is then computed. Using an appropriate evaluation function,
the objective value for each chromosome is converted into its fitness. The objective
function is problem-specific and can have intent of maximization or minimization of the
function under consideration. An evaluation function plays the role of the environment,
and rates the chromosomes in terms of their fitness. The total fitness of the population is
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gauged by a summation of the individual fitness function values of all chromosomes in
the population.
Selection
A significant question is how to select chromosomes from the population to be
parents for crossover, while keeping the population size constant. According to Darwin’s
theory of evolution, the best ones survive to create new offspring. Some examples of
chromosome selection methods are: roulette wheel, Boltzman, tournament, rank and
steady state selection. In the roulette wheel or proportionate selection approach, selection
of chromosomes is proportional to their fitness values. The basic idea is to determine
selection or survival probability for each chromosome, which is the ratio of its fitness
value to the total population fitness. The chromosomes with larger fitness values will
clearly have a higher chance of being selected. However, due to duplication of fit or
super chromosomes, the succeeding generations may rapidly converge to local optima
and stagnate. It is also desired that the diversity of the population is preserved so that
constant generation pool can contain much more information for genetic search.
Deterministic sampling techniques like (µ+λ), (µ, λ), truncation selection, block selection
and steady state selection prevent duplication of fit chromosomes [30]. A modification to
the generational replacement approach is to replace n worst old chromosomes with
offspring, where n is the number of offspring. Whitley’s [66] ‘GENITOR’ system uses
ranking selection to create an offspring that replaces the worst individual of the
population. Rank selection ranks the population first and then every chromosome
receives fitness value determined by this ranking. There is always a chance of losing the
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fittest chromosomes during the selection process. Elitism or elitist strategy is an approach
that first copies the few best chromosomes to the new population, without being altered
by the genetic operators. The fitness of the population in the subsequent generations is
thus maintained. The rest of the population is constructed in other ways as described
above.
Recombination / Crossover
The selection process is followed by the crossover operation which generates
offspring’s by combining the chromosomes, called parents, chosen during selection.
Crossover is the main genetic operator. A simple way to achieve crossover is to choose a
random cut-point and generate the offspring by combining the segment of one parent to
the left of the cut-point with the segment of the other parent to the right of the cut-point.
The performance of the GA’s depends to a great extent, on the performance of the
crossover operator used. Two-point or multiple-point crossover can be performed by
selecting two or more randomly selected crossover sites. The crossover rate controls the
expected number of chromosomes to undergo the crossover operation. It is the ratio of
the number of offspring produced in each generation to the population size. A high
crossover rate allows exploration of a greater area of the solution space thereby reducing
the chances of converging in local optima. On the other hand, a very high crossover rate
results in the wastage of computation time in exploring bleak (unproductive) regions of
the solution space. The offspring produced inherit some advantages from their parents
because of this recombination process. The new offspring solutions thus enable the GA to
extend the search in new unexplored regions.
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Many different crossover operators have been proposed such as, partially mapped
crossover (PMX), order based crossover (OBC), cycle crossover (CX), and edge
combination crossover (ERX). The PMX operator proposed by Goldberg [34] involves
building the offspring by swapping randomly selected section of genes from the parent.
The Position based crossover (PBC) proposed by Syswerda [63] copies several random
genes from one parent to the offspring. The remaining genes are inherited in the order in
which they appear in the other parent. In OBC, [63], the order of the randomly selected
genes in one parent is inherited while the remaining genes are copied from the other
parent. Crossover can be quite complicated and depends mainly on the encoding of the
chromosomes. The performance of GA’s depends, to a great extent, on the performance
and type of crossover operator used.
Mutation
Mutation is the next step following a crossover. It is intended to prevent the fall of
all solutions of the population into a local optimum of the solved problem. A mutation
operation randomly changes the offspring resulting from crossover. It introduces some
variability in the characteristics of the offspring, thereby overcoming the problems of loss
of individual identity and stagnation of the population. Crossover alters one or more
genes with a probability equal to the mutation rate. A high mutation rate varies the
offspring to such an extent that it loses the traits it inherits from the parents, thus
randomizing the GA search. Mutation maintains the diversity of the population thereby
helping the GA to take small leaps towards reaching the best solution in the search space.
Several mutation operators have been proposed for permutation representation such as
43
inversion, insertion, reciprocal exchange and scramble mutation. Inversion mutation
selects two positions within a chromosome at random and then inverts the substring
between these two positions. Insertion mutation selects a gene at random and inserts it in
a random position. Reciprocal exchange mutation selects two positions at random and
then swaps the genes on these positions. Scramble mutation randomly reorders the
section of genes.
2.3.3 Application of genetic algorithms to job-shop scheduling problems
Job-shop scheduling is one of the most interesting and challenging applications for
GA’s. The major difference among different genetic algorithm applications is in
chromosome representation. A very important issue in building a GA for the job-shop
problem is to devise an appropriate representation of solutions together with problemspecific genetic operations. This enables all chromosomes generated, in either the initial
phase or the evolutionary process to produce feasible schedules. This crucial phase
affects all the subsequent steps of genetic algorithms. GA’s have been used for a wide
variety of problems such as machine learning, cellular manufacturing, combinatorial
optimization, inventory control, traveling salesman and game playing. The application of
GA’s in scheduling was first introduced by Davis [21]. Liepins et al. [52] investigated the
simplest scheduling problem of a static queue of jobs with specified due dates and run
times without precedence constraints. Gupta et al. [35] studied a single machine model
with an objective to minimizing flow time variance. Lee and Kim [51] examine the
performance of a parallel GA for a model in which earliness and tardiness penalties are
44
allowed to be arbitrary and jobs share a common due date. Cheng et al. ([15], [16])
consider a model of identical parallel machines where the objective is to minimize the
maximum weighted absolute lateness about an unrestricted common due date.
The chromosome representation which is an important step can be classified into
two basic encoding approaches namely: direct and indirect. In the direct approach, a
schedule (the solution of the JSSP) is encoded into a chromosome and genetic algorithms
are used to evolve these chromosomes to determine a better schedule. Representations
such as operation-based, job-based, job-pair-relation-based, completion-time and
random-key belong to this class. In the indirect approach, such as priority-rule-based
representation, a sequence of dispatching rules for job assignment (schedule) is encoded
into a chromosome, and genetic algorithms are used to evolve those chromosomes to
determine a better sequence of dispatching rules. A schedule is constructed with the
sequence of dispatching rules. The representations that belong to this class include
preference-list-based, priority-rule-based, disjunctive-graph-based and machine-based.
Tsujimura et al. [65] adopted a direct encoding approach. An example of the indirect
encoding approach is a priority-rule-based representation which was proposed by
Dorndorf and Pesch [23]. They use priority based rules to deal with the conflicts
generated by Giffler and Thompson [31]. A detailed discussion about the different
representations and considered approaches has been given by Gen and Cheng [30].
A schedule builder is introduced to handle the resource allocation process in
scheduling problems for the application of genetic algorithms. It transforms the
45
scheduling problem to a simple sequencing problem. It performs the task of interpreting
the encoded solution to real schedules. Embedding heuristics into the schedule builder
helps to generate better active schedules, thereby reducing the solution space of feasible
schedules and improving the efficiency of the GA. Dispatching heuristics such as Leftshift [5] and First-In-First-Out (FIFO) are often used as schedule builders. A FIFO
dispatching rule allocates resources by operation sequence and generates semi-active
schedules while the Left-shift rule adjusts operations by moving them to where they can
commence earlier without delaying other operations and generates active schedules. The
performance of the schedule builder depends on the heuristic implemented for the
scheduling process. The fitness function is very critical and guides the GA in selecting
and combining good solutions to uncover the best solution.
2.4 Knowledge Discovery in Databases (KDD) and Data Mining
The fundamental characteristic of human intelligence is the ability to learn and
thereby accumulate knowledge. Data are the fundamental building blocks to information
and hence knowledge. The true value of data relies on the ability to extract information
useful for decision support or investigation. Data is only as valuable as the user’s
capability to experiment on it. As put by Fayyad et al. [25] the value of the stored data
lies in the user’s ability to extract reports or spot remarkable trends and utilize the
analyses to operational or scientific advantage. Conventional methods of data-analysis
rely on human-expertise and understanding. As the magnitude of data grows
exponentially, such manual analysis becomes impractical due to the associated expense,
46
slow probing speeds and highly subjective nature. This remarkable growth of databases
requires the development of a new set of tools and techniques for automatic and sharp
database investigations. These tools and techniques are the subject of the rapidly
emerging field of knowledge discovery in databases.
2.4.1
The process of Knowledge Discovery in Databases
The term KDD can be used to refer to the overall process of discovering useful
knowledge from data. Data mining is a particular step in this process and is concerned
with the application of specific algorithms for extracting patterns (models) from data. The
steps involved in the process are data preparation, data selection, data cleaning,
incorporation of appropriate prior knowledge and proper interpretation of the results of
mining to ensure that useful knowledge is derived from the data. Fayyad et al. [24] define
KDD as a non-trivial process of identifying valid, novel, potentially useful and ultimately
understandable patterns in data. The basic steps involved in the interactive and iterative
process of KDD are shown in Figure 2.6 from Fayyad et al. [24] and can be described as
follows:
1. Identify with the application domain, relevant prior knowledge and end-user goals.
2. Create a target data set and focus on data samples on which discovery are to be
performed.
47
3. Clean and Preprocess data by removing noise or outliers replace information to
account for noise, decide on strategies to handle missing information and
accounting for time sequence information and known changes.
4. Reduce the effective number of variables under consideration using dimensionality
reduction or transformation methods.
5. Define the goal of the KDD process and the data mining tasks to be performed.
6. Select appropriate data mining algorithm and decide on apt models and parameters.
7. Perform data mining and search for patters on interest.
8. Interpret mined patterns and return to any of the above steps for further iteration.
9. Document, report or incorporate this knowledge into the performance system.
Interpretation/
Evaluation
Data Mining
Transformation
Knowledge
Preprocessing
Selection
Patterns
Preprocessed
Data
Data
Transformed
Data
Target Data
Figure 2.6 An overview of the steps comprising the KDD process
(Adapted from Fayyad [24])
48
2.4.2 An Overview of Data Mining
Data Mining is a particular step in the KDD process and can be described as the
science of extracting useful information from large data sets or databases. Hand et al.
[40] describe data mining as the analysis of larger observational data sets to find
unsuspected relationships and to summarize the data in novel ways that are both
understandable and useful to the data owner. It is a new discipline and encompasses a
wide variety of topics such as statistics, machine learning, data management and
databases, pattern recognition and artificial intelligence. Mining can identify concealed
patterns; provide presumable descriptive information or rules that can be applied to
analyze new sets of data [45].
The data mining method used should be specific to the problem and the desired
objective of the KDD process. The two primary goals of data mining are prediction and
description. The two goals are achieved by using the following primary data mining
tasks:
1. Classification is a learning function that classifies or maps a data item into several
predefined categorical classes.
2. Regression is a learning function which maps a data item to a real-valued prediction
variable.
3. Clustering is a common descriptive task which seeks to identify a finite set of
categories or clusters to describe the data. The categories can be mutually exclusive
and exhaustive. The classes are determined from the data and are not predefined.
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4. Summarization involves methods for finding a compact description for a subset of
data. These techniques are often applied to interactive exploratory analysis and
automated report generation.
5. Dependency Modeling consists of finding a model which describes significant
dependencies between variables. These models exist at two levels: structural, which
specifies which variables are locally dependent and quantitative, which provides the
strengths of the dependencies using some numerical scale.
6. Change and Deviation Detection focuses on discovering the most significant
changes in the data from previously measured values.
Data mining is not the only technique and any method that can help obtain more
information can be used. Each method offers its own advantages and disadvantages.
Some commonly used methods include statistical methods, case-based reasoning, neural
networks, decision trees, rule induction and Bayesian Belief networks [33]. The issue is
not which technique is better that the other, but rather which technique is suitable for the
problem at hand.
2.4.3
Attribute-Oriented Induction in Data Mining
Due to the increase in research activities and application demands on knowledge
discovery in databases, attribute-oriented induction method has been developed as a
technique for mining knowledge from data [39]. It summarizes the information in a
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relational database by repeatedly replacing specific attribute values with more general
concepts according to user defined concept hierarchies.
The induction process integrates machine learning techniques, like learning from
examples, observations and database operations to discover rules. The two types of
induction rules developed by the process are learning characteristic rules and learning
discrimination rules.
Learning characteristic rules generalize an attribute through the relatively highlevel concept and eliminates an attribute that contains a large set of distinct values
without relevance to a higher-level concept [48]. The steps involved in mining
characteristic rules are: initial data collection, derivation of the generalization plan for
each attribute and prime relation derivation. These steps integrate relational database
operations with attribute oriented simplification and leads to an efficient induction
process [39]. Characteristic rules learned from the induction do not take into account the
concepts present in other classes that also satisfy the necessary conditions. A
characteristic rule is a statement which characterizes the relevant concepts of the target
class. Characteristic rules summarize general characteristics of a data-set that satisfy
certain criteria’s.
The occurrence of the generalized concepts in the target class may not be indicated
sometimes due to their presence in both the target class and the contrasting class. Hence,
the classes being learned cannot be distinguished due to overlapping properties of these
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generalized conditions. A discrimination rule is a statement that discriminates the
relevant concepts of the target class from the contrasting classes. In learning
discrimination rules, both the target and contrast classes may share tuples, called
overlapping tuples. Due to class ambiguity of these overlapping tuples, they should be
marked and not generalized. Further generalization or attribute removal should rely on
unmarked tuples. The steps involved in mining discrimination rules are as follows:
(Adapted from Han et al. [38])
1. Separate the task-relevant data into the target class and the contrasting class
2. Generalize the concepts in both the classes synchronously and mark the overlapping
tuples. Repeat the generalization process until the number of unmarked tuples in the
target class is less than the threshold value
3. Transform the generalized relation to obtain discrimination rule
In order to present the discriminating ability of a generalized concept in the target
class from that of the contrasting class, quantitative information can be introduced in the
discrimination rule. The weight factor measures the probability of occurrence of a
generalized concept in the target class and can be defined as the ratio of number of tuples
covered by the generalized concept in the target class to the total number of tuples
covered by it in both the target class and the contrasting classes. A higher weight factor
indicates that the concept belongs to the target class while a lower factor indicates that
the concept belongs to the contrasting class.
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Attribute oriented induction can support incremental learning by integrating the
concepts of old data with those derived from new data. It is a bottom-up, data driven
induction method [22]. It is an efficient and robust method with wide applications and
extensible to knowledge discovery in advanced database systems, including objectoriented, deductive, and spatial databases [39]. Apte and Weiss [3] describe the use of
decision tree and rule induction in data-mining applications. They describe these
representation techniques as symbolic and interpretable which makes them attractive for
data-mining applications.
Han and Fu [39] define attribute-oriented induction as a set oriented database
mining method that extracts the broad features from data by generalizing the taskrelevant data, attribute-by-attribute and compressing it into a generalized relation. The
generalization process is guided by the concept hierarchy, which represent crucial
background knowledge. A concept hierarchy categorizes the task relevant data in a
relation by defining a sequence of mappings from a set of concepts to their higher-level
correspondences. They can be directly provided by users, implicitly stored in the
database, or constructed automatically based on clustering behavior and data statistics. A
concept tree for an attribute of a data item consists of leaf nodes that represent the
taxonomy of distinct attribute values. The concept tree uses a “bottom-up” hierarchical
approach in which the terminal node (leaf) represents the most generalized concept and
corresponds to the attribute to which the concepts have been mapped. The concept
hierarchy enables the discovered rules to be concisely represented in terms of generalized
concepts that can now be clearly understood by the users [37].
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Figure 2.7 (a) from Han et al. [37] specifies the available concept hierarchies of the
attributes for a departmental store database. The attributes for a commodity relation
comprise of the following four attributes: price, color, category of the product and place
of manufacture. Figure 2.7 (b) and (c) from Han et al. [36] show the concept hierarchy
representing background knowledge for a student relation in a university database. The
concept hierarchy table shows the primary classifications for students which are based on
their major, their status i.e. freshman, sophomore, graduate, etc., the place they are from
and their GPA. The concept tree represents taxonomy of concepts of the values in the
student attribute field. Additional examples on concept hierarchies and concept trees are
provided by Han et al. [38].
54
{ 1 .. 99 } ⊂ cheap
{ 100 .. 499 } ⊂ moderate
{ 500 .. 5000 } ⊂ expensive
{ cheap, moderate, expensive} ⊂ Any (Price)
{ red, black, brown, blue } ⊂ dark_color
{ white, yellow, pink, silver } ⊂ light_color
{ light_color, dark_color } ⊂ Any (color)
{ love_seat, sofa_set, ... } ⊂ sofa
{ dining table, kitchen table, ... } ⊂ table
{ queen_size_bed, twin_size_bed, ... } ⊂ bed
{ sofa, table, bed, ... } ⊂ furniture
{ video_recorder, video_camera, ... } ⊂ video_equipment
{ stereo_equipment, AM/FM_radio, ... } ⊂ audio_equipment
{ video_equipment, audio_equipment, ... } ⊂ elec_equip
{ skirt, fur, blouse, ... } ⊂ ladies’_wear
{ tie, sports_coat, ... } ⊂ men’s_wear
{ sleeper, pants, ... } ⊂ children’s_wear
{ ladies’_wear, men’s_wear, children’s_wear, ... } ⊂ clothing
{ furniture, elec_equip, clothing, ... } ⊂ Any (Category)
{ Seattle, Chicago, LA, ... } ⊂ USA
{ Tokyo, Osaka, ... } ⊂ Japan
{ Shanghai, Beijing, Tianjin, ... } ⊂ China
{ USA, Japan, China, ... } ⊂ Any (Place)
(a)
Concept hierarchy for a department store database
(Adapted from Han et al. [37])
55
(b) A concept hierarchy table of a university database
{ B io lo g y , C h e m is tr y , M a th , ..., P h y sic s }
{ L ite ra tu re, D ra m a , M u sic, ..., P a in tin g }
{ S cien c e, A r t }
A N Y (M a jo r)
⊂
⊂
⊂
S c ie n ce
A rt
{ F re sh m a n , S o p h o m o r e, J u n io r, S en io r } ⊂ U n d er g r a d u a te
{ M .S ., M .A ., P h .D . }
⊂ G ra d u a te
{ U n d erg ra d u a te , G r a d u a te } ⊂
A N Y (S ta tu s)
{ B u r n a b y , ..., V a n c o u v er , V icto r ia } ⊂ B r itish C o lu m b ia
{ C a lg a r y , ..., E d m o n to n , L eth b r id g e } ⊂ A lb er ta
{ H a m ilto n , T o r o n to , ..., W a ter lo o }
{ B o m b a y , ..., N e w D e lh i } ⊂ In d ia
{ B e ijin g , N a n jin g , ..., S h a n g h a i } ⊂
⊂ O n ta r io
C h in a
{ C h in a , In d ia , G e r m a n y , ..., S w itz e rla n d } ⊂ F o r eig n
{ A lb e rta , B r itish C o lu m b ia , ..., O n ta r io }
{ F o re ig n , C a n a d a } ⊂
{ 0 .0 — 1 .9 9 }
{ 2 .0 — 2 .9 9 }
⊂
⊂
⊂
⊂
{ 3 .0 — 3 .4 9 }
{ 3 .5 — 4 .0 }
⊂ C anada
A N Y (P la ce )
poor
a v er a g e
good
ex c e lle n t
{ p o o r , a v e ra g e, g o o d , ex ce lle n t }
⊂ A N Y (G ra d e )
(c) A concept tree for student status attribute
ANY (status)
undergraduate
Freshman Sophomore
Junior
graduate
Senior
M.A.
M.S. Ph. D.
(Adapted from Han et al. [36])
Figure 2.7 Examples of concept hierarchy representation
56
2.4.4 Application of Data mining to job-shop scheduling problems
Job-shop scheduling can be viewed as defining the ordering between all operations
that must be processed on the same machine, i.e. to fix the precedence constraints
between these operations. This ordering of operations affects the scheduling process
significantly, as it results in creating good or bad schedules. Attributes such as processing
time, machine load and remaining processing time play a vital role in the ordering op
operations in the schedule. Dispatching rules, which usually determine the ordering of
operations, rely heavily on information about these characteristics while making the
decisions. Priorities are assigned by dispatching rules to the jobs based on the
characteristics of the schedule, thus building sequences that determine the quality of the
schedules.
GA’s do not demonstrate repeatability or provide an explanation of how a good
schedule was developed. It is however evident that some relationship exists between the
characteristics of a particular operation and its corresponding position in the schedule. A
detailed study of these relationships can thus enable us to understand the mechanism of
priority assignment for operations as carried out by the GA while developing good
schedules. Data mining can help discover patterns or derive relationships present in the
solutions, thus providing us with a better picture of the decision making criterion of
genetic algorithms.
Data mining has been employed earlier to mine information from large job-shops.
An attribute-oriented induction methodology was applied by Koonce and Tsai [48] to
57
explore the relationship between an operation’s sequence and its attributes. The approach
followed was to observe the operational sequence of the GA scheduler and use attribute
oriented induction to find patterns in the sequence. The rules prioritized the operations in
different jobs to yield sequences that were transformed to schedules, based on the
characteristics of the operations. The performance of the GA was duplicated by the rulebased scheduler for the learned scenario. In addition, the results produced were better
than those provided by traditional dispatching rules when applied to similar scenarios.
This work was extended by Koonce and Kantak [45] to develop supplementary rule sets
for job-shop scheduling by mining data collected from additional scenario. Inaccuracies
in operation ranking resulting from the previous research were resolved due to the
creation of addition rule sets. It is evident from the research that performance of the rule
sets is enhanced by mining multiple data sets. It was also observed that the performance
was not affected by rules learned from additional data sets, beyond a threshold number.
2.5 Regression Analysis
Regression analysis can be defined as a statistical technique for investigating the
relationship between two or more variables. It is the process of fitting equations to data.
The general purpose of regression is to learn more about the relationship between several
independent or predictor variables and a dependent or criterion variable. Regression
analysis is used to estimate quantitative functional relationships between dependent
variables and one or more independent casual variables from actual data, when the
relationship from the variables is statistical in nature rather than exact. A statistical
58
relationship means that the dependent variable’s observed values are generated by a
probability distribution that is a function of other casual variables.
Linear Regression is an example of multivariate modeling techniques. It is useful
when we are using several variables to predict the values of a continuous dependent
variable. Various predictor variables can be included and then linear regression can be
used to indicate the unique influence of each predictor on the desired variable, controlling
for the influence of all the other predictors. The relative importance of each predictor can
be understood by examining all of the predictor’s simultaneous influences on the
dependent variable. So regression is a more powerful way of sorting out multiple
influences than visual-analysis of output from separate cross-tabs or other simple
bivariate techniques. It generates exact coefficient for each predictor, and shows us what
proportion of the variability of the dependent variable is uniquely explained by each
individual predictor. This makes it possible to make a predictive model which has
predictor coefficients that can be used to “score” the records. Multivariate analysis is a
much more powerful data-mining technique than univariate or bivariate analysis.
The regression equation in cases on simple linear regression can be expressed as
representing a line in two dimensional or two variable spaces. A simple linear regression
model is shown in equation 2.1.
Y = β0 + β1 * x + ε
(2.1)
59
In the equation, the dependent variable ‘Y’ is expressed in terms of a constant ‘β0’,
slope ‘β1’ times the independent variable ‘x’ and a random error term ‘ε ’. The constant is
referred to as the intercept and the slope as the regression coefficient. It is called as a
simple linear regression model as it has only one independent variable or regressor. It is
assumed that each observation can be described by the model with the above coefficients
and with mean zero and variance ‘σ 2 ’. Fitting a regression equation to a data-set is done
to describe the data and predict the response from the independent variable. For
predictive purposes it is always desired that the predicted values obtained by using the
fitted regression line should be close to the actually observed values, i.e., the residues
should be small. Hence while assessing the fit of a line; the vertical distances of the
points to the line are the only distances that matter. The parameters ‘β0’ and ‘β1’ in
equation 2.1 are estimated in order to minimize the sum of squares of the vertical
deviations. This criterion for estimating the regression coefficients is called the method of
least squares. It is a general method of finding estimated values of parameters such that
the sum of squared values between the fitted values and the corresponding observed
values is as small as possible.
The straight-line regression model is often found to be inappropriate as the true
regression function is non-linear. Visual analysis tools such as scatter plots can aid in
determining nonlinearity. Data transformations are performed in regression analysis to
achieve linearity and/or constant variance normality about the regression equation.
Commonly used transformations include square, square-root, logarithmic, inverse and
60
exponential. Sometimes several transformations may be applied jointly to linearize a
function.
Multiple regression models are used in situations where more than one regressor
variable is required. In the multivariate case where there is more than one independent
variable, the regression line cannot be visualized in two dimensional spaces. There is a
continuous random variable called the dependent variable ‘Y’ and a number of
independent variables, x1, x2, ...., xp. The aim is to predict the value of the dependent or
response variable using a linear function of the independent variables. The values of the
independent or regressor variables are known quantities for the purpose of prediction. A
multiple linear regression model is represented in equation 2.2.
Y = β0 + β1 * x1 + β2 * x2+.......+βp * xp + ε
(2.2)
The ε or the noise variable is a normally distributed random variable with mean
equal to zero and standard deviationσ. The values of the coefficients β0, β1, β2 ,....βp are
estimated from the available data.
In non-linear regression, the fitted value of the response variable is a non-linear
function of one or more independent variables. A non-linear function is one that cannot
be made into a linear function by transforming the dependent variable. Two of the most
61
common type of non-linear relationships are represented in quadratic cubic forms and can
be modeled through polynomial regression. Polynomial regression simply adds terms to
the original equation to account for non-linear relationships. A second-degree polynomial
regression model in one variable can be given by equation 2.3.
Y = β 0 + β 1 * x + β 2 * x2 + ε
(2.3)
Polynomial regression models are widely used when the response follows a
curvilinear trend. They can be useful to create a standard curve for interpolation, or to
create a smooth curve for graphing.
62
CHAPTER 3. METHODOLOGY
3.1 Approach
The steps involved in the procedure are as follows:
•
collect job-shop problems and their GA solutions,
•
transform data in the required format along with desired attributes,
•
create the Regression Model using the design part of the data set,
•
validate the equation generated by applying to the remaining problem set,
•
apply Attribute-oriented induction based rules to schedule the problems set,
•
use Shortest Processing Time (SPT) heuristic to schedule the problem set and
•
compare the effectiveness of the regression approach against the induction rules and
the heuristic
A strategy for selecting a model or pattern is based on its score function. However
it is often desired to know how well this model or pattern will perform on new unseen
data. The models are often chosen based on their external validation. The basic idea is to
split the data randomly into two mutually exclusive parts, a design part and a validation
part. The design part is used to construct the models and estimate the parameters. The
score function is then recalculated using the validation part. These validation scores are
used to select models or patterns. If any of the same data that have been used for selecting
a model or used for parameter estimation are then also used again for performance
63
evaluation, then the evaluation will be optimistically biased [40] i.e. the model would be
selected only on the basis of its good performance on this particular data set. Since the
two data sets are independently and randomly selected, for a given model the validation
score provides an unbiased estimate of the score value of that model for new data points.
Thus the bias in estimates that inevitably arises with the design component is absent from
the independent validation estimate.
An attempt to achieve this objective was made by splitting a test data-set of
nineteen problems into a design set of seven problems and a validation set of twelve
problems. The regression equation was created from the schedules generated for the 7
problems each of which was run 100 times in the GA. The sequences for each run was
collected and used to construct the design database.
3.2 Attribute-Oriented Rule Sets
The rule sets developed by Kantak [44] were applied to the sequences generated
from the problem sets. Due to its superior performance, the rule set containing 47 rules
learned from 10 test cases was selected. The rule set was learned by mining the schedules
obtained from GA for the job-shop scenarios. The various decision criteria for selecting a
job and operation are described below.
64
Concept Hierarchies
The attributes and concepts necessary for construction of concept hierarchies can
be obtained from the Operation relation. The concept hierarchy tables for the following
attributes are based on the 6 x 6 job-shop problem.
Priority
In a 6 x 6 job-shop, there are 36 sequential positions available in the solution for
any operation to occupy. It therefore becomes difficult to discover a relationship with the
exact position of an operation. This task can be much easier if the choice is increased
from the exact position to a range of positions. The Priority attribute determines the range
of positions in the sequence solution within which a given operation lies. The range of
positions is classified into six classes:
Priority 0 - {1, 2, 3, 4, 5, 6}
Priority 1 – {7, 8, 9, 10, 11, 12}
Priority 2 – {13, 14, 15, 16, 17, 18}
Priority 3 – {19, 20, 21, 22, 23, 24}
Priority 4 – {25, 26, 27, 28, 29, 30}
Priority 5 – {31, 32, 33, 34, 35, 36}
Operation
This attribute determines the operation number of the job and is classified into four
classes as follows:
First – {1}
65
Middle – {2, 3}
Later – {4, 5}
Last – {6}
Since the operations are performed in order, it’s obvious that the first operation is
the most critical and delaying it would result in the delay of the whole job causing a
corresponding increase in the makespan of the job.
ProcessTime
This attribute represents the processing time for a particular operation of the job
while the RemainingTime attribute represents the sum of the processing times for the
remaining operations of that job. The three classes (clusters) for these attributes are
determined from the data and sorted as:
{First 1 / 3rd}
– Short
{Second 1 / 3rd} – Middle
{Last 1 / 3rd}
– Long
Load
This attribute determines the machine loading and represents the machine
utilization in time. Machine loading is determined from the data and is categorized into
two classes as:
{First 1 / 2} – Light
{Last 1 / 2} – Heavy
66
Rule Network Tree
The number of classes for the attribute Priority, ProcessTime, RemainingTime and
Load are four, three, three and two respectively. The number of possible rules that can be
deduced from these attribute combinations is 4 x 3 x 3 x 2 = 72 rules. The rule network
tree that covers the range of possible attribute combinations is as shown in Figure 3.1,
(Adapted from Kantak [44]). The actual number of rules, however, is less than 72 due to
infeasible combinations. A maximum of 58 rules were learned from the induction
process.
Figure 3.1 A Rule Network tree
67
Rule-Set
A rule set based on the average weighted priorities was computed to assign a single
priority to the rules. The average priority enables the rules to rank the operations. As the
average priority represents the expected value, a smaller value ranks the operation earlier
in the schedule. Table 3.1 contains Kantak’s [44] set of generic rules, which were
computed for those operations, in the new scenarios that failed to fire learned rules. The
rules were created by calculating the mean of the average priorities for each concept of
the Operation attribute and generalized concepts (Any) of other attributes, thus
transforming the rule set to a generalized rule set.
Table 3.1 Generic rules for the rule-set
(Adapted from Kantak [44])
Operation ProcessTime
First
Any
Middle
Any
Later
Any
Last
Any
Load RemainingTime Priority AveragePriority
Any
Any
Any
0.3893
Any
Any
Any
1.4380
Any
Any
Any
3.4062
Any
Any
Any
4.7613
Ten test cases for a 6 x 6 job-shop scenario generated Koonce and Tsai [48] were
scheduled 100 times each by Kantak [44], using the genetic algorithm. The schedules
generated by the all the scenarios were mined by applying the attribute-oriented
technique to obtain 47 discrimination rules. The increase in the number of rules is
attributed to the more operations in the test cases covering more concepts, compared to a
68
single test case. The rule set, X, learned from the test cases is shown in Table 3.2
(Adapted from Kantak [44]).
69
Table 3.2 Rule set of 47 rules learned from 10 cases
Operation ProcessTime
First
Long
Middle
Short
Middle
Long
Middle
Short
Later
Long
Middle
Short
Last
Long
Middle
Short
Load RemainingTime AveragePriority
Heavy
Long
0.3841
Middle
1.0186
Light
Long
0.1443
Middle
0.7450
Short
0.6900
Heavy
Long
0.1240
Middle
0.2400
Light
Long
0.4467
Middle
0.2700
Heavy
Long
0.2750
Light
Long
0.2264
Middle
0.1750
Heavy
Long
1.3457
Middle
1.7359
Short
2.4000
Light
Long
1.2200
Middle
1.9275
Short
2.4150
Heavy
Long
0.7220
Middle
1.7442
Short
3.0140
Light
Long
1.0671
Middle
1.9014
Heavy
Long
0.5550
Middle
1.1318
Short
2.7900
Light
Long
0.9588
Middle
1.5000
Short
2.0667
Heavy
Middle
3.0717
Short
3.3421
Light
Middle
2.1950
Short
3.6000
Heavy
Middle
2.2420
Short
3.9383
Light
Middle
2.3450
Short
3.5069
Heavy
Middle
2.5200
Short
3.7468
Light
Middle
2.6960
Short
3.8686
Heavy
Short
4.3436
Light
Short
4.3590
Heavy
Short
4.8058
Light
Short
4.7690
Heavy
Short
4.4317
Light
Short
4.7150
70
3.3 Procedure
The solutions for nine problems were obtained from by using a GA tool from an
ongoing research at Ohio University involving Shah and Koonce [60]. The GA solutions
for the remaining ten problems were provided by Kantak [44]. The approach followed
was to generate a database of solutions from a part of the test problem dataset and then
test it on the remaining problems. The training data set comprised seven problems whose
GA solutions were generated by Koonce and Gandhi [46]. Each scenario comprised of a
total of 36 allocations for the six jobs, each with six operations, performed one on each
machine with a specific processing time. For each of the seven scenarios 100 solutions
were generated, resulting in different job sequences and similar or varying makespans for
each solution. The solutions were extended to include calculation of other attributes such
as Remaining Processing Time and Machine load; the already included other four details
were job-number (J), operation-number (O), machine (M) and processing time (PT). The
attribute remaining processing time (RPT) was the summation of the processing times for
all the remaining operations of a particular job. The attribute machine load (ML) for any
machine was the summation of the processing times of all jobs processed on that
particular machine. The attributes were then generated for each of the 100 solutions for
each scenario.
The resulting solutions were used to create the database to obtain the regression
equation. The regression equation was obtained by using MINITAB to perform ANOVA
on the training dataset. This thus became the score function for all the data sets. Using
71
this equation the problems were scored and the results were then sorted in an ascending
order giving priority to the job with least score value. It was noticed that the general
nature of ordering was based on performing the whole set of particular operation number
together for all the jobs. Thus mostly the first operation of all jobs was performed
sequentially, followed by operation number two and so on. This was followed by plotting
Gantt charts to obtain the makespans for each new sequence of the 19 test problems. The
Gantt chart was developed manually on graph paper using the left-shift rule.
The second step was an analysis using a heuristic for comparing the obtained
results. The Shortest processing time (SPT) dispatching rule was selected for the analysis.
The SPT rule orders the jobs in the order of increasing processing times. Whenever a
machine is free, the shortest job ready at that time will begin processing. The machine
and processing time attributes being the same, new sequences were generated for each of
the test cases using the SPT criteria. This was done manually for all the 19 problems, and
new order of job processing was obtained. This was followed by Gantt charts of these
sequences to determine the makespans obtained using the heuristic. The consistency for
both cases was maintained by using the left-shift rule in this case as well.
The third step was to test the rule sets developed by Kantak [44]. The new
sequences for all the 19 tests cases were determined with the rule sets. The makespans
were obtained in a format similar to the procedure carried out before, by plotting a Gantt
chart. The results from various methodologies being obtained, analysis was initiated to
72
determine the viability of regression alongside GA, attribute-oriented-induction rules and
heuristic dispatching rule.
The makespans generated by all the four techniques were plotted as a Normal
probability plot using MINITAB. The purpose of this plot is to check if the data are
approximately Normally distributed. The data were plotted against a theoretical normal
distribution in such a way that the points form an approximate straight line. Departures
from this straight line indicate departures from normality. The probability plots for
makespans obtained by GA’s, regression analysis, induction rules and SPT techniques are
provided in Appendix E. The points on this plot form a nearly linear pattern, which
indicates that the normal distribution is a good model for this data set.
73
CHAPTER 4. RESULTS
4.1 Statistical Analysis
The design set of test problems comprised of seven 6 x 6 job-shop scenarios. Each
was run 100 times with the GA resulting in a large set of job-shop sequences (solutions).
This data set was subject to regression analysis by using MINITAB. The regression
equation was generated by this analysis. Error! Reference source not found. shows the
results of the above analysis.
74
Figure 4.1 ANOVA statistics for the design database
The equation assigns a sequential position to each job based on the contribution of
each attribute. The attribute ‘Operation’ has a large positive coefficient which indicates
that earlier operations are assigned greater priority and processed earlier. This was
intuitive, as operations in a job-shop always have to abide precedence constraints wherein
they can be processed only in a given sequential order. The ‘Processing Time’ attribute
was determined to have a negative coefficient. Therefore, operations having high
processing times are assigned a lower position and processed before similar operations
75
having smaller processing times. While counter to SPT; this is intuitive as the makespans
of jobs can be improved if jobs having higher processing times are handled in the earlier
phase of the processing. The next attribute was ‘Remaining Processing Time’ which was
assigned a negative coefficient. The jobs having longer remaining processing times were
assigned earlier priority over others. The last attribute was ‘Machine Load’ which was
assigned a positive coefficient. The operations on machines with lesser loads would thus
be assigned a later priority over those on machines with heavy loads. This was intuitive
as it is important to tightly schedule bottleneck machines.
This scoring function was then used to determine the new schedules for the test
problems. A scenario picked at random out of the many solutions, resulting from multiple
runs of the GA was subject to the regression equation to obtain the score values. These
values were arranged in an ascending order to obtain the job-operation-sequence for each
scenario. This resulted in a database of nineteen 6 x 6 job-shop solutions. The makespans
were obtained by plotting Gantt charts of the new schedules. It was observed that the
makespans were identical to slightly higher than those obtained by the GA.
The SPT heuristic was then applied to the test problems to determine the new
operation scheduling. The job-operation with the shortest processing time obtained the
priority. The makespans obtained by the Gantt chart plot of SPT sequences were found to
be much higher than the makespans generated by the regression analysis.
76
The induction rules were applied next to the problems. The rules assigned priorities
to the operations as per the developed classification rule-set, thereby assigning the
sequential positions. The priorities were sorted in an ascending position to obtain the
sequences for the problems. This was followed by plotting these sequences on graph
paper with the left-shift rules to obtain the makespans. It was noticed that the makespans
were better or worse than those obtained by regression. However on an average, the
regression model produced results much closer to the GA makespans as compared to the
induction rules.
The makespans produced by all the four techniques are provided in Table 4.1
below. It is clear from the average values that regression comes much closer to the good
schedules produced by the GA then both the induction rules as well as the SPT heuristic.
In one case (S2) regression produces the exact makespan as given by the GA scheduler.
77
Table 4.1 Makespans obtained by GA, Regression, SPT and Induction rules techniques
Case
2
3
4
5
6
7
8
9
10
S1
S2
S3
S4
S5
S6
S7
S8
S9
S10
Avg.
Difference
GA Regression (Regr. & GA)
51
60
9
50
56
6
43
45
2
50
60
10
57
60
3
49
54
5
44
50
6
42
52
10
54
59
5
46
52
6
53
53
0
60
64
4
48
55
7
55
59
4
54
59
5
51
65
14
67
78
11
54
58
4
60
64
4
52
58.053
Difference
Difference
SPT
(SPT & GA) Rules (Rules & GA)
72
21
59
8
69
19
56
6
52
9
49
6
56
6
59
9
101
44
57
0
59
10
50
1
74
30
53
9
58
16
52
10
80
26
63
9
65
19
50
4
83
30
56
3
80
20
67
7
65
17
57
9
79
24
62
7
79
25
61
7
57
6
67
16
83
16
84
17
80
26
61
7
68
8
65
5
71.579
59.368
The next step was to perform a statistical analysis to obtain the confidence in the
variation shown by the techniques. A blocked ANOVA was performed on the makespans
obtained by the various scheduling techniques to determine whether the techniques varied
considerably. The ANOVA showed a very high confidence variation in the results. The
obtained p-value was 3.28 x 10-13. Figure 4.2 below shows the results obtained for
blocked ANOVA performed on the results.
78
Blocked Anova
SUMMARY
GA
Regression
SPT
Rules
Count
19
19
19
19
Source of Variation
Blocks
Columns
Error
Total
SS
3697.5
3838.25
1848.5
9384.25
Sum Average
988
52
1103 58.053
1360 71.579
1128 59.368
df
18
3
54
75
Variance
39.778
49.719
152.368
66.246
MS
F
P-value
F crit
1279.417
34.231
37.375
3.27501E-13
2.776
Figure 4.2 Blocked ANOVA showing difference in the techniques
The high variability in the makespans obtained by the SPT technique could have
influenced the confidence in the difference showed by various techniques. Note Figure
4.2 shows inflated variance for SPT, possibly violating the equal variance assumption. It
was decided to observe the results by performing a blocked ANOVA without including
the results produced by SPT. The results of the blocked ANOVA without SPT are shown
in Figure 4.3. It can be seen that significant difference still exists between the techniques.
79
Blocked Anova
SUMMARY
GA
Regression
Rules
Count
19
19
19
Sum
988
1103
1128
Source of Variation
Blocks
Columns
Error
Total
SS
2584.877
586.842
218.491
3390.211
df
18
2
36
56
Average Variance
52
39.778
58.053
49.719
59.368
66.246
MS
F
293.421
6.069
48.346
P-value
F crit
6.342E-11 3.259
Figure 4.3 Blocked ANOVA showing difference in the techniques without SPT
The database of sequences from the test cases was subject to transformation in
order to evaluate the chances of a possible increase in the explained-error percentage.
Graphs between various attributes were plotted to observe the existing relationship
between the two and thereby the possibility of enhancement by transformation. The only
interesting relationship or the possibility of improvement was observed between the
obtained sequence and the remaining processing time (RPT). ANOVA was performed on
the solutions and the Adjusted R-Square (error coefficient) term was found to have a
value of 73.53 %. The two transformations applied to the remaining processing time were
Inverse (1 / x) and Exponential (e -x). Least squares regression was then performed with
these transformed attributes and the sequences. The explained error term however didn’t
show improvement as compared to the untransformed values. The Adjusted R-square
with Inverse transform was found to be 6.325 %, while the one with Exponential
transform was 39.368 %.
80
CHAPTER 5. CONCLUSIONS AND FUTURE RESEARCH
5.1 Conclusions
Data mining techniques have been successively applied to the job-shop problem,
thus illustrating it as a novel and distinctive approach for solving job-shop scheduling
problems. Genetic algorithms most often provide fast and near-optimal solutions to
conventional mathematical problems as compared to heuristics. They are a good source
of sequence generators and have provided good results for the 6 x 6 job-shop scenarios
within a short time. The recognition of the GA solutions as optimal is difficult as the
optimality of job-shop problems can be proven only by an enumerative approach.
Pruning search methods which eliminate lower level splits or entire sub-trees in a
decision tree while searching to the solution can also be incorporated. Some examples
are, Apriori, which is a bottom-up breath-first counting based approach on horizontal data
layouts and Éclat, which is a depth first intersection based approach on vertical data
layouts. It can however be attempted to replicate the performance of the GA by applying
data mining techniques. The regression technique helped the data mining process
generalize the scheduling information contained in the GA solutions, providing
knowledge that can be applied to similar job-shop scenarios.
The trial job-shop scenarios were subject to various scheduling techniques, each
producing a different schedule and corresponding makespan. The GA always produced
81
the minimum makespan while regression analysis provided the next best results. The
makespans of the regression schedules were close to those of the GA and in one case
matched the performance of the GA. The rule sets developed by previous researchers
were also tested and regression analysis was found to better those results. Regression
analysis provided much better results than a well known and tested dispatching rule
shortest processing time (SPT) heuristic. The research was thus able to successfully apply
regression analysis as an efficient and useful data mining technique. The aim to explore
better ways to find optimal techniques was thus fulfilled by the results.
5.2 Future Research
This research focused primarily on discovering new approaches that can match the
GA’s performance in solving machine scheduling problems. Significant improvements
can be made by modifying the goals of this research and adopting new data mining
techniques to extend the knowledge of job-shop problems. Some suggestions for future
work are as follows:
1. Investigate other job-shops
This research dealt specifically with the classical 6 x 6 job-shop scheduling
problem with the objective of minimizing the makespan. This research can be extended
by applying data mining techniques to solutions obtained from the following problems to
develop a universal rule set.
82
•
Benchmark classic job-shop instances with known optimality like the 10 x 10,
15 x 15, 20 x 10 problems
•
Dynamic and stochastic job-shop problems which incorporate attributes such
as non-zero ready times, inter-travel time, multiple operations on a job by one
machine, random arrival of jobs, multiple scheduling criteria, multiple job
routes simultaneously, assign weights to jobs and uncertain processing times.
These features increase the search space of solutions and make the task of
scheduling very complicated.
•
Single machine scheduling problems with minimizing objective function for
various performance measures like mean flowtime or completion time,
tardiness, number of tardy jobs, etc
2. Efficiency and cost of data mining
The performance of analyses can most possibly be strengthened by the use of
multiple data sets. The generated regression equation can be made more efficient by
using more problems to develop it. However, the time and cost involved in generating
data sets and performing data mining is very high. Erroneous data sets can also induce
noise in the data leading to imprecise results and reduced efficiency. It is therefore
necessary to deal with the efficiency and cost of incorporating multiple data sets for jobshop scheduling problems.
83
3. Incremental Learning
Another approach to cover schedule scenario is used as incremental learning
process. Incremental learning, also known as parallel mining discovers new knowledge
by updating existing knowledge when new data is merged with the old data. While
discovered knowledge cannot describe a schedule case, the incremental learning process
may be employed to discover new knowledge and combine it with the old by knowledge
generalization.
84
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88
APPENDIX A TEST PROBLEMS
Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4,9
3,1
3,2
4,1
2 , 10
2,1
3,6
3,8
6,9
2,6
1,1
1,9
2 , 10
3,2
3,7
4,1
4,2
1,4
5,1
2,9
2,3
1,2
5 , 10
5,8
2,6
2,1
2,8
3,6
4,7
4,2
3,3
4 , 10
4,3
5,6
5,5
5,3
6,3
6,3
6,4
6,4
6,7
6 , 10
1,2
1,2
3,6
1,6
5,6
5,4
5,7
1,7
1,1
6,1
6,6
4,1
1,2
5,6
5,5
6,3
1,5
1 , 10
6,8
6,5
6,8
6,9
4,8
4,2
3,9
6,9
6,9
5,7
4,7
1,2
2,1
6 , 10
6,8
4,6
2,9
2,9
4,5
5,1
2,5
5,7
3,5
3,4
5,5
5,7
5,2
1,7
3,2
5,7
3,1
2,4
2,1
2,1
3,2
3,3
2,8
2,6
3,2
4,7
1,6
1,4
1,2
4,1
4,7
3,4
1,4
4,5
3,3
1,9
1,5
1,7
4,7
4,9
3,1
1,3
4,9
4,5
3 , 10
3,5
3,7
2,7
2,9
2,3
3,2
3,3
2,1
2,8
2,9
2,2
3,2
3,2
4,8
2,6
2,1
5,7
4,2
4,3
4,7
1,9
1,8
5,4
4,8
5,9
1,8
5,4
5,9
5,4
6,6
6,1
1,6
5,2
6,2
6,1
6,7
6,2
6,1
5,8
5,8
6,1
6,8
1,2
2,4
6,2
6,1
2,4
5,5
5,7
6,1
6,6
1,5
4,4
4,1
4,6
3,2
6,2
6,1
4,6
5,4
2,8
1,3
5,1
5,9
4,5
3,2
3,9
5,3
4,1
3,1
3,1
1,9
1,3
5,2
4,4
4,6
3 , 10
3,6
6,9
5,6
2,6
2,6
1,6
1,6
1,6
2,6
1,6
5,6
6,2
3,5
3,4
6,5
2,8
2,7
1,2
2,6
4,7
6,4
3,3
3,6
2,5
2,8
2,9
1,1
2,3
4,7
4,9
2,8
2,1
4,6
1,1
1,7
5,9
4,8
3 , 10
1,6
1,4
1,1
3,7
4,5
4,8
2,4
3,1
5 , 10
2,5
4,2
4,7
5,3
3,2
3,2
4,6
5,9
2,4
3,8
6,1
5,6
1,7
5,6
5,9
3 , 10
6,1
6,6
5,2
5,4
6,7
6,4
6 , 10
6,7
6,1
1,1
1,1
3,1
4,7
4,4
2,4
6,7
6,1
6,4
1,7
1,9
4,7
3,8
3,4
3,6
3,6
3,4
4,1
3,3
3,9
6 , 10
2,5
2,9
4,2
5,6
5,1
3,1
2 , 10
2,3
5,7
1,6
1,4
6,8
2,8
2,9
5,2
4,8
4,1
1,6
5,6
5,5
5,8
4,5
4,5
1,1
5,2
6,1
2,2
6,6
6,1
1,7
1,4
1,7
2,4
6,1
5,1
89
Problem S1 Problem S2 Problem S3 Problem S4 Problem S5 Problem S6 Problem S7 Problem S8 Problem S9 Problem S10
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
2,9
2,9
5,4
4,6
6,5
6,7
6,2
4,9
4,1
2,8
3,7
5 , 10
4,1
5,7
2,5
4,1
5,2
1,9
3,9
5,7
4,1
1,3
6,8
6,4
3,7
3,5
2,3
2,9
5 , 10
1,3
6,8
6,1
3,1
3,8
4,7
2 , 10
3,4
5 , 10
1,9
4,2
1,4
3,8
2 , 10
1,1
5,3
5,3
1,3
6,2
2,9
6,6
5,2
4,1
1,8
2,6
1,2
1,5
4 , 10
3,7
6,8
3,6
4,9
4,4
2 , 10
5,8
1 , 10
6 , 10
5,3
2,7
6,1
3,4
2,7
6,5
5 , 10
4,5
5,5
5,8
2,8
5,3
1 , 10
2,5
6,7
5,7
3,6
2,5
4 , 10
1,8
4,6
6,4
4,9
5,2
3,1
1,7
6,5
1,4
3 , 10
3,2
3 , 10
1,3
5,8
1,1
5,4
3,1
4,1
6,1
6,8
2,2
1,4
3,9
2,3
4,5
1,4
2,5
1 , 10
3,3
2,2
4 , 10
6,2
4,7
3,1
6,9
5,1
5,8
1,4
4,2
6,2
4 , 10
6,3
4,1
3,6
4,7
1,6
1,3
4,9
6,1
3,4
2,8
4 , 10
3,9
1,6
1,1
2,3
3 , 10
2,5
5,8
5,5
1,8
3,3
6,3
2,7
3,6
3,6
2,1
3,9
2,3
4,7
6,7
1,2
1,9
5,5
5,8
4,4
6,6
5,9
1,4
1 , 10
5,5
5,6
2,9
4,3
6,2
6,5
4 , 10
6,4
3,3
2 , 10
3,3
2,3
5,1
6,1
2,6
3,6
6 , 10
3,2
5,4
1,3
2,9
5 , 10
2,7
1,5
1,6
6,4
2,1
2,5
1,1
5,2
6,8
6,1
4,1
5,1
3,3
1,5
1,9
1,9
6 , 10
2,6
5,7
2 , 10
6,9
3,7
5,9
5,6
3 , 10
4,8
4,2
3,7
3,2
1 , 10
1 , 10
2,7
4,7
4,7
4,5
5,6
2,8
4,5
1,5
3,1
3,8
4,5
2,7
2,4
5,6
6 , 10
3,7
6,5
4,3
4,6
5,4
6,4
6,6
2,2
4,7
2,7
2,8
3,7
1,2
3 , 10
4,6
2,6
5,7
4,4
3,4
3,4
1,4
6 , 10
2,7
2,5
2,4
6,6
3,9
1,7
6,9
4,7
6,5
2,1
3,9
1,5
5 , 10
1,5
6,4
5,2
1,3
6,1
5,2
5,4
5,1
6,1
3,7
5,5
2,7
3,6
5,2
1,7
3,2
1,4
6,1
5,7
1,8
4,8
1,8
6,5
2,7
5 , 10
4,4
4,3
4,2
4,1
6,9
3,7
4 , 10
5,7
1 , 10
1 , 10
2,3
3,8
2,9
4,9
3,1
6,9
3,9
1,1
2,4
5,1
4,1
5 , 10
4,5
6,3
2,9
4,1
1,8
2,9
4,4
6,3
5,3
4,4
3,9
3,3
6 , 10
2,4
6 , 10
4,4
3,9
4,1
6,6
2 , 10
1,4
5,4
1,9
1,3
4,1
6,4
5,5
3 , 10
3,8
6,1
5,1
1 , 10
4,4
5,1
5,5
3,8
6 , 10
2,4
1,3
1,4
6 , 10
2,1
5,9
3,5
2 , 10
90
APPENDIX B GA SOLUTIONS AND MAKESPANS (IN LAST ROW)
PROBLEM 2
Soln.1
3
2
2
1
6
1
3
3
4
5
6
3
5
3
1
5
6
2
4
2
4
2
5
1
1
6
3
6
2
4
1
6
4
4
5
5
52
Soln.2
1
2
1
6
5
6
5
1
4
3
3
2
2
6
1
2
6
5
1
3
4
1
2
3
4
6
2
5
4
4
5
6
3
5
4
3
55
Soln.3
4
2
3
1
1
3
6
3
1
1
3
2
2
4
6
5
4
2
1
3
4
5
6
5
6
6
5
6
2
4
2
4
3
1
5
5
55
Soln.4
2
2
2
1
3
6
1
5
1
6
5
4
4
1
2
6
5
3
1
4
6
5
3
2
3
6
4
1
4
2
4
5
3
6
3
5
55
Soln.5
2
3
1
3
4
5
3
2
6
1
3
2
4
6
4
1
2
4
3
6
5
5
2
4
1
5
6
5
6
1
1
3
5
2
6
4
55
Soln.6
1
4
1
3
5
3
2
1
3
5
6
3
2
6
1
4
3
5
4
2
1
6
3
2
4
2
5
1
5
6
2
4
4
6
6
5
55
Soln.7
5
3
4
2
6
1
3
6
2
6
1
3
1
3
2
5
3
1
2
6
5
4
5
4
2
1
1
4
5
4
6
2
5
3
4
6
54
Soln.8
3
1
3
4
6
2
3
1
6
3
2
5
6
2
1
5
2
4
1
3
2
3
1
5
5
4
4
6
1
5
6
4
2
5
6
4
53
Soln.9
3
2
1
1
6
1
3
3
5
3
6
2
3
5
4
4
2
2
2
5
6
1
6
4
1
5
2
6
4
1
4
3
5
6
5
4
52
Soln.10
4
5
3
3
2
3
6
3
1
6
1
3
2
5
2
5
2
2
1
6
3
1
4
4
5
2
6
4
6
1
5
4
5
4
1
6
54
Soln.11
1
1
3
3
6
6
2
3
2
5
3
1
2
6
5
3
1
2
2
5
4
4
6
2
4
1
1
5
6
4
3
4
5
6
5
4
54
Soln.12
3
2
4
1
1
3
2
6
4
6
5
3
1
2
3
5
4
2
4
4
6
3
1
5
1
5
6
2
2
3
6
5
4
5
1
6
55
Soln.13
1
2
3
2
6
1
2
3
3
3
1
5
2
6
6
5
1
4
3
2
4
5
4
4
1
6
6
1
5
2
4
5
4
5
3
6
53
Soln.14
5
6
1
2
3
2
2
6
1
4
5
1
5
4
2
6
3
4
1
2
2
5
6
3
1
1
3
4
6
5
3
4
4
5
6
3
55
Soln.15
1
6
2
4
3
4
1
1
3
4
5
2
3
3
6
6
1
2
5
1
3
4
3
5
2
5
6
2
6
5
4
6
2
4
5
1
55
91
Soln.16
5
6
1
2
3
3
3
6
3
2
5
6
2
1
1
3
2
6
4
4
2
1
1
3
5
4
4
6
4
2
5
6
5
1
4
5
51
Soln.17
5
2
6
6
3
3
1
3
3
3
2
4
5
6
5
1
1
2
2
1
6
4
2
4
4
1
6
2
5
1
5
4
5
6
3
4
55
Soln.18
3
6
2
4
2
3
3
6
2
1
5
5
3
6
3
1
5
6
1
5
1
2
4
4
1
2
4
5
4
5
6
6
1
4
2
3
54
Soln.19
4
2
3
5
1
6
4
3
6
1
5
2
3
1
2
3
4
6
2
3
5
5
6
1
3
6
4
4
2
2
6
1
4
5
5
1
55
Soln.20
4
1
4
6
3
2
3
2
3
5
1
4
6
5
3
3
5
2
5
2
1
1
6
4
2
1
2
3
6
5
4
6
1
4
6
5
55
Soln.21
2
2
1
3
3
3
5
4
6
6
5
3
2
4
2
2
6
1
3
5
4
2
1
1
6
4
4
5
6
1
1
6
3
4
5
5
51
Soln.22
3
5
4
1
2
2
1
3
6
4
3
1
2
3
3
4
2
6
6
5
1
2
4
1
5
5
6
4
4
2
6
6
5
1
5
3
55
Soln.23
1
3
5
3
2
6
6
2
1
2
5
3
1
3
4
4
6
2
3
6
2
5
5
4
1
2
4
4
5
1
6
1
6
3
5
4
53
Soln.24
2
3
4
1
3
2
3
2
3
6
4
6
2
1
5
4
5
1
1
6
4
3
5
2
6
2
4
3
1
1
5
6
5
6
4
5
55
Soln.25
4
1
1
1
3
3
3
6
2
2
2
5
3
6
1
6
5
3
1
4
2
6
2
5
4
1
5
4
5
2
4
5
3
4
6
6
54
Soln.26
1
3
2
3
2
3
3
1
6
2
1
5
3
6
4
5
4
5
6
2
6
4
5
1
2
1
2
6
4
4
5
3
4
1
5
6
51
Soln.27
2
1
4
5
3
3
3
2
6
2
3
6
3
1
5
1
1
4
5
6
2
2
4
5
4
6
6
1
2
5
3
6
4
1
5
4
51
Soln.28
6
1
2
2
6
2
1
4
5
1
1
6
5
2
5
6
3
4
3
4
5
1
3
3
5
2
5
6
4
1
4
4
3
3
6
2
55
Soln.29
2
2
3
3
3
2
3
4
1
2
5
1
6
6
5
1
4
6
2
3
1
5
2
1
6
4
4
4
5
5
6
3
5
1
4
6
54
Soln.30
2
1
2
3
4
6
1
2
3
5
3
3
1
6
3
2
6
5
6
5
3
5
4
1
4
2
1
6
2
4
6
4
4
1
5
5
53
92
Soln.31
3
1
6
2
2
4
6
5
2
5
4
1
6
5
2
1
4
5
1
6
4
2
5
3
3
2
3
6
5
1
1
4
6
4
3
3
55
Soln.32
2
6
1
4
2
1
3
3
3
6
5
3
5
6
4
1
3
2
1
2
5
6
4
4
1
2
1
2
6
3
4
5
5
4
5
6
54
Soln.33
1
3
4
2
6
2
5
3
6
6
1
3
3
2
1
2
1
5
3
1
2
3
5
4
4
1
6
2
6
5
6
4
4
4
5
5
54
Soln.34
3
6
1
2
4
1
3
1
5
4
3
4
4
2
6
3
6
2
5
4
2
1
3
6
5
5
6
2
1
1
4
5
6
5
3
2
55
Soln.35
2
4
6
1
3
4
4
2
3
1
4
2
3
1
3
2
5
6
1
2
5
3
2
6
1
5
6
3
4
5
6
4
6
1
5
5
55
Soln.36
6
6
2
5
2
3
3
1
2
5
3
3
3
6
5
4
2
1
1
3
4
4
6
5
1
2
1
6
1
2
4
6
5
4
4
5
52
Soln.37
6
3
2
1
3
2
1
3
4
6
3
2
5
2
1
5
3
4
1
2
5
6
3
4
6
4
4
2
1
5
6
1
5
4
6
5
52
Soln.38
5
2
3
1
4
6
3
2
3
1
3
6
3
2
6
2
5
1
4
5
4
1
2
2
6
6
4
3
1
5
5
1
4
4
6
5
51
Soln.39
2
6
1
4
4
1
2
3
3
4
6
5
3
6
1
5
3
1
2
3
5
2
3
4
1
2
5
6
4
1
5
2
6
4
5
6
55
Soln.40
3
2
1
3
4
3
6
1
2
2
6
5
3
1
5
3
1
2
6
4
5
4
2
5
4
3
6
1
6
5
1
6
4
5
4
2
51
Soln.41
3
2
1
3
4
6
3
6
5
1
2
3
3
6
2
1
2
5
2
5
5
4
1
4
2
4
1
6
5
6
6
3
1
4
5
4
51
Soln.42
3
2
3
3
5
4
2
6
1
3
3
1
2
6
5
1
2
6
6
1
5
5
4
3
4
1
4
2
5
2
6
4
5
1
6
4
54
Soln.43
1
1
5
2
6
3
2
3
1
3
2
3
6
3
2
5
2
5
4
1
6
4
5
4
1
6
6
4
4
1
3
6
4
5
2
5
52
Soln.44
3
5
4
6
1
1
2
3
3
1
3
4
4
6
2
2
3
2
5
6
2
1
1
5
3
6
4
2
6
4
5
5
6
4
1
5
55
Soln.45
3
2
2
1
3
4
1
3
6
5
3
2
6
6
3
5
4
1
1
3
1
5
4
2
2
6
5
4
1
6
2
6
4
5
5
4
54
93
Soln.46
3
1
1
3
3
5
4
2
2
6
3
2
1
2
3
6
6
4
5
4
5
2
1
5
6
4
5
3
2
4
6
1
5
4
1
6
55
Soln.47
3
2
1
5
1
1
5
2
2
3
2
5
1
6
4
1
1
3
6
4
2
5
6
3
2
5
6
3
4
6
4
6
4
3
4
5
55
Soln.48
3
1
5
4
6
1
2
3
3
1
2
3
2
6
2
5
3
5
2
1
4
6
1
4
5
4
3
1
5
6
4
5
6
4
6
2
54
Soln.49
1
3
3
2
6
2
4
3
1
3
3
2
5
6
5
1
2
6
5
5
2
4
4
1
4
3
2
1
6
6
4
5
1
4
6
5
51
Soln.50
6
1
5
2
1
3
1
2
3
1
3
3
6
5
2
6
3
2
1
4
5
4
4
2
6
5
6
4
4
6
4
5
3
2
5
1
52
Soln.51
3
1
2
3
2
3
3
4
1
3
6
5
2
2
6
5
1
6
4
5
6
4
1
1
4
5
6
2
4
4
3
5
2
1
5
6
53
Soln.52
3
2
1
1
6
3
2
5
3
3
1
6
3
1
5
6
1
5
2
2
3
6
4
5
4
4
4
2
6
5
4
5
6
2
1
4
52
Soln.53
1
4
3
3
1
2
2
6
5
3
6
6
1
3
1
4
3
5
5
2
2
2
6
4
3
6
4
5
5
1
4
2
5
1
4
6
55
Soln.54
4
5
1
3
2
3
2
4
6
1
3
2
1
6
4
6
3
1
3
5
5
4
2
3
4
1
2
5
6
5
5
2
4
1
6
6
55
Soln.55
5
6
3
1
1
3
5
3
2
1
1
6
3
2
5
2
1
4
4
2
1
4
6
6
2
2
4
3
6
4
3
4
5
5
5
6
55
Soln.56
2
1
3
2
3
5
1
6
1
6
3
2
6
3
3
5
4
1
4
5
2
6
4
5
4
2
6
1
4
5
3
4
5
2
6
1
52
Soln.57
3
2
2
1
5
3
6
6
3
2
1
3
2
6
5
3
5
1
6
1
4
5
4
2
4
1
4
6
3
5
6
2
1
4
4
5
51
Soln.58
4
3
1
2
3
6
1
1
6
3
3
3
2
1
4
5
2
4
6
2
2
4
5
6
2
5
5
1
4
6
1
5
6
4
5
3
55
Soln.59
6
2
3
1
4
1
4
3
2
1
6
3
3
5
6
4
3
5
5
1
2
1
2
6
2
4
3
5
4
2
4
6
6
5
5
1
55
Soln.60
1
2
2
1
3
6
1
3
3
6
5
5
3
2
4
2
2
1
4
6
6
1
3
5
5
4
6
3
4
5
6
4
5
4
1
2
52
94
Soln.61
1
3
1
2
5
2
3
3
6
6
2
1
3
6
5
3
3
1
1
5
2
4
4
2
4
6
2
6
4
5
5
4
1
6
5
4
52
Soln.62
1
1
3
3
2
2
3
6
6
5
3
4
3
4
5
4
2
6
1
1
1
2
5
1
2
3
5
2
4
6
4
4
5
5
6
6
55
Soln.63
3
1
3
5
6
6
2
3
3
4
2
3
6
3
1
2
1
2
1
6
2
5
1
4
5
4
6
5
2
4
5
4
5
1
6
4
52
Soln.64
2
1
3
1
3
3
3
4
2
6
1
3
6
5
5
6
2
1
2
4
5
2
1
6
4
5
4
6
6
1
5
3
2
4
4
5
52
Soln.65
1
3
6
1
3
6
2
2
3
5
6
3
2
4
2
3
5
3
1
4
1
5
4
4
6
5
6
1
1
2
4
5
6
2
4
5
52
Soln.66
1
5
1
1
3
6
4
3
5
1
2
2
6
5
2
4
1
3
6
2
3
1
6
4
2
3
2
3
6
4
5
5
4
6
5
4
55
Soln.67
1
1
2
3
3
2
1
5
3
2
6
4
6
3
5
3
4
2
1
6
2
5
5
4
6
3
2
1
6
6
4
5
4
4
5
1
52
Soln.68
2
1
6
2
5
3
3
1
3
6
3
2
6
3
5
4
2
1
2
6
3
4
5
4
1
6
1
4
5
4
5
1
2
5
6
4
52
Soln.69
3
1
1
2
2
3
3
3
4
6
6
5
2
2
5
6
2
1
3
5
4
1
5
6
4
4
6
4
5
3
6
5
1
2
1
4
52
Soln.70
3
6
4
1
3
3
3
1
2
2
5
6
2
5
4
1
3
1
5
6
2
4
5
1
6
6
2
2
4
5
4
6
3
1
5
4
51
Soln.71
6
3
4
1
2
5
5
1
6
2
1
5
1
3
2
3
1
6
2
1
3
4
4
3
4
3
6
4
6
2
5
5
4
5
6
2
54
Soln.72
6
3
6
3
5
3
2
1
2
3
6
4
5
1
3
6
2
1
4
1
2
5
4
5
4
2
4
2
6
1
5
5
4
6
1
3
52
Soln.73
4
1
3
1
6
2
2
3
6
5
3
3
2
5
4
1
6
3
2
5
2
3
5
1
4
4
6
4
6
5
6
1
2
1
4
5
52
Soln.74
1
6
3
2
3
3
3
6
3
4
2
1
1
2
5
5
4
5
6
2
2
3
1
6
1
4
5
6
4
4
6
5
1
2
4
5
52
Soln.75
1
4
3
3
6
3
5
2
3
2
2
6
2
1
3
6
5
1
6
5
5
4
1
4
1
3
6
4
4
2
5
6
5
1
2
4
51
95
Soln.76
4
1
3
2
6
5
2
3
1
3
2
4
6
3
2
6
4
5
3
1
6
1
2
3
1
4
4
5
5
6
1
5
2
6
5
4
55
Soln.77
1
1
2
1
3
3
6
5
3
4
2
1
2
3
6
5
3
2
4
6
3
5
4
6
5
2
4
1
2
4
6
6
4
5
1
5
52
Soln.78
3
4
2
6
1
3
2
3
6
1
5
3
1
1
3
5
6
3
4
5
4
1
2
4
2
6
2
5
4
5
4
6
2
1
6
5
51
Soln.79
6
6
4
5
6
4
1
2
4
3
1
6
5
4
2
3
2
1
3
2
1
2
1
3
6
5
5
6
4
5
2
5
4
1
3
3
52
Soln.80
1
1
3
2
3
6
6
3
2
1
3
5
2
6
1
3
5
2
2
4
5
4
6
5
4
1
6
3
1
4
5
4
4
2
5
6
52
Soln.81
1
3
3
5
2
2
4
4
6
1
3
1
6
3
4
2
5
2
2
4
6
2
3
1
1
6
5
5
6
3
6
4
4
1
5
5
55
Soln.82
6
6
3
2
1
6
6
1
5
3
4
6
5
3
2
2
6
1
3
2
5
4
4
3
1
4
5
2
4
1
3
5
1
4
2
5
54
Soln.83
4
3
1
6
3
1
4
2
4
2
6
3
2
3
4
5
6
5
2
1
3
2
1
5
6
6
1
5
2
4
1
5
6
4
5
3
55
Soln.84
3
1
3
4
6
2
6
3
2
2
1
5
3
4
6
1
2
3
1
3
4
5
1
2
4
6
4
5
4
5
5
2
1
6
6
5
55
Soln.85
3
6
2
1
1
2
3
6
5
3
3
5
6
2
1
3
5
2
4
1
4
6
1
2
5
4
6
4
2
6
3
4
4
5
1
5
52
Soln.86
1
3
4
3
3
5
4
2
3
4
2
6
6
1
2
3
1
1
5
5
5
2
4
2
6
1
2
6
4
1
4
6
6
3
5
5
55
Soln.87
2
1
6
2
4
3
1
3
6
5
3
3
2
3
1
5
4
6
1
3
5
1
2
6
4
2
2
4
5
4
5
4
6
5
6
1
52
Soln.88
6
3
6
2
4
5
2
3
3
6
1
3
2
6
1
5
3
1
4
5
1
4
5
1
2
6
2
4
4
2
4
6
5
1
5
3
52
Soln.89
5
1
4
1
3
3
2
2
4
6
2
4
3
6
3
1
1
3
2
6
5
5
5
4
2
5
3
1
4
2
1
6
5
6
6
4
55
Soln.90
6
4
2
1
1
6
2
1
5
5
1
4
6
3
5
1
2
2
4
2
1
3
5
3
6
6
3
4
4
2
3
4
5
6
3
5
55
96
Soln.91
5
1
3
1
6
2
3
3
3
1
2
1
4
2
3
6
5
2
4
6
5
2
6
5
3
4
2
1
4
6
4
1
6
5
4
5
52
Soln.92
1
4
5
3
2
3
1
3
3
6
2
6
1
5
3
6
2
1
5
4
2
5
3
4
4
6
2
2
6
5
4
1
4
5
6
1
52
Soln.93
2
3
5
4
1
4
2
1
3
1
6
3
6
4
3
5
3
4
2
5
2
6
4
1
5
2
5
3
1
6
1
4
5
6
6
2
55
Soln.94
4
3
5
2
6
1
3
2
3
6
2
5
1
3
3
2
6
1
5
2
4
4
2
1
6
1
3
4
6
5
5
4
1
6
5
4
51
Soln.95
3
2
1
3
2
6
4
5
3
6
3
1
6
1
3
5
2
5
1
2
6
3
1
4
4
4
5
2
2
1
4
6
5
5
4
6
53
Soln.96
4
2
1
4
4
6
3
3
3
3
6
5
2
1
3
2
5
4
5
5
1
1
6
2
1
6
2
4
6
6
4
5
5
3
2
1
55
Soln.97
3
2
1
3
6
4
3
3
2
2
6
1
5
6
3
1
2
5
2
6
1
4
5
4
4
4
1
5
5
6
1
5
6
2
3
4
51
Soln.98
6
1
3
2
4
2
3
5
6
6
1
1
2
3
5
3
2
2
1
4
1
3
5
4
4
2
6
5
6
5
4
4
5
3
6
1
53
Soln.99
6
3
4
1
3
2
1
3
2
5
1
3
6
5
6
2
1
4
2
3
1
5
6
4
3
6
4
2
4
2
5
5
1
6
5
4
52
Soln.100
3
1
3
6
2
3
3
3
6
1
1
1
4
2
5
2
5
5
6
2
6
4
1
3
2
5
4
6
4
5
1
6
4
5
4
2
52
97
PROBLEM 3
Soln. 1
1
4
5
5
1
2
6
3
1
4
2
5
2
4
6
5
3
5
4
1
1
6
3
5
6
4
2
2
2
3
6
3
6
4
1
3
51
Soln. 2
4
1
2
1
5
6
4
1
3
6
2
5
6
4
1
2
3
4
6
3
2
3
5
4
6
1
6
3
5
2
2
1
5
3
4
5
50
Soln. 3
2
5
4
6
3
1
1
4
2
5
5
6
2
1
6
5
4
1
5
3
1
2
3
6
4
5
6
2
2
4
4
3
3
6
3
1
51
Soln. 4
6
4
1
4
5
3
5
1
2
2
1
5
5
6
3
4
6
1
4
5
5
6
2
3
2
2
6
4
3
1
3
6
4
2
1
3
51
Soln. 5
5
6
2
1
3
5
4
2
1
4
1
6
3
4
5
5
4
6
5
1
5
2
6
6
2
3
3
2
3
2
4
4
1
6
3
1
51
Soln. 6
5
5
3
4
1
6
2
3
1
2
5
5
4
4
5
1
1
2
4
3
1
6
6
3
3
2
5
2
6
6
3
1
2
4
6
4
51
Soln. 7
5
1
2
6
4
5
2
6
1
3
1
4
3
2
6
1
4
5
5
1
6
4
6
6
3
5
2
2
2
4
3
3
3
1
4
5
51
Soln. 8
3
4
5
5
1
6
4
6
2
1
1
2
4
5
5
3
4
1
6
3
3
2
6
5
2
5
2
1
4
6
6
3
3
2
4
1
50
Soln. 9
2
5
1
6
2
4
1
4
5
3
4
6
2
1
3
5
6
2
5
6
3
2
1
4
5
1
4
3
2
6
4
3
5
6
1
3
51
Soln. 10
6
1
2
1
6
2
4
1
4
2
6
5
1
4
3
3
2
6
4
5
3
3
5
2
6
2
4
5
1
4
1
6
3
5
3
5
51
Soln. 11
5
6
3
2
5
1
3
1
6
4
5
4
2
4
1
5
2
5
3
2
4
5
6
6
3
3
4
2
6
1
4
2
3
1
1
6
51
Soln. 12
4
2
1
5
6
2
1
4
1
6
5
3
3
5
6
2
6
4
6
4
5
5
1
3
6
2
5
2
3
4
3
1
4
2
1
3
52
Soln. 13
2
1
1
4
4
5
3
1
6
5
4
5
2
6
2
3
6
6
6
5
2
5
1
3
5
2
4
1
3
2
4
3
1
6
3
4
51
Soln. 14
4
6
2
4
1
4
5
1
6
2
3
3
4
1
5
6
5
1
2
5
2
3
6
4
6
3
1
6
4
3
5
2
1
5
3
2
50
Soln. 15
3
2
5
1
1
5
3
6
5
2
6
4
2
1
4
4
2
3
4
3
5
3
5
6
3
1
4
5
6
1
4
2
6
2
6
1
51
98
Soln. 16
2
5
1
3
6
5
2
1
6
3
1
6
5
1
4
4
6
3
4
6
5
3
2
1
2
2
5
4
6
4
3
5
1
4
2
3
51
Soln. 17
1
2
5
1
3
6
6
1
5
4
6
3
4
4
5
2
5
6
4
1
3
3
2
2
1
6
2
3
5
2
1
4
6
3
5
4
50
Soln. 18
1
6
1
3
4
4
2
5
1
6
4
2
2
5
4
3
6
5
1
6
2
3
5
1
5
6
4
3
5
3
2
1
6
3
4
2
51
Soln. 19
2
4
5
6
3
1
4
1
5
4
6
2
5
5
1
6
5
3
4
2
1
5
6
3
4
6
1
3
6
2
3
3
2
1
4
2
50
Soln. 20
6
1
4
2
1
5
3
1
2
5
5
4
4
5
2
5
6
1
3
6
3
5
6
6
3
2
3
1
2
6
1
4
3
2
4
4
50
Soln. 21
4
1
2
6
1
1
3
4
4
5
1
5
1
6
4
3
2
6
6
2
1
3
6
2
4
3
5
5
2
3
5
5
4
6
3
2
51
Soln. 22
5
5
4
3
6
1
4
1
2
5
6
2
2
1
3
2
4
6
3
4
1
5
6
4
6
1
3
2
6
4
5
3
1
5
3
2
51
Soln. 23
6
4
1
1
5
2
4
1
1
6
2
5
6
1
4
2
1
3
4
6
4
5
3
4
3
5
6
6
3
2
5
2
3
2
5
3
51
Soln. 24
6
2
2
2
1
1
5
6
3
1
5
4
6
1
5
4
5
4
6
1
3
2
6
3
3
6
3
5
2
3
1
5
4
4
4
2
50
Soln. 25
6
4
2
1
5
1
2
6
1
3
5
2
3
5
6
2
5
3
2
5
4
1
1
4
6
5
4
3
4
6
3
4
6
2
3
1
50
Soln. 26
1
5
6
6
4
2
1
4
4
3
1
5
4
2
5
1
1
1
3
6
3
5
2
6
2
6
3
5
3
4
5
6
3
2
2
4
51
Soln. 27
2
5
6
3
4
1
4
1
2
6
5
3
1
2
6
1
2
4
5
4
6
6
1
3
6
2
3
5
2
5
3
4
4
3
1
5
51
Soln. 28
5
3
1
4
6
5
1
4
3
1
5
2
2
5
4
1
4
6
3
5
3
5
2
2
6
4
6
3
3
2
1
2
6
1
6
4
51
Soln. 29
1
1
1
5
6
4
2
6
3
3
4
4
2
5
1
6
5
5
3
2
6
5
2
3
6
1
4
2
4
6
3
1
2
4
5
3
50
Soln. 30
2
5
3
1
6
4
5
4
2
2
1
3
5
6
1
4
3
3
4
6
5
2
6
2
3
5
2
3
5
6
1
1
6
4
1
4
51
99
Soln. 31
3
4
5
1
6
2
2
6
1
4
5
3
4
5
1
6
1
3
2
6
2
6
2
3
1
4
3
6
1
2
5
5
4
4
5
3
51
Soln. 32
4
6
1
4
1
5
2
6
2
2
1
5
3
4
5
5
6
3
3
6
1
5
2
5
4
1
3
2
6
3
6
1
4
4
2
3
50
Soln. 33
5
1
5
6
2
2
3
4
1
5
5
4
1
1
4
5
3
4
6
3
3
3
6
5
6
2
2
1
6
1
4
2
3
2
4
6
51
Soln. 34
2
4
1
1
6
2
5
3
1
2
4
1
5
5
6
6
4
3
6
2
3
2
1
5
4
3
6
3
1
4
4
6
5
5
3
2
50
Soln. 35
6
2
1
5
1
3
4
1
2
6
5
5
1
3
6
3
6
4
4
3
2
4
1
5
6
3
1
6
2
5
2
5
4
2
3
4
50
Soln. 36
5
1
1
6
4
4
5
2
6
1
6
3
4
5
2
6
1
2
3
2
1
5
3
3
6
6
5
4
3
3
2
1
2
5
4
4
51
Soln. 37
6
1
5
2
1
4
1
3
4
2
2
6
5
5
6
5
4
3
6
6
4
3
1
1
5
3
2
3
2
6
5
2
4
4
3
1
50
Soln. 38
4
6
1
2
5
1
6
2
3
1
4
3
5
5
2
6
1
3
4
1
6
6
4
2
3
2
3
5
5
4
5
4
1
6
3
2
51
Soln. 39
5
1
4
6
1
5
4
2
3
1
2
6
1
6
4
2
5
5
3
2
6
3
5
6
3
1
4
3
5
2
2
1
4
6
3
4
51
Soln. 40
6
1
1
5
1
2
2
3
6
1
2
4
4
3
5
4
1
6
5
3
5
6
6
5
4
6
3
3
2
2
4
2
5
1
3
4
50
Soln. 41
1
1
5
4
4
1
4
3
2
5
6
1
1
4
5
2
6
3
6
6
1
4
2
2
4
6
3
5
2
3
5
6
3
3
5
2
51
Soln. 42
4
3
6
5
5
1
1
1
2
4
6
4
2
2
3
5
6
3
5
4
2
5
4
6
5
1
2
6
3
3
2
1
6
1
3
4
50
Soln. 43
5
1
5
3
1
3
1
4
5
6
2
4
2
1
2
4
3
6
5
4
1
3
5
6
3
6
5
2
6
1
4
3
6
2
4
2
51
Soln. 44
1
4
2
1
5
6
1
4
2
2
6
3
5
6
5
1
3
4
1
6
6
6
3
3
2
4
3
5
5
2
2
1
4
5
3
4
50
Soln. 45
4
6
4
2
1
5
1
2
3
1
6
4
4
3
5
5
1
6
6
4
1
1
5
2
3
5
6
3
2
6
3
2
5
2
3
4
51
100
Soln. 46
1
5
6
6
2
1
4
5
1
4
2
2
3
3
5
5
5
4
5
4
1
1
6
6
2
3
6
6
3
3
2
4
4
2
3
1
51
Soln. 47
6
2
3
1
2
5
1
5
4
6
2
5
3
1
4
4
4
5
6
3
6
5
1
1
2
3
5
2
6
4
3
2
4
6
1
3
51
Soln. 48
5
1
4
2
6
3
4
1
5
1
3
2
6
3
3
2
2
4
5
6
5
3
4
5
1
3
1
2
6
5
6
6
4
2
4
1
51
Soln. 49
5
4
6
1
3
1
6
5
2
4
1
5
6
2
4
5
3
1
2
3
5
2
4
5
3
6
2
1
4
6
6
3
3
4
2
1
50
Soln. 50
4
5
5
4
1
6
3
2
1
1
2
3
6
4
3
2
1
6
3
4
5
5
4
1
2
4
5
3
5
6
6
3
2
2
6
1
51
Soln. 51
5
1
6
2
6
2
3
1
5
1
4
5
6
4
5
1
5
2
1
3
3
5
3
4
6
6
6
3
2
1
2
4
3
4
2
4
51
Soln. 52
6
4
1
5
2
1
6
3
1
5
3
6
2
5
3
4
5
1
2
6
1
3
4
5
2
6
3
6
3
4
5
2
4
4
1
2
50
Soln. 53
5
6
1
3
5
4
2
6
3
1
4
2
1
2
5
6
5
4
6
2
6
1
3
2
1
3
3
1
4
4
5
4
2
6
3
5
51
Soln. 54
4
2
1
5
1
6
3
5
2
6
6
4
5
2
5
4
6
4
1
1
5
3
2
3
5
3
2
6
3
2
1
6
4
3
1
4
52
Soln. 55
2
1
1
5
3
1
6
5
6
5
4
2
4
1
5
2
6
6
4
3
4
5
3
2
1
6
2
4
3
4
5
3
6
2
3
1
51
Soln. 56
6
5
3
5
1
2
4
6
1
4
2
3
1
6
1
2
1
4
4
2
5
6
5
3
2
4
5
3
6
3
5
1
6
3
2
4
51
Soln. 57
4
6
1
1
3
1
4
5
4
4
5
6
6
1
2
1
2
5
2
1
3
5
2
6
4
3
3
5
2
4
6
6
3
3
2
5
51
Soln. 58
2
6
1
6
1
2
1
3
6
1
5
4
2
3
5
4
6
3
3
2
1
4
4
2
6
4
1
4
6
5
5
3
5
2
3
5
51
Soln. 59
2
5
4
3
1
1
6
2
6
3
5
1
5
1
2
5
1
4
6
3
3
2
5
2
6
4
4
6
3
6
2
4
1
3
4
5
51
Soln. 60
6
5
2
1
4
5
4
3
6
1
5
3
1
2
5
4
6
2
1
1
4
6
6
2
3
3
3
5
4
6
2
1
5
3
4
2
51
101
Soln. 61
3
5
4
6
4
1
2
4
5
1
3
1
6
3
2
1
2
5
6
1
4
5
3
2
5
3
4
3
5
6
4
6
2
6
2
1
51
Soln. 62
1
2
5
1
6
3
1
5
2
2
6
3
4
5
6
6
1
2
3
4
1
4
5
6
4
2
4
5
2
6
5
3
3
3
1
4
51
Soln. 63
1
1
6
2
1
2
3
5
5
5
6
2
4
6
3
1
3
4
6
4
1
2
5
4
5
2
4
4
2
3
6
3
6
1
3
5
50
Soln. 64
4
5
2
6
3
5
1
1
5
4
4
1
6
3
6
2
4
6
5
3
2
1
6
3
5
2
2
2
3
1
1
4
5
4
6
3
51
Soln. 65
6
4
1
2
1
5
4
1
6
2
5
1
2
4
1
3
6
3
4
6
2
6
3
3
3
6
4
5
4
5
1
2
2
5
3
5
50
Soln. 66
6
1
4
1
4
2
5
2
1
2
6
3
5
1
5
4
5
3
6
6
3
4
3
5
6
1
3
2
2
6
3
5
2
4
4
1
50
Soln. 67
1
1
6
2
4
1
5
5
3
4
6
3
2
6
5
6
1
4
2
3
5
3
2
1
6
3
2
1
5
4
6
4
3
4
5
2
50
Soln. 68
4
1
6
1
4
4
2
1
5
2
3
6
1
2
5
3
6
4
3
2
3
5
1
5
6
4
6
3
4
1
2
5
6
3
5
2
51
Soln. 69
4
6
2
1
1
1
3
5
6
5
5
2
1
3
5
6
2
5
4
2
5
3
2
1
6
4
4
6
2
3
6
4
3
1
3
4
51
Soln. 70
6
1
1
4
6
1
3
5
6
2
4
4
3
4
6
5
3
5
4
6
1
1
2
2
5
1
5
3
2
4
6
2
3
5
2
3
52
Soln. 71
6
1
4
4
2
1
5
3
1
4
6
6
1
3
5
4
2
3
1
6
5
2
1
4
2
5
5
6
6
2
5
3
2
3
3
4
52
Soln. 72
2
1
6
5
2
1
1
1
5
3
6
6
4
3
2
2
4
1
5
6
4
4
3
2
5
4
6
4
3
5
3
2
6
1
3
5
50
Soln. 73
5
6
2
1
1
4
6
4
3
5
3
2
1
2
4
4
6
1
3
2
6
3
4
2
5
4
2
1
6
6
5
3
5
3
1
5
51
Soln. 74
1
1
6
5
4
1
3
4
6
1
2
3
1
5
4
4
2
6
5
1
6
3
4
5
2
6
3
2
2
5
4
2
3
5
6
3
51
Soln. 75
1
1
4
2
3
5
6
1
4
4
4
5
6
3
6
2
1
6
3
5
4
2
6
3
2
1
3
2
6
2
5
5
4
1
5
3
50
102
Soln. 76
2
1
1
4
6
1
2
3
5
4
5
1
2
6
5
3
6
4
3
6
3
1
6
2
6
4
3
2
5
5
4
1
3
5
4
2
50
Soln. 77
5
5
1
2
6
4
4
3
2
1
5
1
6
6
5
2
4
5
3
1
1
5
4
2
3
4
2
6
6
3
3
6
2
1
4
3
50
Soln. 78
6
1
2
2
1
4
1
5
6
3
2
5
1
5
3
6
6
3
5
4
2
4
4
2
5
4
5
6
3
1
3
2
6
4
1
3
50
Soln. 79
5
3
4
1
1
6
2
5
2
1
5
2
3
5
4
4
6
3
5
5
3
4
1
2
6
6
4
2
2
3
3
6
1
1
6
4
51
Soln. 80
4
2
3
5
4
1
1
6
4
5
2
3
2
1
3
1
6
2
4
5
6
2
3
5
4
5
3
5
6
1
3
6
1
6
4
2
51
Soln. 81
1
6
1
2
3
1
5
4
6
5
5
2
3
6
1
5
2
1
2
4
4
3
6
3
5
2
6
6
4
2
3
4
1
5
3
4
50
Soln. 82
6
4
2
4
1
1
3
5
6
2
4
1
4
3
1
5
5
5
1
6
6
3
2
2
6
5
6
5
4
3
4
3
2
1
2
3
52
Soln. 83
1
1
2
6
5
2
1
4
4
5
3
6
1
6
3
4
5
6
3
2
5
5
4
5
2
6
1
2
3
3
4
1
6
3
4
2
50
Soln. 84
1
2
5
6
4
1
2
1
3
6
3
1
5
5
5
2
6
6
4
4
3
2
5
4
6
2
4
1
3
3
5
4
1
2
6
3
50
Soln. 85
4
3
6
5
1
1
2
5
4
6
2
1
1
5
2
5
3
4
2
6
6
5
3
1
6
3
6
3
3
2
5
1
4
2
4
4
51
Soln. 86
6
2
5
1
5
6
4
3
3
1
4
1
5
4
2
1
4
2
6
5
6
5
3
4
6
3
2
3
5
1
2
2
1
4
6
3
51
Soln. 87
5
3
4
1
5
1
6
5
3
2
4
2
6
1
4
3
2
4
6
6
4
1
3
1
2
4
5
3
5
2
2
5
3
6
6
1
51
Soln. 88
5
4
1
6
6
3
3
5
4
2
1
1
4
5
2
6
4
3
5
2
1
2
3
6
4
6
2
3
1
1
5
6
4
2
5
3
51
Soln. 89
6
2
5
1
6
3
4
1
4
4
5
3
1
5
6
2
4
6
2
6
3
1
2
4
5
1
6
4
3
2
3
2
1
5
3
5
51
Soln. 90
1
5
6
4
5
2
4
1
3
1
2
5
5
4
6
4
3
6
2
6
3
2
6
1
1
5
4
2
2
4
3
3
5
1
6
3
52
103
Soln. 91
3
2
5
1
4
1
6
5
5
3
1
2
4
4
6
3
1
4
6
3
2
6
3
5
4
2
2
1
5
5
4
6
3
2
6
1
51
Soln. 92
5
1
6
3
1
4
2
4
2
5
6
4
4
5
1
2
1
1
6
3
5
6
3
2
5
3
2
4
6
2
3
5
6
1
4
3
51
Soln. 93
2
4
5
5
6
2
4
3
1
6
3
1
2
2
5
5
1
4
1
6
3
2
6
3
4
1
2
6
3
1
6
5
5
3
4
4
51
Soln. 94
6
1
2
1
5
6
4
1
5
3
3
6
5
2
6
1
5
2
2
4
1
3
4
3
2
6
3
4
4
2
5
5
4
6
3
1
50
Soln. 95
5
2
5
1
1
6
2
6
3
4
5
2
1
5
6
5
4
1
3
4
4
1
2
3
5
4
3
3
3
6
6
4
6
2
1
2
52
Soln. 96
3
5
1
3
1
2
4
5
6
2
5
1
3
2
4
4
5
1
6
5
2
1
4
5
6
3
3
6
6
3
6
2
2
1
4
4
51
104
PROBLEM 4
Soln.1
5
5
3
4
2
4
1
6
2
5
5
3
6
3
4
5
2
3
1
3
6
4
5
6
1
4
1
2
2
4
2
1
6
1
3
6
45
Soln.2
5
6
4
3
2
2
5
4
6
5
4
1
6
3
3
2
5
6
5
1
4
5
2
1
3
4
1
4
6
6
1
2
3
2
1
3
46
Soln.3
4
5
2
6
3
4
4
2
5
3
3
5
1
2
6
5
1
5
6
5
1
4
3
3
2
6
4
1
2
1
3
6
6
4
2
1
43
Soln.4
6
1
3
4
4
5
5
2
3
5
1
2
6
4
3
2
6
5
3
5
5
2
1
4
6
1
3
2
1
6
1
2
4
3
4
6
45
Soln.5
2
6
4
1
3
5
4
6
5
2
4
6
1
3
3
2
2
5
6
5
5
6
4
5
3
1
1
3
2
1
1
6
3
2
4
4
46
Soln.6
4
5
6
2
5
4
3
3
6
2
1
3
2
4
5
5
2
6
1
3
2
4
5
1
6
5
1
4
1
3
3
6
6
2
1
4
43
Soln.7
3
2
4
5
1
6
5
4
5
1
3
2
3
5
4
2
5
1
6
3
3
1
4
6
6
2
4
1
1
5
6
6
2
3
4
2
46
Soln.8
4
3
1
5
5
2
6
5
3
3
1
4
2
5
6
4
2
4
1
3
2
5
2
6
4
1
6
1
3
4
3
2
5
1
6
6
45
Soln.9
1
3
2
5
3
5
6
2
4
1
5
4
5
2
1
1
2
4
3
2
6
5
1
3
5
6
1
4
6
2
3
4
3
4
6
6
47
Soln.10
3
5
3
4
6
2
4
1
5
3
6
5
6
1
4
5
2
2
2
1
2
3
5
4
6
1
5
3
6
3
4
1
4
1
6
2
44
Soln.11
4
3
4
2
5
3
2
2
6
5
5
3
6
1
1
5
2
4
3
5
5
6
1
2
3
4
3
4
1
4
6
6
1
6
1
2
45
Soln.12
6
5
5
4
2
3
6
5
4
4
2
1
5
5
3
6
2
3
1
6
6
5
3
2
1
1
2
6
3
1
1
4
4
2
4
3
44
Soln.13
2
3
4
4
2
6
5
1
5
3
5
4
5
1
4
3
1
2
3
5
2
1
1
2
5
6
3
6
4
3
6
4
2
1
6
6
45
Soln.14
3
5
6
4
5
5
6
4
2
1
2
6
3
3
5
1
4
5
3
1
2
1
5
6
4
6
2
4
2
2
3
1
6
4
1
3
44
Soln.15
6
2
4
5
5
3
1
4
3
6
5
3
2
2
4
5
6
4
3
5
1
6
2
5
1
3
3
4
1
1
4
6
2
6
2
1
45
105
Soln.16
4
4
6
5
2
3
2
5
3
2
1
3
4
5
6
1
4
6
2
6
5
2
3
1
2
5
3
1
4
5
1
3
6
4
6
1
44
Soln.17
5
3
5
2
3
6
1
4
4
3
5
2
2
1
4
6
5
4
5
3
1
2
2
3
6
1
1
6
3
4
2
1
4
5
6
6
45
Soln.18
6
5
6
2
4
1
4
5
3
2
5
1
2
6
6
5
3
3
4
5
2
6
4
5
2
3
1
1
2
4
4
1
3
1
6
3
44
Soln.19
2
4
6
3
5
5
5
6
3
6
3
1
6
4
1
2
5
4
1
5
6
2
5
3
3
1
1
6
2
3
1
2
4
4
2
4
46
Soln.20
6
3
5
3
4
6
5
4
5
5
2
2
1
4
1
5
3
6
2
4
3
5
1
3
4
1
2
3
2
6
6
4
2
1
6
1
44
Soln.21
5
4
5
2
3
4
6
5
3
2
1
2
4
4
3
6
5
6
2
1
6
5
3
3
5
6
1
2
2
1
1
6
1
3
4
4
44
Soln.22
2
3
6
5
5
3
4
4
5
2
4
5
4
3
6
1
5
6
3
1
2
1
1
2
3
6
5
2
6
1
6
2
1
3
4
4
45
Soln.23
5
1
5
6
3
5
1
3
3
2
4
4
5
2
4
3
2
6
4
5
6
5
3
2
6
6
2
1
1
4
2
1
4
3
6
1
45
Soln.24
4
5
2
6
5
4
3
5
4
1
5
4
3
6
3
6
1
2
2
2
5
5
3
1
6
4
1
3
1
2
3
6
6
1
2
4
45
Soln.25
2
5
4
5
6
3
5
3
1
4
3
4
1
6
5
5
4
2
1
6
3
2
4
6
3
1
1
2
3
4
2
1
6
5
2
6
45
Soln.26
4
1
3
5
3
4
5
2
4
6
2
5
2
3
1
5
6
4
3
5
3
1
2
1
1
2
3
6
1
6
5
6
6
2
4
4
45
Soln.27
4
4
3
5
2
5
3
3
2
5
6
4
5
1
1
6
2
3
4
6
5
3
6
1
5
3
2
1
6
4
4
2
1
2
1
6
45
Soln.28
5
1
4
6
6
3
2
5
3
4
5
1
6
5
4
2
4
6
3
1
4
3
2
5
2
2
4
2
5
1
1
6
3
1
6
3
46
Soln.29
6
5
3
2
4
5
6
5
4
6
1
1
3
6
4
5
4
3
2
5
2
3
6
5
1
2
1
4
4
3
6
1
3
2
1
2
44
Soln.30
3
4
4
2
5
6
3
4
5
5
5
4
2
3
2
5
1
1
6
2
6
3
5
1
4
1
3
1
4
3
6
2
6
6
2
1
43
106
Soln.31
3
5
4
3
5
6
6
4
3
1
2
5
1
6
5
4
2
2
3
5
5
1
1
1
4
2
4
6
3
4
2
6
1
3
6
2
43
Soln.32
2
3
5
4
3
4
4
1
5
5
1
2
6
4
5
3
6
6
1
4
5
2
3
5
1
2
2
4
3
6
6
2
1
3
1
6
44
Soln.33
3
5
4
2
5
6
5
1
3
4
6
5
4
1
2
3
3
4
1
5
6
2
1
2
1
4
6
4
2
2
3
5
3
1
6
6
45
Soln.34
5
3
4
5
4
3
6
2
2
5
4
5
6
1
3
6
1
4
2
2
1
6
3
5
1
4
3
2
2
5
6
1
3
4
6
1
44
Soln.35
5
6
6
3
4
5
4
2
1
5
6
2
4
3
3
2
5
1
3
1
5
2
1
6
2
4
5
4
6
6
3
4
1
3
2
1
44
Soln.36
5
1
3
3
6
5
6
5
4
3
4
2
5
4
2
2
4
6
1
6
1
3
2
5
5
1
4
4
3
1
3
6
2
2
1
6
45
Soln.37
1
4
5
2
5
4
3
5
6
1
5
3
3
4
4
2
2
6
6
1
5
6
3
3
5
1
1
3
6
6
1
4
2
2
4
2
45
Soln.38
4
5
2
4
4
3
5
5
6
1
3
2
5
1
3
4
2
5
6
6
3
1
1
2
3
6
5
6
2
1
2
4
4
6
1
3
45
Soln.39
1
4
2
3
5
5
1
5
3
4
2
4
6
6
5
3
2
5
3
6
2
5
3
4
1
6
4
3
2
6
1
1
4
6
2
1
47
Soln.40
4
5
3
2
6
3
5
4
6
5
4
2
1
5
1
4
2
3
5
2
6
1
4
3
2
4
5
1
1
6
2
3
6
3
1
6
43
Soln.41
3
5
6
4
6
1
3
4
2
2
5
5
5
6
1
6
3
2
1
4
5
5
6
2
3
1
6
1
2
3
4
4
3
2
4
1
44
Soln.42
2
5
3
5
3
6
4
4
4
6
2
1
3
5
1
5
2
5
6
5
2
3
1
3
4
6
3
6
1
1
6
1
2
2
4
4
44
Soln.43
1
6
2
5
2
4
3
3
4
1
5
5
4
3
5
6
6
2
4
6
5
3
1
2
1
2
4
2
4
5
3
1
6
3
1
6
46
Soln.44
4
2
5
4
5
3
4
5
1
6
3
5
4
6
1
5
3
2
2
3
6
1
4
2
4
3
1
6
2
5
1
6
2
6
3
1
45
Soln.45
5
2
4
4
1
3
4
5
5
3
2
5
1
3
2
6
5
6
4
3
5
2
1
6
6
2
3
4
2
1
1
3
1
4
6
6
45
107
Soln.46
6
3
5
4
3
5
5
6
4
1
2
4
5
5
2
3
1
2
2
6
3
3
4
6
1
1
2
6
4
5
3
1
4
2
1
6
44
Soln.47
2
5
4
5
1
3
3
4
4
5
6
2
6
5
3
4
6
5
1
1
2
5
3
1
2
4
3
1
6
4
6
1
2
3
6
2
45
Soln.48
4
5
2
4
5
4
1
6
3
3
6
2
5
5
3
1
6
3
4
5
1
4
2
2
4
5
1
1
3
6
3
2
6
6
2
1
45
Soln.49
5
3
4
5
5
2
4
3
1
3
1
2
4
2
6
6
5
2
6
4
5
1
5
2
3
6
3
4
1
2
6
1
4
6
3
1
44
Soln.50
4
3
2
6
5
3
4
2
2
4
5
6
3
6
1
5
5
3
4
2
5
1
5
1
6
1
2
3
6
3
1
6
2
4
4
1
45
Soln.51
3
4
5
5
3
4
5
3
2
1
6
4
4
5
2
1
6
2
6
2
6
5
5
3
1
2
3
6
3
1
4
2
1
6
4
1
45
Soln.52
2
4
6
3
5
6
2
5
6
1
4
5
5
4
6
3
2
1
5
3
3
6
1
1
4
5
2
6
3
1
3
4
2
1
2
4
46
Soln.53
4
6
1
5
3
3
4
6
2
5
3
4
5
4
2
5
2
6
3
1
2
1
2
5
5
4
3
3
1
6
1
4
6
1
6
2
45
Soln.54
2
4
3
5
3
5
4
5
1
2
6
4
3
4
1
6
5
6
3
1
2
2
1
5
2
3
6
4
2
4
5
1
6
6
3
1
44
Soln.55
5
2
3
5
4
1
4
4
3
3
2
2
5
5
1
6
3
1
6
4
5
1
3
6
6
2
1
1
2
3
4
6
5
2
4
6
45
Soln.56
2
3
4
4
5
5
3
6
1
4
2
5
6
2
3
2
6
2
5
3
1
1
5
1
5
4
3
4
6
1
4
2
6
3
1
6
44
Soln.57
4
5
4
5
3
3
3
5
6
1
2
5
4
6
2
1
6
1
2
2
5
4
6
3
1
5
2
1
2
3
4
6
3
1
6
4
43
Soln.58
3
2
4
5
4
5
3
5
3
2
1
2
6
1
4
5
2
3
1
6
2
6
5
4
3
2
6
1
1
5
3
4
4
1
6
6
45
Soln.59
1
1
6
5
5
4
6
5
2
1
4
6
2
2
5
4
3
4
3
3
2
5
1
6
2
1
4
1
3
5
6
4
2
6
3
3
46
Soln.60
4
5
3
6
6
4
5
2
2
2
6
1
3
5
1
3
4
1
5
2
2
3
6
5
4
6
4
5
6
1
2
4
1
3
1
3
44
108
Soln.61
5
6
2
3
5
3
4
5
3
4
2
5
2
1
6
2
4
3
5
6
1
4
1
5
3
6
6
2
2
1
1
1
6
3
4
4
43
Soln.62
4
6
3
2
5
4
3
4
5
2
1
5
1
5
6
2
3
6
1
3
5
5
1
1
4
6
4
3
2
6
6
1
3
4
2
2
43
Soln.63
3
5
2
1
4
4
5
2
3
5
1
2
6
4
3
5
1
3
5
3
2
1
6
4
2
6
1
4
1
3
2
6
6
5
6
4
45
Soln.64
4
5
3
3
5
6
1
4
5
2
4
5
2
6
3
6
1
2
5
4
2
1
3
1
3
4
5
4
6
1
2
3
6
6
1
2
43
Soln.65
4
4
5
2
3
4
6
5
3
3
5
4
5
2
2
6
1
6
3
1
5
5
3
1
6
6
1
2
4
2
1
6
3
2
1
4
43
Soln.66
4
6
3
4
2
5
2
3
6
5
1
2
5
3
5
4
6
1
2
1
5
4
6
4
3
3
5
3
1
2
2
1
6
1
4
6
44
Soln.67
4
6
2
5
6
5
4
5
2
3
5
6
3
4
2
5
6
1
1
4
3
3
5
1
2
2
6
1
4
3
1
6
2
4
1
3
46
Soln.68
4
4
1
3
5
2
5
6
3
2
5
5
2
4
6
1
3
4
3
1
1
2
1
2
6
2
4
5
5
3
3
6
4
1
6
6
45
Soln.69
2
4
3
5
5
5
4
3
5
1
1
2
6
4
6
4
3
1
3
6
2
5
4
1
2
1
4
6
2
3
2
1
5
3
6
6
45
Soln.70
3
5
1
3
4
5
4
2
2
5
1
5
2
3
6
2
3
4
1
2
6
5
4
1
2
6
5
3
1
3
6
4
1
4
6
6
45
Soln.71
3
6
2
4
5
5
5
4
4
5
3
6
3
4
6
2
1
1
1
3
1
2
5
2
6
4
5
3
3
6
2
2
4
1
1
6
45
Soln.72
4
3
5
6
6
1
5
6
5
3
4
2
2
2
1
2
6
5
4
4
3
1
5
5
6
3
1
2
4
1
4
6
3
2
1
3
44
Soln.73
2
3
5
5
4
3
6
5
4
4
2
1
5
6
5
3
2
1
6
4
5
2
3
1
6
3
4
1
4
6
1
2
6
1
2
3
43
Soln.74
4
3
2
5
6
5
4
3
3
2
5
4
5
4
6
1
5
3
6
3
2
1
5
1
1
3
4
2
2
6
6
2
1
4
6
1
45
Soln.75
2
5
4
5
6
1
2
3
4
2
3
3
5
2
4
5
3
5
2
1
5
3
6
4
2
6
1
1
1
6
3
6
4
4
1
6
46
109
Soln.76
5
3
5
3
2
5
4
4
6
1
6
2
4
5
3
6
1
3
5
1
2
4
3
2
1
2
4
5
1
6
6
2
1
4
3
6
44
Soln.77
1
3
4
3
2
5
6
6
1
2
4
3
4
3
5
1
5
2
4
6
5
1
2
1
6
4
3
2
4
5
6
3
5
2
1
6
46
Soln.78
2
4
4
5
3
6
5
3
3
5
1
4
1
2
5
6
3
5
2
6
2
3
4
6
6
2
1
5
1
1
2
4
4
6
3
1
44
Soln.79
5
5
6
6
4
3
1
5
4
2
2
3
2
6
5
1
3
1
4
2
6
5
4
2
2
5
6
3
3
1
1
1
3
4
4
6
44
Soln.80
4
2
5
6
5
4
3
3
5
5
3
1
4
5
6
2
6
4
1
1
2
5
3
6
2
2
3
4
1
1
2
3
6
4
6
1
45
Soln.81
1
4
4
2
5
5
2
6
3
3
4
2
5
6
5
3
4
1
5
6
6
2
3
5
2
3
1
3
1
1
6
1
4
2
4
6
45
Soln.82
5
5
3
4
4
3
2
1
3
4
2
6
6
2
5
2
4
5
3
6
1
5
5
1
6
3
1
1
2
6
4
2
4
1
3
6
44
Soln.83
3
5
5
4
3
1
6
5
4
5
2
1
4
2
6
2
4
5
3
6
3
5
6
2
1
3
1
2
6
1
3
2
1
6
4
4
43
Soln.84
6
4
5
5
4
2
6
3
1
5
1
5
4
2
6
6
5
3
4
3
1
3
2
2
6
1
2
4
4
5
1
6
3
2
1
3
44
Soln.85
5
4
3
3
2
1
6
5
4
5
2
2
1
4
6
3
6
4
1
5
6
5
3
3
5
2
1
1
6
3
4
6
1
4
2
2
43
Soln.86
1
6
4
5
5
4
3
2
4
1
3
2
2
6
5
4
3
6
5
1
5
3
4
5
1
2
6
3
4
2
6
1
6
2
1
3
46
Soln.87
5
4
3
5
3
5
2
6
4
4
5
3
6
2
4
1
1
6
2
3
2
1
5
5
4
4
3
1
6
2
6
3
2
1
1
6
45
Soln.88
5
4
3
6
4
2
5
3
4
5
6
5
2
5
3
2
6
1
5
1
3
1
4
2
2
3
4
6
3
1
4
6
2
1
1
6
44
Soln.89
6
5
3
3
4
4
5
5
2
5
2
1
6
3
6
4
2
2
5
4
3
3
1
1
1
4
5
6
3
1
2
6
2
4
6
1
45
Soln.90
4
5
2
5
3
5
6
3
4
1
3
6
5
2
4
5
3
6
2
3
2
1
4
6
6
5
1
3
2
1
4
1
1
6
2
4
44
110
Soln.91
4
6
5
4
5
6
5
1
6
3
2
4
3
3
5
6
5
2
1
4
2
6
5
6
3
2
1
1
2
4
3
1
3
2
4
1
46
Soln.92
4
5
6
5
3
3
2
5
6
4
4
5
3
6
5
1
2
2
1
5
2
3
1
4
2
6
3
4
1
2
1
6
4
1
6
3
43
Soln.93
4
3
5
3
5
2
6
3
1
4
6
2
2
2
5
6
4
1
5
3
1
1
4
5
2
4
5
4
3
6
3
6
1
2
1
6
44
Soln.94
5
2
6
3
3
5
4
5
1
4
4
6
1
2
3
6
6
5
4
5
3
1
2
2
3
1
4
6
5
2
2
1
3
4
1
6
47
Soln.95
2
6
4
3
5
6
4
5
1
5
3
5
2
6
2
2
3
4
5
3
1
6
4
1
1
4
6
2
5
3
4
1
2
6
1
3
44
Soln.96
5
2
6
3
5
3
4
5
4
6
1
4
1
3
5
5
2
6
3
2
5
1
1
3
1
4
2
6
4
3
6
4
1
2
6
2
43
Soln.97
4
5
6
5
4
6
5
2
3
2
6
5
1
1
2
4
3
2
3
5
1
2
3
6
1
5
4
6
2
3
1
4
6
1
3
4
44
Soln.98
1
1
5
6
5
3
4
2
4
6
1
4
5
1
6
2
3
1
2
3
1
5
4
5
6
3
2
6
4
4
2
5
6
2
3
3
47
Soln.99
6
2
4
5
2
1
5
3
3
6
4
2
5
3
4
5
5
2
1
4
2
3
3
5
6
6
1
6
4
1
1
1
2
6
3
4
46
Soln.100
4
5
3
3
4
5
5
1
6
2
3
5
2
4
1
2
6
4
5
2
3
5
6
6
1
3
2
1
4
6
4
1
2
3
1
6
43
111
PROBLEM 5
Soln.1
6
1
2
6
6
6
3
4
6
5
1
5
4
1
3
5
6
3
5
4
1
4
2
3
1
3
1
3
5
2
2
2
5
4
4
2
53
Soln.2
5
2
1
2
6
5
4
3
2
1
6
4
5
2
6
1
6
4
1
5
1
5
2
4
6
4
1
4
5
3
6
3
2
3
3
3
53
Soln.3
6
2
4
5
1
6
3
6
5
3
1
6
4
5
1
1
6
4
1
3
4
4
2
3
5
4
1
5
2
3
2
5
3
6
2
2
52
Soln.4
2
1
5
4
6
2
4
5
2
6
6
4
2
2
5
4
3
6
1
4
5
6
2
1
5
3
1
6
4
3
3
1
3
3
5
1
54
Soln.5
6
4
5
6
1
6
5
2
1
3
4
2
5
1
1
4
6
3
4
6
4
5
2
3
3
2
5
1
5
2
4
6
2
1
3
3
54
Soln.6
6
4
3
5
2
6
1
6
1
4
2
5
2
5
6
4
1
6
5
1
5
1
5
3
4
3
3
1
2
3
4
2
6
2
4
3
53
Soln.7
6
6
5
4
6
3
1
3
2
4
1
5
1
5
4
6
1
1
3
6
6
5
5
3
4
1
2
5
2
3
3
4
4
2
2
2
53
Soln.8
5
6
2
5
2
1
6
1
5
2
6
4
2
5
1
1
3
4
6
4
2
6
5
4
1
2
3
4
3
5
1
6
3
4
3
3
53
Soln.9
3
6
6
5
1
6
5
6
1
1
3
5
6
2
5
6
5
1
1
1
5
4
3
3
4
4
4
3
4
3
2
2
2
4
2
2
53
Soln.10
6
5
4
6
6
1
3
2
4
4
4
5
6
1
4
6
3
3
2
3
3
4
2
1
5
1
5
1
2
6
1
5
5
2
2
3
53
Soln.11
5
1
5
2
2
4
2
4
6
6
1
5
2
4
6
2
3
4
3
3
6
1
5
1
4
5
6
1
4
3
1
6
3
5
3
2
53
Soln.12
5
2
1
4
2
5
2
6
2
3
1
4
6
2
5
2
1
1
6
6
5
4
4
6
4
1
3
3
3
1
3
4
5
5
3
6
52
Soln.13
2
5
3
2
5
6
4
2
2
2
6
6
1
5
2
5
4
3
1
5
6
1
6
4
4
1
5
1
3
3
3
3
4
1
4
6
54
Soln.14
6
4
6
5
2
1
4
3
2
6
3
4
1
5
4
4
3
1
6
6
1
2
2
6
5
1
4
5
2
3
2
1
3
5
5
3
53
Soln.15
2
6
4
5
6
1
1
3
1
4
6
4
5
4
6
2
2
6
3
3
2
5
4
1
6
5
4
1
5
3
2
5
3
3
1
2
50
112
Soln.16
6
6
4
4
6
2
5
3
4
2
6
1
1
5
4
3
4
6
2
3
6
4
3
1
2
5
2
5
3
2
1
1
5
5
3
1
51
Soln.17
6
5
4
3
6
5
6
3
2
3
5
5
1
4
3
4
5
2
1
6
3
4
1
3
6
1
1
2
2
5
4
6
2
1
2
4
53
Soln.18
6
1
2
4
6
4
6
2
1
6
4
3
1
3
4
2
5
1
5
6
3
2
5
4
1
2
2
6
5
3
5
4
3
3
5
1
54
Soln.19
4
6
4
4
1
5
2
1
4
6
6
6
2
5
5
3
1
4
6
1
3
2
1
5
5
3
4
5
1
2
2
6
3
2
3
3
54
Soln.20
6
5
3
4
6
6
1
6
3
1
5
4
1
2
3
4
6
5
5
2
4
5
1
1
4
3
3
5
6
2
2
1
2
3
2
4
53
Soln.21
5
5
1
2
4
6
2
6
2
4
5
1
6
2
3
1
1
2
3
6
4
2
1
3
4
5
4
5
4
3
6
6
1
3
5
3
52
Soln.22
5
1
2
6
5
2
2
1
3
2
6
5
6
4
1
4
2
5
5
4
3
2
4
4
6
6
6
1
1
4
3
5
1
3
3
3
54
Soln.23
4
3
4
3
4
5
6
1
5
6
4
5
2
6
6
4
5
3
2
3
3
1
4
1
3
5
6
1
1
6
2
5
2
1
2
2
54
Soln.24
2
5
5
6
2
1
4
6
4
5
2
3
6
2
6
4
1
5
1
4
1
5
6
2
1
5
4
6
3
4
2
1
3
3
3
3
52
Soln.25
6
6
6
4
3
2
6
5
1
1
4
2
6
5
3
1
4
5
3
2
4
4
5
3
6
1
3
1
2
1
4
5
3
2
2
5
54
Soln.26
2
6
1
6
6
4
4
4
2
5
1
4
3
1
3
4
6
1
4
3
5
6
6
5
2
3
2
2
2
1
5
5
3
3
1
5
52
Soln.27
6
6
2
4
4
5
1
1
2
6
3
5
4
5
6
1
4
2
1
3
6
2
1
5
4
3
3
1
2
4
6
5
5
3
3
2
54
Soln.28
3
4
1
5
2
6
1
2
5
2
1
2
4
6
6
5
2
3
4
6
4
4
1
2
5
3
1
6
5
4
1
5
3
3
6
3
53
Soln.29
6
4
6
5
6
6
1
2
6
1
5
4
3
3
3
5
5
1
4
2
4
3
5
5
3
2
3
1
4
6
2
2
1
4
2
1
54
Soln.30
1
6
5
4
3
1
6
6
3
4
1
5
5
6
2
3
2
4
4
4
1
6
6
3
4
2
5
5
3
2
2
3
1
2
5
1
53
113
Soln.31
1
2
6
5
4
3
1
6
2
4
6
6
5
5
6
1
1
6
1
3
4
3
2
3
4
4
4
5
1
2
5
2
2
5
3
3
53
Soln.32
5
6
4
2
2
1
6
4
5
2
1
6
3
5
1
4
4
3
5
3
5
6
1
4
2
3
6
3
2
1
5
1
6
3
2
4
54
Soln.33
2
5
4
6
3
5
1
2
2
4
6
1
5
3
1
2
1
6
3
4
4
4
6
2
5
6
5
1
6
1
4
3
3
5
3
2
53
Soln.34
2
6
6
4
3
1
5
2
6
6
5
1
4
5
1
1
4
6
6
3
2
5
5
1
2
4
4
3
2
2
3
1
5
4
3
3
54
Soln.35
6
6
2
4
6
4
2
5
6
1
3
3
4
4
6
5
5
4
1
2
2
6
3
1
5
3
4
2
1
1
3
2
5
5
3
1
51
Soln.36
4
6
3
5
4
3
6
2
4
5
4
3
5
6
1
5
4
2
5
1
1
6
1
3
3
2
3
4
1
6
2
2
6
1
5
2
54
Soln.37
4
4
6
6
1
6
2
3
2
4
5
6
5
4
2
3
1
5
6
4
1
4
2
5
1
3
6
3
5
1
3
5
1
3
2
2
53
Soln.38
6
3
4
6
6
3
2
6
6
1
5
1
4
1
3
5
4
2
5
4
1
6
3
4
1
4
3
2
5
5
3
2
5
2
1
2
52
Soln.39
6
4
3
1
5
6
6
4
4
2
4
1
4
5
2
1
6
6
2
3
6
5
2
5
4
1
5
1
1
2
2
5
3
3
3
3
54
Soln.40
6
2
5
6
1
6
4
4
6
3
2
4
5
1
1
3
4
6
5
2
3
3
2
4
5
6
4
1
3
1
5
2
3
5
1
2
52
Soln.41
5
5
6
1
2
2
5
2
3
2
4
6
4
6
3
5
5
6
4
4
1
1
3
6
1
1
5
2
4
3
3
2
6
1
3
4
54
Soln.42
5
2
4
2
1
5
2
6
2
1
4
6
6
5
3
4
5
1
5
4
6
1
2
4
2
1
6
4
6
5
1
3
3
3
3
3
53
Soln.43
6
1
6
3
6
5
5
2
2
6
4
4
4
6
5
5
2
1
1
1
3
4
5
4
3
1
3
5
2
1
4
2
3
6
3
2
53
Soln.44
6
3
1
6
4
6
6
1
4
4
6
1
2
4
5
3
1
6
1
1
5
2
5
3
4
3
5
5
4
5
3
2
2
3
2
2
54
Soln.45
6
4
2
3
6
1
5
4
4
4
3
6
2
1
6
5
5
1
6
3
1
3
5
2
4
1
4
6
3
3
2
5
5
1
2
2
54
114
Soln.46
5
6
2
2
1
5
3
6
4
5
6
5
1
4
2
4
3
2
2
6
5
4
3
3
5
6
3
2
1
4
1
6
1
3
1
4
54
Soln.47
6
2
6
3
4
6
6
4
5
2
1
1
5
6
4
1
5
2
1
3
4
1
5
4
3
5
6
4
5
3
1
2
2
2
3
3
52
Soln.48
6
6
3
3
5
4
6
3
6
5
1
4
2
4
1
5
1
4
2
3
5
1
2
1
6
4
3
5
5
2
6
2
3
1
2
4
54
Soln.49
6
2
5
6
6
6
3
4
1
3
5
4
6
1
4
4
5
2
1
1
6
3
4
3
1
4
5
5
3
2
5
3
2
2
2
1
53
Soln.50
6
6
6
1
4
4
2
3
6
3
6
5
1
5
5
2
1
1
4
3
3
5
1
3
4
5
3
4
2
2
1
4
2
6
5
2
53
Soln.51
5
6
2
4
2
3
5
2
1
5
6
6
1
4
1
2
3
5
1
6
5
1
4
4
2
5
3
6
3
4
6
2
3
4
3
1
54
Soln.52
3
6
1
6
6
5
5
4
1
3
6
6
4
2
5
1
5
1
3
4
1
4
3
3
1
5
4
2
3
4
5
6
2
2
2
2
53
Soln.53
4
2
5
5
1
2
6
3
6
2
2
1
4
5
6
2
4
3
1
5
1
3
4
6
1
5
4
6
4
3
6
5
1
3
2
3
54
Soln.54
6
6
2
3
4
4
5
1
6
4
3
1
2
4
1
6
4
5
1
6
1
6
2
5
3
2
5
3
3
4
2
3
2
5
1
5
53
Soln.55
6
6
6
4
1
5
3
4
2
1
1
6
5
2
6
4
2
5
1
4
1
6
2
5
3
3
2
4
4
1
5
2
5
3
3
3
53
Soln.56
3
4
6
4
6
4
6
2
4
1
4
6
5
1
1
6
5
6
2
3
2
5
3
2
5
3
5
1
3
4
2
2
1
3
5
1
51
Soln.57
2
1
5
4
3
2
6
4
5
6
5
2
1
3
6
6
1
4
2
2
4
5
3
2
4
1
5
1
6
1
6
3
5
4
3
3
53
Soln.58
5
4
6
1
5
2
2
6
4
2
2
1
3
5
6
4
1
4
6
1
3
2
1
5
4
6
4
5
1
2
3
3
5
6
3
3
52
Soln.59
6
2
6
4
5
1
4
3
2
6
4
4
1
1
5
6
3
4
5
2
3
6
2
1
5
5
6
2
3
2
1
3
4
1
3
5
54
Soln.60
6
3
4
6
1
3
5
1
6
5
3
2
4
4
1
6
3
6
5
1
5
6
2
1
4
5
3
2
1
2
2
3
5
4
4
2
53
115
Soln.61
1
6
3
6
4
3
5
6
5
1
3
4
5
4
4
6
3
2
2
2
6
4
5
1
3
2
1
1
4
6
5
2
3
5
1
2
53
Soln.62
6
6
4
5
1
4
2
1
2
6
3
5
1
5
1
6
6
3
4
4
1
3
5
4
2
3
5
4
6
5
1
2
3
2
2
3
52
Soln.63
3
6
6
5
1
6
2
4
6
5
2
1
2
4
4
6
4
3
3
3
1
5
1
6
1
4
2
4
5
5
3
2
5
1
2
3
53
Soln.64
3
6
4
6
2
4
5
4
6
3
1
4
6
5
2
4
3
1
6
4
5
1
5
3
3
1
1
2
2
1
5
5
6
2
3
2
54
Soln.65
6
4
6
1
6
3
2
5
4
4
4
6
3
6
3
4
4
1
5
3
1
2
6
5
3
2
5
1
2
2
1
5
1
5
2
3
52
Soln.66
6
6
2
1
3
4
4
5
6
4
2
4
6
5
1
1
5
1
6
2
5
1
3
4
5
6
2
4
3
1
5
2
2
3
3
3
54
Soln.67
4
4
4
1
2
6
6
6
4
3
2
5
5
6
3
6
1
4
1
2
1
5
1
3
6
2
2
3
5
2
3
1
5
4
3
5
54
Soln.68
6
4
2
6
6
5
1
3
4
5
4
1
1
3
6
6
2
2
3
6
1
4
4
1
5
2
5
4
5
1
3
5
2
3
3
2
54
Soln.69
2
5
4
1
2
6
4
5
6
2
1
6
3
5
4
2
3
1
2
6
4
4
1
5
3
3
6
5
1
1
4
5
6
3
2
3
53
Soln.70
2
6
4
6
3
5
4
6
6
3
1
5
5
1
4
1
6
5
5
2
1
3
3
2
4
5
3
1
2
3
4
4
6
2
1
2
53
Soln.71
6
1
6
4
2
3
5
1
4
6
3
6
4
4
6
3
5
2
5
4
4
3
1
3
2
6
5
2
5
1
1
2
3
5
1
2
51
Soln.72
3
6
6
4
6
4
5
1
2
5
1
2
3
6
1
4
4
6
5
2
4
6
3
2
2
1
3
5
4
2
5
3
1
3
5
1
54
Soln.73
6
6
4
5
2
1
3
1
3
3
4
6
5
4
6
6
5
4
3
2
6
1
1
2
3
5
1
3
4
4
2
5
2
1
2
5
54
Soln.74
2
6
6
1
6
4
5
3
1
5
6
6
2
4
4
5
4
5
2
1
1
4
1
5
3
4
2
2
3
1
2
5
6
3
3
3
53
Soln.75
4
4
3
2
5
1
3
5
6
1
3
4
6
4
6
5
4
2
6
3
5
1
4
5
6
2
3
5
2
6
1
3
1
1
2
2
54
116
Soln.76
6
4
6
1
5
2
2
6
5
4
1
6
3
1
4
6
5
4
2
5
4
5
2
1
1
4
3
6
3
2
2
3
1
3
5
3
52
Soln.77
1
6
6
4
5
4
2
3
3
6
5
5
1
3
4
4
1
2
5
6
6
3
2
4
5
1
3
4
1
6
2
2
5
3
1
2
53
Soln.78
6
2
1
6
4
4
6
4
2
3
5
3
4
6
5
1
2
3
6
2
5
2
4
6
4
1
5
1
3
5
1
3
5
2
3
1
51
Soln.79
6
4
3
6
6
1
5
5
3
4
3
4
4
3
5
1
5
1
6
2
5
6
4
2
1
3
6
5
2
2
1
4
3
1
2
2
53
Soln.80
5
5
2
4
3
2
2
6
4
4
4
6
1
6
4
5
6
3
5
1
4
5
2
3
1
1
3
1
2
5
6
3
1
6
2
3
54
Soln.81
4
6
5
3
1
4
3
3
4
6
5
1
2
4
5
6
5
3
6
1
6
3
2
5
5
3
4
2
1
6
2
1
4
1
2
2
54
Soln.82
6
6
4
1
4
2
6
5
6
3
5
4
6
1
2
1
4
3
5
3
4
2
2
5
1
1
1
5
2
2
4
3
3
6
5
3
54
Soln.83
6
1
4
6
5
3
6
4
2
4
1
4
3
4
1
2
5
3
6
1
3
6
6
2
4
3
5
1
5
3
2
2
1
5
5
2
53
Soln.84
2
4
4
6
4
1
5
3
3
6
5
4
5
6
5
1
6
4
4
3
1
3
5
2
6
6
1
3
1
5
3
2
2
1
2
2
54
Soln.85
1
6
3
4
6
6
2
4
3
5
4
1
1
3
6
4
5
6
4
2
5
6
3
3
1
2
5
1
2
4
5
5
2
2
3
1
52
Soln.86
6
4
6
1
4
3
2
6
5
1
2
5
6
6
4
1
2
4
3
2
3
3
5
4
1
2
2
4
1
5
3
5
6
3
5
1
52
Soln.87
2
4
5
6
4
1
3
5
3
3
4
1
6
5
6
1
2
4
5
6
4
6
3
3
1
2
1
4
2
5
1
6
5
3
2
2
54
Soln.88
2
1
6
6
2
5
3
1
1
6
5
6
4
6
5
2
5
1
4
4
2
3
3
5
4
2
1
5
1
4
4
3
3
2
3
6
53
Soln.89
6
6
6
3
1
4
6
5
5
6
4
3
4
5
2
1
5
3
1
3
4
6
1
4
2
5
1
3
1
2
4
2
3
2
2
5
52
Soln.90
3
1
2
2
5
5
6
2
6
6
5
4
4
2
6
5
1
2
3
1
1
3
5
6
5
4
4
4
3
1
4
2
6
3
1
3
54
117
Soln.91
6
6
3
4
5
1
6
1
5
2
4
4
3
4
3
5
5
3
6
2
4
1
6
5
3
4
2
6
1
2
2
3
1
5
1
2
53
Soln.92
3
6
6
5
6
6
4
2
3
1
1
4
6
5
5
5
2
1
4
3
6
3
4
3
2
3
1
1
4
5
2
4
2
1
5
2
53
Soln.93
6
1
6
5
6
2
4
3
6
6
2
4
3
1
5
1
2
2
5
1
4
1
4
3
6
3
5
4
3
2
5
3
1
4
5
2
52
Soln.94
6
4
2
6
6
4
1
3
6
3
5
4
4
3
2
4
6
1
5
3
1
3
5
1
4
2
5
2
1
3
6
2
5
1
5
2
51
Soln.95
2
4
6
4
1
3
6
4
6
4
2
3
6
6
5
1
1
4
4
6
5
5
3
2
1
2
1
5
3
2
3
1
5
3
2
5
51
Soln.96
6
1
3
6
6
5
4
6
5
3
4
4
2
1
3
5
1
6
5
3
1
1
4
3
2
1
6
3
4
2
2
5
4
5
2
2
52
Soln.97
1
5
3
2
2
4
4
5
6
2
4
1
1
4
2
6
6
5
5
3
3
1
2
4
6
6
1
3
1
4
5
2
6
5
3
3
53
Soln.98
4
4
4
3
2
1
6
3
5
6
4
5
3
1
6
5
2
1
6
3
2
5
4
3
6
2
5
1
1
3
1
2
6
4
5
2
54
Soln.99
6
6
6
1
4
5
2
6
3
1
5
4
2
4
1
3
2
6
5
4
3
6
3
1
1
2
5
3
1
3
4
4
5
5
2
2
52
Soln.100
5
5
1
6
4
1
2
2
4
3
2
6
2
1
6
5
1
4
4
5
4
6
2
6
1
1
4
2
5
6
3
3
3
3
5
3
53
118
PROBLEM 6
Soln.1
1
2
3
5
1
4
6
3
5
5
2
1
4
2
6
4
5
1
6
2
1
6
4
3
5
3
4
3
1
5
2
2
6
4
3
6
57
Soln.2
1
6
5
1
6
2
4
5
3
5
1
1
4
2
5
3
4
3
2
4
1
3
4
6
5
6
4
3
2
3
1
5
2
6
2
6
57
Soln.3
4
6
1
1
2
5
4
3
6
5
5
2
3
1
6
2
6
1
3
3
5
1
5
2
4
4
4
6
2
4
1
3
3
5
6
2
57
Soln.4
1
1
2
5
6
5
1
3
6
2
4
5
3
4
3
6
2
3
1
4
5
2
1
2
6
3
4
2
4
1
5
3
6
5
6
4
57
Soln.5
6
2
4
1
5
5
1
3
3
6
1
6
4
4
2
4
1
3
5
4
5
1
2
3
3
2
6
4
5
6
3
2
1
6
2
5
57
Soln.6
4
2
1
6
5
5
1
1
4
4
3
5
1
6
6
3
2
3
1
3
2
5
4
6
2
5
4
3
3
2
1
2
5
6
6
4
57
Soln.7
1
4
5
2
6
1
5
4
2
1
3
4
4
6
3
3
2
6
1
2
1
4
6
5
5
2
5
3
3
2
4
5
1
3
6
6
57
Soln.8
1
3
2
6
3
1
2
4
4
5
4
3
5
4
1
3
1
6
1
6
2
5
2
4
6
1
6
5
6
4
5
3
3
2
2
5
57
Soln.9
2
1
2
3
4
6
1
6
5
3
2
5
1
5
6
4
3
5
3
4
2
6
1
1
4
2
5
6
3
4
1
5
2
3
6
4
57
Soln.10
2
6
1
5
4
3
4
5
1
1
6
2
3
3
5
1
6
4
2
6
3
1
5
4
6
4
3
5
1
5
2
6
2
4
3
2
57
Soln.11
4
1
5
1
2
6
2
1
3
6
4
5
3
4
6
4
1
3
3
2
5
5
6
4
6
1
4
2
5
3
1
5
2
3
2
6
57
Soln.12
2
1
4
6
1
4
2
5
4
1
6
3
4
3
5
2
3
5
3
4
6
5
5
1
2
1
3
2
5
3
1
2
6
6
4
6
57
Soln.13
1
2
2
3
5
6
1
5
6
3
2
5
3
1
4
6
3
2
4
6
1
1
4
4
4
5
4
6
5
1
3
3
6
5
2
2
57
Soln.14
4
6
2
1
1
4
3
3
4
1
4
6
2
3
3
5
5
2
1
5
4
6
1
5
3
1
2
5
6
5
4
3
6
6
2
2
57
Soln.15
4
1
5
3
6
2
1
4
6
5
3
5
2
1
6
1
4
3
2
3
6
4
1
6
4
1
3
2
5
2
2
6
5
4
5
3
57
119
Soln.16
1
1
4
5
2
3
6
1
5
5
4
6
3
3
2
4
1
6
5
3
2
6
4
4
1
5
3
2
1
4
3
2
2
5
6
6
57
Soln.17
4
3
1
1
6
4
5
3
4
2
2
1
5
6
5
1
2
3
3
5
6
2
1
5
6
3
4
4
3
6
1
6
4
5
2
2
57
Soln.18
1
1
4
3
6
4
3
4
5
2
3
1
4
2
5
4
1
6
2
6
5
5
2
3
2
2
1
5
6
3
4
6
5
3
1
6
57
Soln.19
6
4
1
4
2
3
2
4
4
1
3
3
1
5
3
5
1
6
4
1
5
6
2
1
6
6
2
6
5
2
5
3
4
2
3
5
57
Soln.20
4
4
1
6
3
1
4
4
3
2
6
5
1
2
6
1
2
5
4
6
5
3
5
3
1
4
6
2
2
5
3
1
3
6
2
5
57
Soln.21
1
3
4
5
1
4
4
1
2
4
6
6
2
1
3
3
5
3
5
2
6
6
2
6
1
3
4
5
5
3
5
1
4
2
2
6
57
Soln.22
3
1
5
1
3
6
4
4
2
5
2
4
6
3
2
4
1
3
1
1
6
4
1
5
2
6
5
2
3
5
4
6
6
3
5
2
57
Soln.23
2
3
1
6
4
1
5
5
2
1
5
3
4
6
3
2
1
6
5
3
5
1
2
4
4
3
6
3
2
6
4
4
2
1
6
5
57
Soln.24
1
4
6
1
5
2
5
3
4
6
4
1
3
5
1
4
3
6
2
1
2
3
4
6
2
4
5
1
6
6
5
3
2
2
3
5
57
Soln.25
3
1
4
2
4
1
6
4
6
1
3
2
5
6
4
4
1
3
1
2
3
5
6
4
5
2
3
1
5
5
3
6
6
2
2
5
57
Soln.26
4
1
1
2
6
5
4
5
3
2
3
1
6
6
2
4
3
4
1
1
5
3
4
6
1
6
4
2
5
2
5
3
5
6
2
3
57
Soln.27
1
1
4
1
6
2
5
2
3
5
6
4
3
2
4
3
4
6
1
2
5
1
6
3
4
5
3
5
6
3
4
2
2
1
6
5
57
Soln.28
4
1
1
2
5
6
6
3
1
6
5
2
3
2
4
5
3
2
6
4
1
5
2
4
1
2
3
4
1
6
5
3
4
3
6
5
57
Soln.29
4
1
6
1
5
2
5
1
4
3
4
2
6
3
2
4
1
3
5
6
2
3
6
5
1
4
5
3
3
2
1
4
5
2
6
6
57
Soln.30
4
3
2
1
1
5
2
6
1
5
3
5
2
6
1
3
4
5
6
5
4
3
4
1
2
6
3
3
2
4
4
6
5
1
6
2
57
120
Soln.31
1
3
3
4
5
6
2
4
1
5
1
4
4
6
3
1
2
3
1
2
4
5
6
2
3
5
5
6
2
6
1
3
4
2
5
6
57
Soln.32
1
6
4
3
5
1
4
4
1
6
3
3
2
4
6
1
5
5
2
1
2
5
3
2
6
4
6
4
3
2
2
5
3
5
1
6
57
Soln.33
4
4
4
1
3
5
1
2
6
5
2
1
5
3
6
2
1
6
4
6
3
2
5
5
3
1
2
6
3
2
3
4
1
4
5
6
57
Soln.34
6
1
4
1
2
3
5
1
4
6
3
3
5
5
1
6
2
2
6
1
4
4
6
5
3
2
5
4
3
1
2
2
6
3
4
5
57
Soln.35
4
1
2
4
6
5
4
3
3
4
5
1
1
5
6
6
2
1
2
3
2
6
4
3
5
1
5
2
6
3
1
6
2
4
5
3
57
Soln.36
4
2
1
1
4
5
3
4
5
4
3
5
5
4
2
3
6
1
1
3
5
6
2
6
2
1
2
3
3
6
1
4
6
2
6
5
57
Soln.37
2
1
6
3
1
5
5
6
3
4
3
4
5
2
1
1
2
5
6
5
6
3
4
4
1
3
4
2
2
3
1
5
6
2
4
6
57
Soln.38
4
6
1
1
5
6
2
5
1
2
3
5
3
4
6
3
2
5
6
4
1
2
3
4
4
1
4
3
5
1
6
6
2
5
2
3
57
Soln.39
1
4
1
6
2
4
3
5
3
4
5
4
2
1
6
3
3
2
4
6
5
2
6
1
4
1
3
5
2
2
1
5
3
5
6
6
57
Soln.40
1
4
1
3
6
2
5
5
5
1
4
4
1
3
6
6
2
2
1
6
4
3
2
3
5
3
6
2
5
4
1
6
5
3
2
4
57
Soln.41
2
1
1
3
6
6
5
3
2
4
4
4
5
1
5
6
3
5
1
1
2
3
3
5
2
2
4
6
6
5
3
4
1
2
4
6
57
Soln.42
2
4
3
6
1
1
5
5
3
1
5
6
1
6
4
2
5
4
2
3
1
3
5
3
4
2
4
2
3
1
2
6
6
5
4
6
57
Soln.43
6
3
1
1
4
5
4
6
4
2
5
4
1
2
1
6
5
3
2
1
3
3
2
5
3
4
1
6
5
6
2
4
6
3
5
2
57
Soln.44
4
4
6
2
1
4
3
5
5
1
5
3
5
1
5
3
2
6
6
1
3
1
2
4
2
6
6
4
3
1
6
2
4
5
2
3
57
Soln.45
1
4
1
4
6
1
2
3
4
6
5
2
5
6
4
5
4
3
3
5
1
2
1
3
2
5
2
6
6
4
3
5
3
2
1
6
57
121
Soln.46
4
1
6
3
5
1
6
1
2
4
4
5
3
2
4
1
6
2
3
1
5
6
3
6
2
5
3
4
1
2
5
6
5
3
4
2
57
Soln.47
6
1
3
1
4
1
2
2
5
5
3
6
6
4
5
3
1
4
1
4
5
2
3
3
6
4
6
2
4
5
1
6
2
3
2
5
57
Soln.48
1
1
2
6
5
4
3
4
5
5
6
2
1
4
3
1
1
3
2
6
3
4
5
3
6
4
2
2
6
4
2
1
5
6
3
5
57
Soln.49
1
4
4
2
6
3
1
3
5
4
1
6
4
6
2
4
5
3
2
1
5
6
2
5
5
3
2
1
1
6
2
3
6
5
3
4
57
Soln.50
2
4
1
3
4
1
6
6
5
5
2
1
3
5
5
6
4
5
1
4
1
3
3
2
2
4
6
6
3
2
1
4
2
5
3
6
57
Soln.51
3
2
4
1
1
4
6
5
3
5
2
5
1
6
1
3
5
4
2
1
3
6
6
4
4
6
2
5
2
1
4
2
3
5
6
3
57
Soln.52
1
6
3
2
4
5
4
1
1
3
2
5
6
3
5
1
4
4
1
2
3
5
6
4
6
2
1
2
2
6
4
5
6
3
5
3
57
Soln.53
6
2
1
3
1
4
5
1
5
6
5
3
4
6
4
3
2
2
3
5
1
1
2
3
6
4
2
5
4
6
3
1
6
5
4
2
57
Soln.54
1
4
2
5
6
4
5
4
3
1
1
4
5
3
6
3
4
1
2
6
2
6
1
3
3
1
2
6
5
2
2
4
6
5
5
3
57
Soln.55
1
1
2
6
5
4
5
2
3
4
5
1
6
3
3
5
6
4
2
1
1
5
3
2
4
4
3
6
4
1
3
6
2
5
6
2
57
Soln.56
2
1
6
4
3
1
5
3
5
6
1
5
1
4
2
3
6
3
2
4
1
5
2
5
6
4
3
3
4
6
1
5
4
6
2
2
57
Soln.57
1
4
5
2
1
4
6
1
4
5
3
2
3
6
3
4
1
1
4
2
6
6
3
5
1
6
5
2
4
3
5
5
2
6
3
2
57
Soln.58
2
1
1
6
5
1
5
3
4
5
6
1
2
4
3
3
6
2
3
1
5
5
4
4
4
2
6
3
6
2
1
4
5
2
6
3
57
Soln.59
6
4
1
2
4
3
4
1
6
4
5
3
5
3
2
1
3
2
1
6
1
5
4
3
1
6
2
5
4
5
6
6
5
2
3
2
57
Soln.60
1
2
6
5
4
4
3
1
4
5
4
6
2
1
3
3
5
4
5
2
6
6
1
5
2
1
3
3
4
6
3
1
6
2
2
5
57
122
Soln.61
4
1
1
2
6
3
5
4
4
2
1
3
6
6
5
2
2
3
5
1
4
3
6
4
1
6
1
4
5
2
2
3
5
3
5
6
57
Soln.62
4
2
1
1
6
5
5
6
4
3
5
5
3
4
1
1
2
3
3
6
4
2
6
2
2
4
1
3
6
5
1
3
5
4
2
6
57
Soln.63
1
1
4
4
3
3
2
6
5
4
1
4
1
4
6
5
3
2
5
6
1
2
6
5
2
3
4
2
2
5
6
1
6
3
5
3
57
Soln.64
4
2
1
4
5
1
1
3
2
4
5
3
4
6
5
4
1
6
3
1
5
6
2
6
3
1
5
3
6
2
6
3
4
2
5
2
57
Soln.65
1
4
6
5
1
4
2
3
5
4
2
3
4
2
6
3
1
1
5
5
4
6
3
2
5
1
6
3
3
4
6
2
1
5
2
6
57
Soln.66
4
2
1
3
6
5
3
1
5
2
1
5
4
6
5
4
1
3
3
2
4
6
1
3
4
1
6
2
2
5
5
4
6
3
2
6
57
Soln.67
2
6
3
1
1
5
3
1
6
4
5
3
2
5
3
4
4
1
1
6
5
5
2
4
6
4
4
1
2
3
6
5
3
6
2
2
57
Soln.68
1
1
3
6
6
3
4
4
4
5
2
1
4
3
3
2
6
1
2
1
2
5
5
5
5
6
4
1
3
2
2
4
5
6
6
3
57
Soln.69
4
4
1
2
5
1
5
3
1
3
2
5
1
4
4
6
6
3
2
6
1
3
5
3
6
4
4
6
5
2
2
1
2
3
5
6
57
Soln.70
1
4
6
1
2
6
3
5
4
1
5
3
2
4
2
3
4
6
1
1
5
5
2
3
2
3
5
6
6
4
3
1
6
2
4
5
57
Soln.71
3
1
1
2
4
1
6
5
6
3
5
2
5
6
5
5
3
1
4
1
4
2
3
4
4
6
1
2
4
6
3
5
2
3
6
2
57
Soln.72
2
4
4
6
2
1
4
3
1
4
3
6
1
4
3
5
1
6
5
2
3
6
2
5
5
1
1
3
2
5
2
6
3
6
4
5
57
Soln.73
2
1
5
4
1
2
6
1
4
5
3
6
3
5
1
6
3
3
5
4
2
2
1
4
3
6
6
5
4
5
3
4
1
6
2
2
57
Soln.74
1
4
1
3
5
6
3
1
2
6
4
5
2
5
1
2
6
3
3
5
4
6
2
2
4
5
4
2
1
3
1
3
6
6
5
4
57
Soln.75
6
1
4
4
1
6
2
4
3
5
4
5
3
2
4
1
1
2
3
6
2
6
1
5
5
2
6
3
3
4
1
5
5
6
3
2
57
123
Soln.76
1
1
6
4
1
4
5
2
6
4
2
3
4
5
3
3
1
5
4
5
6
2
2
1
6
2
3
6
3
5
1
2
5
3
6
4
57
Soln.77
3
4
6
1
1
5
2
3
4
5
4
1
2
5
5
3
5
6
2
6
2
1
3
1
4
3
2
6
6
4
4
2
1
3
6
5
57
Soln.78
4
4
4
1
3
2
1
1
5
6
5
4
3
6
5
1
2
2
3
1
6
6
5
4
3
3
6
1
2
5
3
2
4
2
6
5
57
Soln.79
4
1
1
6
4
3
3
4
6
1
2
5
4
1
5
6
2
3
4
2
5
5
2
2
3
6
1
6
2
1
3
5
5
4
3
6
57
Soln.80
3
6
4
1
1
4
3
6
5
4
1
4
2
2
1
3
6
3
5
4
5
2
6
4
1
5
2
3
5
2
2
3
1
6
6
5
57
Soln.81
6
3
1
2
1
2
4
5
5
1
5
4
3
5
1
5
1
2
3
6
3
4
4
6
6
3
4
2
1
5
4
6
6
2
3
2
57
Soln.82
3
4
1
4
2
6
5
5
3
1
2
4
1
6
6
5
2
5
5
1
3
3
1
3
6
4
2
2
2
5
4
3
6
1
4
6
57
Soln.83
1
2
3
1
6
5
4
2
6
1
5
5
3
6
3
2
5
4
1
3
4
2
6
1
2
4
5
6
3
4
4
1
6
5
2
3
57
Soln.84
4
3
4
1
4
4
3
1
3
6
2
5
2
1
3
1
5
6
6
2
6
4
5
3
1
2
6
1
6
4
5
5
3
2
5
2
57
Soln.85
1
5
2
1
2
6
4
5
4
6
3
1
5
4
1
3
4
6
3
5
5
3
2
1
4
6
6
2
3
5
3
2
4
1
2
6
57
Soln.86
1
4
4
5
2
4
1
3
6
4
6
1
5
3
5
2
6
1
2
6
3
6
3
1
5
4
3
2
5
1
6
2
5
4
2
3
57
Soln.87
2
3
1
5
1
4
4
1
3
6
4
4
2
6
4
1
2
6
5
3
2
1
6
3
5
3
1
5
6
2
5
3
6
4
5
2
57
Soln.88
6
4
3
1
2
5
4
1
5
2
5
3
4
1
6
2
3
6
6
1
5
2
5
3
4
1
3
2
4
5
6
1
6
4
3
2
57
Soln.89
4
4
1
4
3
2
4
6
2
1
4
5
6
3
5
3
1
1
6
2
3
1
5
5
3
6
1
2
6
5
4
2
2
3
6
5
57
Soln.90
4
1
5
4
6
1
6
3
4
3
2
5
1
2
5
3
1
3
6
2
6
1
2
5
4
3
4
5
6
1
3
2
5
2
6
4
57
124
Soln.91
3
1
4
3
1
2
6
4
5
6
2
5
4
1
4
3
1
6
5
5
1
2
2
3
5
4
3
5
6
1
4
3
2
2
6
6
57
Soln.92
3
2
4
4
1
1
6
1
4
2
3
5
4
1
6
5
6
2
5
1
3
6
6
2
5
4
4
5
2
3
3
3
1
5
2
6
57
Soln.93
1
4
5
3
1
1
4
2
6
2
5
4
4
6
6
3
5
4
2
1
6
3
1
5
3
6
2
4
5
2
3
1
3
2
5
6
57
Soln.94
4
3
1
5
6
2
5
4
1
3
1
6
2
4
5
3
1
6
4
2
3
5
2
1
4
1
2
3
5
5
2
3
6
6
6
4
57
Soln.95
2
3
1
1
5
6
4
2
4
1
3
4
5
6
3
5
3
2
1
1
4
4
6
6
5
2
2
3
4
1
5
2
6
6
3
5
57
Soln.96
4
6
3
1
3
1
2
2
4
5
2
4
4
6
1
3
5
1
6
6
3
2
5
4
1
5
5
2
4
6
1
3
2
6
3
5
57
Soln.97
4
2
1
4
1
5
5
1
6
3
5
4
3
1
2
6
3
2
6
1
5
3
4
5
2
2
3
3
6
4
4
5
2
1
6
6
57
Soln.98
6
1
3
5
2
4
4
3
5
4
1
1
4
6
2
6
5
5
5
2
1
3
2
3
1
4
2
3
6
5
1
3
6
4
2
6
57
Soln.99
1
1
2
5
4
2
5
6
3
3
1
6
4
5
6
1
3
2
4
3
4
6
5
2
4
5
1
2
3
4
6
3
1
2
6
5
57
Soln.100
3
1
2
2
5
6
5
1
6
5
3
5
4
3
1
6
3
2
1
5
4
6
1
4
4
4
2
3
4
1
3
5
2
6
2
6
57
125
PROBLEM 7
Soln.1
4
6
4
2
1
2
1
5
1
5
3
3
4
1
5
3
2
6
6
4
5
1
2
3
3
2
6
4
5
1
5
4
6
3
6
2
49
Soln.2
6
4
2
2
1
1
3
5
3
6
1
6
1
4
3
2
2
5
1
4
3
5
5
1
6
5
2
4
6
4
3
4
3
6
2
5
49
Soln.3
1
4
3
2
5
5
6
4
1
3
4
1
1
2
5
6
3
3
6
2
2
6
4
6
1
4
5
4
5
1
6
3
5
2
3
2
49
Soln.4
1
5
6
4
6
2
6
4
1
3
3
1
5
4
3
6
3
5
1
2
5
1
5
4
2
2
4
6
3
2
3
5
1
6
4
2
49
Soln.5
2
4
3
1
6
5
1
5
3
4
1
2
6
3
4
1
5
5
2
3
1
6
1
4
2
5
6
3
3
6
4
4
2
5
6
2
49
Soln.6
1
4
3
1
1
5
6
6
3
2
2
2
4
3
4
1
6
5
5
1
5
3
6
4
1
4
2
5
2
3
6
5
3
6
2
4
49
Soln.7
1
3
5
1
4
6
1
4
3
4
6
2
5
3
2
6
1
1
2
3
5
4
5
3
4
6
6
2
1
2
2
5
4
6
5
3
49
Soln.8
6
3
1
4
5
1
2
4
1
6
1
4
3
2
6
5
2
5
3
3
2
6
4
1
4
4
3
5
1
2
6
5
5
2
6
3
49
Soln.9
2
1
5
4
6
3
4
1
1
3
2
4
6
5
3
2
1
5
6
3
2
6
4
5
1
5
2
4
4
3
3
6
1
5
2
6
49
Soln.10
3
1
4
4
2
1
4
5
1
5
5
3
2
3
6
3
6
1
2
3
6
6
1
5
4
5
2
4
6
2
1
2
5
4
3
6
49
Soln.11
2
6
6
1
5
4
3
4
5
6
1
3
1
4
3
5
2
1
2
2
6
5
1
3
4
3
6
4
1
5
2
3
4
5
2
6
49
Soln.12
3
1
1
6
5
4
3
2
2
1
1
2
6
3
5
6
4
5
4
3
4
1
6
1
3
6
2
5
5
2
5
2
4
4
3
6
49
Soln.13
4
2
1
3
6
6
4
1
5
4
3
1
2
3
5
6
2
5
3
1
6
4
1
2
3
4
4
5
5
1
2
6
5
6
2
3
49
Soln.14
1
3
1
6
5
4
1
2
3
6
1
2
2
6
3
4
5
5
4
5
3
2
2
1
4
3
6
5
3
1
6
4
2
5
4
6
49
Soln.15
1
4
6
1
5
6
3
1
2
5
6
3
4
6
5
1
4
3
4
2
5
1
2
6
5
2
3
2
3
4
5
6
1
4
2
3
49
126
Soln.16
1
3
6
4
2
4
5
1
1
4
5
3
6
2
6
3
3
1
1
5
5
1
2
2
2
6
4
5
6
4
3
5
4
3
2
6
49
Soln.17
4
6
1
1
5
3
2
3
5
4
6
2
3
1
3
1
4
1
3
2
6
5
6
4
5
2
6
1
2
4
5
6
2
4
5
3
49
Soln.18
2
3
4
6
5
6
5
6
4
1
3
1
2
1
2
2
5
3
4
5
5
3
1
4
1
3
5
2
3
6
4
6
6
2
4
1
50
Soln.19
2
1
1
4
3
4
2
6
5
1
5
4
5
3
3
3
3
1
6
6
5
2
6
4
4
1
5
1
6
5
6
2
2
4
2
3
49
Soln.20
4
2
1
3
6
5
3
6
1
1
4
2
3
6
4
5
5
1
1
5
3
4
4
2
6
2
2
6
2
6
1
3
5
4
5
3
50
Soln.21
4
1
1
5
1
6
2
6
6
4
3
3
5
5
4
3
5
6
1
2
2
4
5
2
1
3
4
1
3
2
4
6
6
3
5
2
49
Soln.22
6
1
5
4
2
1
6
5
3
3
4
1
1
4
2
2
5
6
3
1
3
6
2
3
4
6
5
5
5
2
4
4
1
3
2
6
49
Soln.23
4
6
1
1
2
5
3
6
3
1
5
6
4
2
1
2
3
5
4
6
1
4
5
3
2
3
6
5
2
1
4
3
4
6
5
2
49
Soln.24
1
6
2
4
3
1
4
5
3
6
4
6
2
3
5
3
1
5
1
2
5
1
2
3
6
4
6
2
2
4
5
4
3
1
5
6
49
Soln.25
6
3
1
1
4
6
1
5
4
1
2
5
2
4
6
3
3
5
2
5
1
3
1
2
2
4
6
3
5
4
4
5
6
2
6
3
49
Soln.26
2
1
1
4
3
4
5
5
3
2
6
2
1
1
4
6
1
6
5
4
2
3
3
6
1
4
5
3
6
5
2
2
6
4
5
3
49
Soln.27
3
1
4
2
6
1
6
4
5
5
1
3
3
2
4
1
3
6
2
1
5
5
2
4
3
6
1
5
3
5
6
2
2
4
6
4
49
Soln.28
6
4
1
1
3
1
5
2
6
6
4
2
1
3
1
3
2
4
5
5
3
2
4
1
6
5
4
4
3
5
6
2
6
5
2
3
49
Soln.29
6
1
1
5
4
3
2
4
3
6
4
5
2
2
3
6
5
1
2
5
3
1
6
1
4
4
3
1
6
4
6
5
2
5
2
3
49
Soln.30
1
5
6
4
1
1
4
3
4
2
5
3
1
5
6
3
4
2
6
1
2
3
3
1
4
2
5
6
6
6
5
2
3
4
2
5
49
127
Soln.31
1
4
1
6
2
4
2
1
3
5
3
6
5
6
2
4
1
3
2
5
3
1
6
5
4
1
6
4
2
4
3
5
6
2
5
3
49
Soln.32
1
5
1
4
1
3
2
1
5
6
6
4
2
5
3
4
6
1
2
2
3
3
4
3
6
1
6
5
5
4
5
4
6
2
3
2
49
Soln.33
1
2
1
4
6
3
5
5
1
6
1
2
5
3
3
4
1
4
3
4
6
5
1
2
3
2
2
4
5
6
5
6
4
6
2
3
49
Soln.34
4
1
4
6
3
1
5
3
2
1
1
2
6
5
4
3
3
6
2
5
4
6
5
1
6
2
3
4
5
1
2
6
2
3
4
5
49
Soln.35
1
4
6
1
4
2
5
4
5
3
5
1
1
2
6
3
2
3
6
1
5
4
6
5
3
6
3
4
2
1
4
3
6
2
2
5
49
Soln.36
2
1
4
3
6
5
1
6
5
1
4
4
6
3
3
5
2
1
4
3
2
4
5
1
1
3
5
6
6
2
4
2
2
5
3
6
49
Soln.37
3
2
1
4
5
1
6
5
5
3
6
4
1
6
1
4
4
2
1
3
3
2
5
3
5
4
6
2
2
3
6
1
5
6
4
2
49
Soln.38
4
1
4
1
6
5
2
3
6
1
3
4
2
2
5
6
5
3
1
5
5
6
1
2
3
3
4
3
4
1
6
4
2
6
2
5
49
Soln.39
2
6
1
1
5
4
3
3
6
4
2
5
1
4
3
3
6
2
6
5
5
1
1
5
2
3
4
3
4
6
1
6
4
5
2
2
49
Soln.40
1
2
5
5
1
4
3
4
5
1
6
6
3
2
6
4
2
1
3
2
5
3
1
5
3
4
4
6
4
1
6
2
2
3
6
5
49
Soln.41
1
1
6
2
3
4
5
6
3
4
4
1
5
3
5
6
1
2
5
3
4
2
6
6
1
4
3
2
5
4
1
6
5
2
3
2
49
Soln.42
1
6
4
4
1
3
6
2
4
6
5
3
5
1
5
3
3
1
5
2
1
2
4
4
3
2
5
6
2
6
1
2
4
5
6
3
49
Soln.43
3
1
5
1
4
3
4
4
2
6
6
2
5
1
1
3
3
5
5
4
1
2
6
2
1
4
6
2
4
5
6
6
3
2
5
3
49
Soln.44
4
3
2
4
1
1
5
5
3
4
2
3
6
2
6
1
1
6
1
3
2
6
5
4
5
4
1
5
3
6
6
3
2
4
2
5
49
Soln.45
1
5
6
3
2
1
3
4
4
1
1
2
1
4
6
3
5
6
2
3
1
5
2
5
5
6
4
3
4
5
3
4
6
2
6
2
49
128
Soln.46
1
1
4
6
4
4
6
2
5
3
1
5
5
3
3
3
1
6
5
2
2
1
4
5
4
6
3
2
2
4
5
1
3
6
2
6
49
Soln.47
1
5
4
4
3
1
1
3
4
2
5
6
3
2
5
6
1
2
6
4
6
3
4
5
1
4
5
3
1
2
6
3
6
5
2
2
49
Soln.48
1
3
5
4
2
6
3
1
6
1
5
1
2
2
5
6
3
4
4
6
5
1
3
2
3
1
4
2
4
4
6
5
6
3
5
2
49
Soln.49
1
4
5
3
2
3
4
6
5
1
5
1
4
6
6
1
3
4
5
2
5
2
3
2
4
6
1
1
2
6
6
3
2
3
4
5
49
Soln.50
4
1
1
6
5
3
1
2
6
3
6
5
2
1
4
2
3
4
3
5
5
2
1
3
4
1
2
4
6
6
5
2
5
3
4
6
49
Soln.51
6
6
2
1
1
5
3
4
6
4
5
5
3
2
1
4
1
3
4
3
2
2
1
5
2
6
6
5
3
4
5
1
6
4
3
2
49
Soln.52
1
5
4
1
2
3
5
5
1
6
3
6
4
1
6
6
4
2
3
2
4
6
5
3
4
5
1
2
3
3
4
2
6
2
1
5
49
Soln.53
2
1
4
1
3
6
5
3
5
1
1
4
2
4
3
2
6
1
5
3
4
5
6
3
1
2
4
6
2
5
3
6
4
6
2
5
49
Soln.54
1
1
3
5
3
4
2
6
6
4
6
5
4
1
5
1
6
3
2
4
5
3
2
3
6
1
5
2
4
2
6
2
3
4
1
5
49
Soln.55
1
3
1
6
4
5
2
1
6
3
6
4
1
5
5
4
2
4
1
3
2
6
6
2
3
4
1
5
3
5
4
3
5
6
2
2
49
Soln.56
6
1
5
3
4
1
2
3
4
1
2
2
5
6
1
1
5
3
3
4
6
4
2
3
5
2
5
6
5
1
6
4
3
4
2
6
49
Soln.57
1
6
1
2
3
4
5
1
2
6
5
3
1
4
5
3
1
3
4
6
2
3
1
2
4
5
6
5
6
4
2
4
2
6
5
3
49
Soln.58
6
4
1
5
3
1
6
2
4
6
2
5
3
2
4
3
5
1
3
1
5
5
4
4
1
6
2
4
3
6
5
2
2
1
3
6
49
Soln.59
4
1
5
4
2
3
5
1
6
1
2
3
1
6
5
2
4
1
6
3
4
6
3
2
3
5
5
1
4
3
2
4
5
6
2
6
49
Soln.60
2
1
4
4
1
6
5
6
3
4
2
5
3
3
1
3
2
1
5
1
4
6
4
5
6
2
6
1
5
3
5
4
6
2
2
3
49
129
Soln.61
4
1
6
1
1
3
2
5
3
2
6
3
5
3
6
4
3
1
2
1
4
2
5
5
4
6
2
4
6
5
4
5
6
3
2
1
50
Soln.62
1
1
4
4
4
5
3
6
3
2
3
6
1
1
6
2
5
5
2
3
5
4
2
2
4
6
6
5
1
3
1
4
2
6
3
5
49
Soln.63
6
1
1
2
3
5
4
4
2
3
1
5
4
6
5
6
1
2
4
2
3
5
6
5
3
6
1
4
3
4
5
1
2
3
2
6
49
Soln.64
1
4
4
1
3
5
2
4
1
3
3
5
5
2
6
6
1
6
1
3
2
5
5
2
4
6
2
3
4
6
5
4
1
2
3
6
49
Soln.65
2
6
1
4
5
3
1
6
5
4
2
1
6
5
1
3
4
3
1
2
5
5
3
4
2
6
3
4
4
6
1
2
3
5
2
6
49
Soln.66
1
4
1
1
3
2
5
6
3
4
5
1
4
3
2
5
2
4
3
1
6
4
6
6
3
4
2
5
6
1
6
5
2
2
5
3
49
Soln.67
1
4
4
6
1
6
2
5
3
1
3
5
3
2
4
2
1
5
3
6
1
2
4
1
6
5
4
3
2
6
5
3
5
2
4
6
49
Soln.68
1
4
6
3
1
6
5
2
5
5
4
1
4
2
3
1
6
2
1
4
3
3
5
4
2
6
6
1
2
3
6
2
5
4
3
5
49
Soln.69
2
4
4
6
1
1
3
5
3
4
3
6
1
2
5
5
3
1
2
5
4
6
3
5
2
1
5
6
4
3
6
2
4
2
1
6
49
Soln.70
4
6
1
3
5
1
4
3
2
1
6
2
5
1
4
3
1
5
4
2
3
6
6
2
5
5
2
6
3
1
4
3
4
2
5
6
49
Soln.71
6
1
4
1
1
3
2
4
5
4
3
5
2
6
5
3
2
6
1
3
5
6
4
2
5
1
3
4
2
3
4
1
6
6
2
5
49
Soln.72
1
4
1
4
6
2
2
3
6
5
1
4
3
5
1
2
1
3
5
3
1
4
6
6
4
5
4
2
6
3
5
2
3
2
5
6
49
Soln.73
6
1
4
3
1
6
5
1
1
2
4
5
2
5
3
4
4
2
6
1
3
3
5
2
5
6
1
4
3
4
6
2
5
2
6
3
49
Soln.74
1
2
6
1
6
3
4
5
3
1
6
4
4
6
2
5
3
5
2
5
1
1
5
4
3
3
2
2
4
6
3
5
2
6
1
4
49
Soln.75
1
2
5
4
3
1
6
5
3
1
5
1
6
4
4
2
2
1
6
3
3
4
3
4
6
6
5
5
5
2
3
2
2
1
4
6
49
130
Soln.76
1
4
5
1
2
1
6
5
6
4
2
3
6
3
5
4
2
3
3
1
5
1
4
3
4
1
6
5
2
4
5
2
3
6
2
6
49
Soln.77
6
4
4
1
5
1
2
3
2
5
6
1
5
4
3
1
6
3
4
2
6
2
1
5
3
5
4
6
1
2
3
4
3
6
5
2
49
Soln.78
4
4
6
1
1
5
2
6
3
3
5
5
1
2
3
6
2
1
3
6
4
4
1
3
4
6
2
1
5
5
2
2
5
4
6
3
50
Soln.79
1
5
4
4
2
6
1
1
3
5
6
2
3
2
3
3
1
6
4
5
4
1
5
2
1
5
4
3
2
6
4
6
6
3
2
5
49
Soln.80
6
3
6
1
5
3
4
4
2
2
2
1
1
5
5
4
1
3
5
1
6
3
5
4
3
2
6
6
4
3
4
2
5
2
1
6
49
Soln.81
4
2
4
1
1
3
5
1
5
6
3
6
3
2
6
4
2
3
2
1
1
6
5
4
1
3
5
6
4
5
6
5
4
2
3
2
49
Soln.82
1
1
6
5
6
4
2
3
6
4
3
5
5
4
2
1
1
4
1
3
6
5
2
3
4
3
5
6
6
5
2
2
3
1
2
4
49
Soln.83
1
4
6
3
6
5
2
3
5
4
2
1
6
5
1
2
6
1
4
1
3
5
3
4
4
6
1
2
3
5
3
6
4
2
5
2
49
Soln.84
4
1
5
2
4
1
5
3
1
2
6
1
4
3
5
6
2
6
6
3
4
1
3
4
5
2
1
2
3
6
4
5
6
2
3
5
49
Soln.85
1
4
3
5
2
3
5
6
1
4
6
4
1
5
3
6
3
2
1
2
1
3
4
6
5
4
4
6
5
2
1
3
6
2
2
5
49
Soln.86
1
1
5
4
3
3
2
1
6
4
1
6
4
6
2
5
5
2
3
4
3
6
5
1
2
2
1
6
3
4
5
4
6
5
2
3
49
Soln.87
1
5
2
1
6
3
4
6
1
5
4
4
6
1
3
2
1
2
3
5
3
6
4
2
5
4
6
2
5
1
2
4
3
5
3
6
49
Soln.88
6
3
6
6
1
2
4
4
5
5
1
3
4
2
1
5
3
3
5
1
4
6
1
6
2
5
2
4
3
6
1
5
2
3
2
4
49
Soln.89
1
6
6
2
5
5
4
4
2
3
1
6
4
3
6
1
2
5
3
1
4
6
3
4
4
5
5
1
2
3
1
2
2
5
6
3
49
Soln.90
4
2
4
3
1
1
1
4
5
5
2
6
3
5
6
1
6
3
5
3
4
2
2
2
1
4
6
3
1
6
5
2
5
4
3
6
49
131
Soln.91
3
2
1
6
5
4
6
6
4
1
2
3
4
3
5
3
1
5
6
2
1
4
4
4
1
6
5
1
3
5
5
2
2
6
2
3
49
Soln.92
1
4
1
1
5
6
3
4
1
2
5
4
3
3
5
6
2
2
5
6
2
6
1
5
3
2
3
4
4
1
5
4
6
3
2
6
49
Soln.93
4
3
6
1
6
1
6
4
5
4
2
1
3
5
5
2
3
3
6
1
3
5
1
5
2
4
2
3
6
5
4
4
2
6
1
2
49
Soln.94
4
6
4
3
1
6
6
5
3
2
5
5
1
2
4
1
6
1
3
5
3
2
1
4
5
2
4
3
5
2
6
1
6
2
3
4
49
Soln.95
1
6
5
1
4
6
3
3
2
2
4
3
2
5
1
6
5
4
2
3
6
5
1
4
4
6
2
2
6
3
1
4
5
3
5
1
50
Soln.96
4
2
4
1
5
5
2
1
3
3
6
1
2
4
1
3
1
6
5
4
6
2
5
2
3
3
6
1
5
5
6
4
4
3
6
2
49
Soln.97
4
1
3
6
6
5
2
3
4
1
2
6
1
4
3
5
5
1
6
4
2
5
3
4
1
5
6
3
2
2
2
5
1
6
3
4
49
Soln.98
2
1
3
1
4
6
4
1
5
2
3
4
5
3
6
2
1
1
6
5
3
4
1
2
3
4
5
6
6
4
2
5
5
2
3
6
49
Soln.99
1
6
4
5
2
1
1
3
4
2
6
1
3
5
4
5
2
1
3
5
3
4
3
6
2
5
5
4
2
6
1
4
6
6
2
3
49
Soln.100
1
2
5
1
4
5
6
4
3
3
5
6
4
3
1
1
2
6
3
2
4
1
4
6
2
5
5
2
1
3
6
3
2
6
5
4
49
132
PROBLEM 8
Soln.1
4
6
5
1
3
4
5
2
4
1
6
3
4
5
3
6
2
2
2
5
1
3
5
6
4
5
1
4
6
3
3
2
6
1
2
1
45
Soln.2
5
3
4
2
4
3
1
6
3
5
4
5
4
1
6
1
2
2
4
3
6
6
5
2
4
5
5
1
6
1
3
2
2
1
3
6
44
Soln.3
5
4
3
4
4
1
6
2
2
3
5
6
5
3
1
2
3
4
6
6
2
3
6
5
3
1
1
5
5
1
4
2
1
2
6
4
45
Soln.4
4
2
4
4
5
3
1
3
5
6
1
5
4
3
6
1
2
3
6
2
5
2
3
4
5
1
2
3
1
4
1
6
2
6
5
6
44
Soln.5
3
5
4
3
1
4
5
1
3
4
2
6
1
6
2
4
6
1
6
2
3
5
4
2
1
2
5
3
6
5
4
5
3
6
1
2
44
Soln.6
4
2
6
4
1
5
3
4
5
1
3
2
4
1
3
6
5
1
6
3
1
2
2
6
4
6
2
4
5
3
5
1
5
3
2
6
44
Soln.7
4
4
3
2
1
3
5
5
6
4
1
2
1
4
5
3
2
1
1
6
4
2
2
4
6
6
1
5
5
5
6
2
3
6
3
3
44
Soln.8
4
4
4
1
2
3
3
2
1
6
3
5
5
2
6
4
6
3
1
1
5
2
6
6
3
5
3
5
4
1
5
1
2
4
2
6
44
Soln.9
4
1
3
3
2
4
4
5
5
2
1
6
3
4
1
1
6
2
1
5
5
6
4
2
6
5
5
4
2
6
2
3
6
1
3
3
44
Soln.10
4
4
3
2
4
1
3
5
5
6
1
3
2
6
1
2
5
4
2
1
6
1
3
5
5
3
4
2
1
6
2
3
6
5
6
4
44
Soln.11
4
4
3
2
3
6
4
1
5
1
2
5
3
2
3
6
1
3
1
4
5
5
2
1
6
6
4
2
1
4
6
3
5
6
2
5
46
Soln.12
3
1
5
4
4
1
3
6
6
3
4
2
2
1
5
2
3
6
6
4
5
4
2
5
1
5
1
4
6
2
1
6
5
3
3
2
45
Soln.13
4
4
5
3
2
1
4
1
5
3
1
3
5
6
2
2
4
5
4
2
1
6
2
6
3
6
5
5
1
3
6
3
2
6
4
1
44
Soln.14
3
5
1
4
3
4
2
4
5
2
3
1
6
5
3
4
1
4
1
2
2
3
5
3
6
6
2
4
1
6
6
1
5
5
6
2
44
Soln.15
4
3
3
6
2
5
1
4
6
5
6
4
3
2
1
2
5
3
1
5
2
4
4
6
4
1
6
2
3
5
2
3
6
1
5
1
45
133
Soln.16
1
4
2
4
4
3
5
1
5
2
3
3
5
2
6
5
6
4
6
3
2
6
6
1
5
4
5
2
2
6
3
4
1
1
3
1
45
Soln.17
5
3
3
4
4
5
2
3
4
5
6
6
1
4
2
1
2
1
4
2
1
5
2
6
1
6
1
3
5
2
6
5
4
3
6
3
44
Soln.18
4
2
4
5
1
3
1
3
6
5
3
1
6
4
2
4
5
2
4
1
6
3
5
2
2
5
2
1
6
6
6
3
1
4
5
3
44
Soln.19
5
4
4
3
4
1
6
2
6
5
4
1
3
5
6
3
2
2
1
2
3
3
1
4
2
1
6
6
5
1
6
4
5
5
2
3
45
Soln.20
4
3
6
4
2
6
3
5
4
2
3
1
1
5
4
5
2
2
1
3
4
5
6
2
5
4
6
6
5
1
2
3
3
1
1
6
45
Soln.21
4
5
4
3
5
2
1
3
4
3
6
6
5
1
2
1
3
6
2
4
1
4
5
2
2
4
1
6
5
2
6
1
3
6
5
3
45
Soln.22
4
4
3
2
1
1
5
4
2
3
5
3
4
1
3
2
5
4
2
1
3
1
6
1
3
6
4
5
2
6
6
5
2
6
6
5
44
Soln.23
4
1
4
4
3
3
2
1
5
6
5
1
3
4
6
5
2
3
4
2
2
4
6
5
5
6
6
5
1
2
3
3
2
1
1
6
44
Soln.24
3
4
5
3
4
2
1
4
3
2
5
6
4
5
1
3
1
2
5
4
6
2
3
6
2
5
1
3
1
4
2
5
1
6
6
6
44
Soln.25
4
4
2
3
5
5
3
4
1
2
3
6
3
5
6
1
6
1
2
6
4
5
4
2
3
1
1
2
5
6
4
6
2
3
5
1
45
Soln.26
4
6
1
4
5
3
2
5
3
1
1
3
4
2
6
5
2
2
3
4
4
5
3
6
1
2
6
3
1
1
5
2
5
6
6
4
45
Soln.27
4
1
3
5
4
2
6
1
4
3
3
5
2
2
2
5
4
1
3
5
1
4
6
1
2
6
5
5
1
4
2
6
6
6
3
3
45
Soln.28
1
3
4
2
3
4
1
5
3
1
5
4
2
6
4
6
2
6
5
1
1
2
6
3
1
4
5
2
5
4
6
6
2
3
5
3
44
Soln.29
4
1
6
5
4
3
1
2
4
6
2
5
3
3
6
4
1
2
5
5
6
4
2
5
2
3
6
6
2
3
1
4
1
3
1
5
45
Soln.30
5
4
2
3
1
5
4
6
4
3
1
4
3
2
5
1
6
6
3
2
2
4
6
1
2
3
6
1
5
5
5
4
2
1
3
6
44
134
Soln.31
1
5
4
4
6
4
2
3
3
6
2
1
5
6
3
6
2
5
5
4
1
2
1
4
4
2
1
6
1
3
5
2
3
6
5
3
45
Soln.32
4
2
3
4
1
4
3
5
1
5
3
4
2
6
2
6
5
3
1
6
6
5
1
2
4
3
4
3
1
6
2
5
2
1
6
5
44
Soln.33
4
2
3
1
4
5
1
6
4
3
3
5
2
1
2
2
3
1
4
5
6
6
3
5
4
3
4
6
5
1
2
2
6
1
5
6
44
Soln.34
3
4
1
5
3
4
4
6
5
1
5
2
2
6
6
3
2
5
2
4
3
6
5
1
4
3
2
6
6
5
4
1
2
3
1
1
45
Soln.35
3
1
4
3
4
6
5
2
6
5
4
1
2
3
2
2
1
3
1
6
5
6
4
3
4
3
6
4
5
5
5
2
1
2
6
1
45
Soln.36
1
5
6
6
4
6
3
5
2
4
4
1
2
3
1
6
3
4
3
2
5
2
5
4
5
5
6
3
2
1
1
2
3
1
4
6
46
Soln.37
5
3
6
4
6
3
2
4
1
5
4
2
1
5
3
1
2
4
2
3
1
6
4
5
5
4
6
2
6
5
2
1
3
1
6
3
46
Soln.38
3
4
3
1
2
4
1
4
3
1
5
2
1
5
4
2
5
6
3
5
2
6
6
4
6
3
1
1
3
4
2
5
5
2
6
6
44
Soln.39
1
4
2
4
5
3
1
3
5
1
3
4
6
4
5
4
2
6
2
5
2
6
2
6
6
5
3
1
5
3
2
4
3
1
1
6
44
Soln.40
1
4
6
4
1
5
2
3
6
4
3
5
2
6
5
3
2
4
1
2
6
6
3
5
5
4
1
2
1
2
4
5
3
1
6
3
45
Soln.41
3
4
5
2
4
4
3
2
1
5
1
3
4
6
1
6
6
5
2
3
2
5
4
2
5
6
5
2
3
1
6
6
4
1
1
3
44
Soln.42
4
3
2
3
1
4
5
5
4
1
2
1
3
6
2
1
5
2
4
3
1
3
5
4
6
6
1
3
6
5
6
5
2
6
2
4
44
Soln.43
5
4
4
4
3
1
3
1
2
5
6
3
2
1
1
2
2
5
3
1
4
6
5
4
6
5
3
2
6
4
3
6
1
2
5
6
44
Soln.44
3
4
5
4
3
5
5
1
4
2
3
2
1
3
4
6
2
1
6
5
4
6
2
6
2
6
5
4
2
1
3
6
1
5
3
1
44
Soln.45
3
1
5
3
4
4
4
1
2
6
2
3
5
6
3
2
5
4
1
2
6
6
1
5
5
4
6
4
2
1
3
5
3
1
2
6
45
135
Soln.46
4
3
3
5
2
4
1
6
1
5
4
3
1
2
1
4
3
6
5
1
2
1
5
6
4
2
2
3
3
5
6
4
2
6
5
6
45
Soln.47
1
3
4
5
5
6
4
6
4
2
3
1
6
3
2
6
2
1
4
1
5
4
2
1
5
2
6
5
3
1
2
3
4
3
6
5
45
Soln.48
3
4
1
4
4
3
2
1
5
3
2
5
6
6
5
4
3
6
2
1
5
2
4
3
5
5
2
6
6
3
1
4
1
1
2
6
44
Soln.49
5
4
1
4
3
3
5
4
1
2
1
6
2
1
3
3
6
4
2
6
2
6
5
1
4
6
3
1
5
4
5
5
2
3
6
2
44
Soln.50
5
3
3
4
1
4
2
1
6
3
5
5
4
1
2
4
2
3
1
5
2
6
1
6
5
6
6
3
4
1
2
2
5
6
3
4
44
Soln.51
3
3
4
2
5
4
4
3
1
6
6
2
1
5
3
5
4
1
4
6
2
2
5
3
1
5
4
3
5
2
6
2
1
6
1
6
44
Soln.52
5
4
1
2
3
3
6
4
4
5
3
1
2
2
5
1
1
4
2
5
6
6
4
1
5
6
1
6
2
2
3
5
6
3
3
4
46
Soln.53
4
2
3
3
4
5
5
4
1
5
3
2
6
6
1
1
3
2
4
1
6
2
4
6
3
5
2
3
1
5
1
2
4
6
5
6
44
Soln.54
4
1
5
4
3
1
4
3
2
6
3
6
1
2
3
5
2
5
5
4
6
4
2
1
6
1
1
5
3
2
3
6
4
2
6
5
44
Soln.55
3
4
4
2
3
5
4
1
3
6
6
2
1
5
2
4
5
3
2
1
6
1
5
3
4
6
5
5
6
3
2
1
1
4
2
6
44
Soln.56
4
2
3
1
4
1
3
4
2
3
5
5
2
3
4
6
6
6
6
1
5
2
4
3
4
5
2
5
1
6
1
2
5
1
3
6
44
Soln.57
4
6
5
4
4
2
3
3
6
1
5
1
3
2
4
3
1
6
6
5
1
2
2
5
3
5
6
4
2
1
2
6
3
5
4
1
45
Soln.58
3
4
4
1
4
3
2
5
5
1
2
6
3
3
6
2
1
1
5
4
6
2
5
5
3
1
4
5
1
2
2
3
4
6
6
6
44
Soln.59
4
4
3
1
5
3
2
5
1
4
2
1
1
6
5
3
3
2
4
4
6
2
5
6
5
6
3
6
3
5
1
2
6
1
2
4
44
Soln.60
4
3
3
4
1
1
5
5
3
5
2
4
4
2
2
1
6
2
4
4
6
2
6
6
5
5
3
2
6
3
3
5
1
1
1
6
44
136
Soln.61
4
2
4
3
5
1
3
6
1
5
3
6
4
2
1
5
2
4
2
6
4
3
3
5
5
5
1
6
2
1
2
6
4
1
3
6
44
Soln.62
4
3
4
4
2
6
5
5
3
3
1
5
2
1
4
2
3
1
6
1
4
2
6
5
5
5
2
6
3
3
1
6
2
4
6
1
44
Soln.63
4
4
5
3
3
4
2
6
2
1
5
4
1
1
5
3
1
6
2
4
6
2
2
5
3
4
5
1
2
5
1
6
6
3
3
6
44
Soln.64
1
5
4
3
2
4
3
1
4
5
2
1
6
3
6
3
4
2
1
6
2
5
1
5
1
4
2
3
5
6
6
4
5
3
6
2
44
Soln.65
4
2
6
3
3
1
4
1
4
3
2
1
5
1
5
3
1
5
6
4
1
2
5
6
5
2
4
5
2
4
6
6
6
2
3
3
44
Soln.66
4
3
1
5
5
2
4
1
6
4
3
3
1
4
2
6
5
2
6
1
2
5
1
4
5
2
6
2
6
1
6
3
4
3
3
5
44
Soln.67
3
4
3
5
4
2
4
1
5
1
6
3
2
4
6
2
1
5
1
3
4
6
2
1
5
3
3
6
4
5
2
6
6
2
1
5
44
Soln.68
4
4
4
5
1
3
2
3
6
5
4
1
3
5
2
2
3
5
2
1
6
6
6
1
3
6
5
4
5
3
4
2
1
6
1
2
45
Soln.69
5
4
2
4
4
1
3
3
5
1
2
3
3
4
5
2
1
6
6
6
5
6
5
2
4
3
3
1
4
6
1
2
2
5
1
6
44
Soln.70
4
4
3
2
5
1
4
5
4
1
3
3
2
2
1
2
6
1
5
5
3
3
6
1
1
6
4
2
6
3
6
6
4
5
2
5
44
Soln.71
4
3
6
4
6
1
3
3
4
2
1
5
2
5
4
5
6
1
4
2
5
2
5
2
3
2
4
5
6
1
6
3
1
6
1
3
45
Soln.72
3
4
4
1
5
4
2
6
5
1
2
3
3
2
1
4
2
6
5
3
6
1
3
5
3
4
6
2
1
5
2
6
6
5
4
1
44
Soln.73
4
4
1
3
4
3
1
2
2
1
5
6
3
3
5
6
2
1
4
2
6
5
5
6
4
3
5
6
2
2
1
3
1
6
5
4
44
Soln.74
4
3
3
1
4
2
1
4
1
6
5
2
1
5
3
4
3
2
6
2
1
1
6
5
4
6
3
5
6
4
5
3
2
5
6
2
44
Soln.75
1
4
5
3
5
4
4
4
6
2
1
3
2
1
1
3
2
3
6
6
2
5
5
6
5
6
4
3
1
1
3
2
5
2
4
6
46
137
Soln.76
5
1
3
4
3
6
4
6
1
4
2
6
2
6
3
1
5
2
3
5
2
1
5
1
6
4
4
2
1
5
4
2
6
3
5
3
46
Soln.77
4
3
2
5
5
3
1
4
4
3
5
4
2
1
1
3
6
2
2
5
4
6
1
6
3
2
6
6
5
5
4
3
1
6
2
1
44
Soln.78
5
1
3
3
4
4
4
1
6
5
6
2
2
3
5
6
6
4
2
5
1
5
2
4
6
4
2
1
6
5
2
3
3
1
3
1
45
Soln.79
4
3
4
4
1
3
5
5
2
6
2
1
4
3
1
6
6
5
4
2
2
3
1
3
2
6
6
6
1
5
5
2
1
4
5
3
44
Soln.80
3
3
4
5
1
4
5
4
1
4
2
3
1
5
2
6
1
6
6
2
5
3
2
1
6
3
3
1
5
4
4
5
6
2
6
2
44
Soln.81
3
1
3
4
5
4
5
3
4
2
5
2
1
6
3
2
5
4
2
1
4
6
6
2
3
4
6
6
3
1
5
5
6
2
1
1
44
Soln.82
4
3
1
4
1
5
3
4
2
2
5
6
1
3
2
4
6
2
5
3
6
4
5
6
1
5
2
3
5
3
2
1
6
1
4
6
44
Soln.83
5
4
2
3
3
1
4
1
3
4
5
2
6
1
6
1
4
1
5
6
2
3
2
3
4
2
5
4
5
2
5
1
3
6
6
6
45
Soln.84
4
4
3
4
2
3
1
1
2
3
5
2
5
5
6
4
6
1
6
5
3
5
2
5
6
4
6
3
2
1
3
2
4
6
1
1
44
Soln.85
4
2
4
5
3
3
4
1
5
2
5
1
6
3
4
2
3
2
1
3
6
1
6
4
1
4
6
3
5
2
1
6
5
6
5
2
44
Soln.86
3
4
5
4
2
5
3
5
4
1
4
3
1
1
2
6
2
6
1
5
2
3
6
5
4
5
6
1
2
2
6
6
3
1
4
3
44
Soln.87
3
4
1
6
3
5
5
4
2
4
3
1
2
5
6
2
1
4
3
2
3
1
3
4
4
5
6
6
6
1
2
5
5
6
2
1
45
Soln.88
4
3
1
4
2
5
4
3
2
1
1
5
3
5
1
4
6
6
2
1
6
2
5
4
5
4
6
2
1
6
3
3
5
2
6
3
44
Soln.89
5
4
4
3
2
4
1
6
1
5
2
3
3
1
4
2
5
1
2
3
4
2
1
6
6
5
6
5
3
3
5
6
6
1
4
2
44
Soln.90
5
4
1
2
3
3
4
1
3
4
5
2
6
4
1
1
6
6
5
1
2
6
1
5
2
4
6
5
3
2
2
5
3
6
3
4
44
138
Soln.91
4
1
5
2
6
4
1
3
3
3
5
1
4
5
2
2
6
2
5
4
3
4
3
6
6
3
5
4
6
1
2
5
6
2
1
1
44
Soln.92
1
3
4
3
5
2
5
4
4
1
6
3
4
2
2
1
1
3
6
6
2
3
5
5
5
3
1
4
5
6
2
1
6
4
6
2
44
Soln.93
4
4
1
3
2
3
1
4
5
3
1
5
2
4
1
6
5
6
2
5
1
6
5
2
3
5
4
6
4
3
3
6
1
2
2
6
45
Soln.94
5
3
4
1
4
6
4
5
3
1
6
1
3
4
2
2
5
2
2
6
3
1
6
4
5
4
6
1
3
2
6
1
3
5
5
2
45
Soln.95
3
4
5
4
3
1
5
1
2
6
4
3
2
4
5
2
6
6
2
1
3
6
5
3
3
4
1
1
5
6
4
1
5
2
2
6
44
Soln.96
4
5
3
6
2
4
1
6
5
6
3
4
5
3
1
1
2
3
2
5
4
1
2
6
5
4
1
3
2
6
4
1
2
3
5
6
45
Soln.97
1
5
3
3
4
4
1
1
4
2
5
3
2
3
6
6
5
2
2
4
6
5
6
4
5
6
3
2
1
3
1
6
4
1
5
2
44
Soln.98
4
6
3
3
1
4
5
5
1
2
6
4
2
5
3
2
1
1
5
6
4
2
5
6
3
4
5
4
6
1
1
2
2
3
6
3
45
Soln.99
3
4
1
4
4
2
2
1
3
3
6
2
1
5
5
1
4
1
6
5
6
5
4
2
5
6
3
6
2
2
3
6
5
4
1
3
44
Soln.100
4
2
3
5
4
5
6
3
1
3
4
1
6
5
2
2
2
4
3
5
6
6
3
3
1
4
1
6
5
2
2
4
5
6
1
1
44
139
PROBLEM 9
Soln.1
4
6
5
4
1
5
1
2
3
1
3
4
6
1
5
4
3
6
2
1
3
6
2
5
1
4
2
2
3
5
3
2
5
4
6
6
42
Soln.2
4
4
5
6
5
1
5
2
1
3
1
3
6
2
1
1
3
4
6
2
4
4
3
2
1
5
6
5
5
2
3
4
6
3
6
2
42
Soln.3
5
6
4
5
1
3
4
1
1
3
3
2
4
6
1
4
2
1
1
6
4
2
2
3
5
2
6
3
3
5
5
4
6
6
2
5
43
Soln.4
5
5
4
6
4
3
1
1
4
1
5
5
2
2
3
6
4
3
3
1
1
2
5
6
2
2
1
5
4
6
2
3
4
3
6
6
43
Soln.5
5
4
6
5
4
1
3
1
3
6
1
4
1
1
2
4
3
2
5
1
3
2
6
5
3
5
2
6
3
2
4
5
4
6
2
6
42
Soln.6
3
5
4
6
5
4
1
1
5
2
1
3
6
3
2
1
4
2
1
2
4
5
3
6
4
1
2
2
3
6
5
3
5
4
6
6
42
Soln.7
5
1
6
5
1
4
2
6
3
5
4
3
5
5
4
2
3
2
2
1
4
1
3
1
6
6
1
2
3
4
3
6
4
2
6
5
44
Soln.8
2
2
1
3
1
2
5
3
5
6
3
4
1
6
4
1
1
6
5
4
5
3
6
1
5
4
4
3
3
2
5
6
2
2
4
6
43
Soln.9
6
4
1
3
5
5
3
4
4
5
2
5
1
1
4
2
6
3
1
1
3
5
2
6
2
6
5
2
1
3
6
3
4
4
2
6
44
Soln.10
6
1
3
4
5
5
2
1
3
1
6
2
3
4
1
4
2
3
5
1
2
4
1
6
2
5
4
4
2
6
3
5
5
3
6
6
42
Soln.11
4
4
3
5
1
5
6
2
1
4
3
3
5
5
6
1
6
4
1
2
1
2
4
2
3
6
1
5
2
5
3
2
6
3
6
4
44
Soln.12
1
5
4
4
5
1
1
2
4
1
6
3
2
2
3
6
3
3
5
1
4
4
2
5
1
6
5
2
4
2
3
3
6
6
5
6
42
Soln.13
4
5
3
6
1
3
5
2
1
2
4
1
6
1
1
4
5
3
2
1
6
4
5
2
3
5
6
2
3
4
4
3
6
2
6
5
42
Soln.14
3
5
4
1
5
1
1
4
3
1
6
2
3
2
1
1
2
4
5
6
2
6
5
2
4
4
2
5
3
3
6
4
5
6
3
6
42
Soln.15
1
5
5
3
2
4
3
1
1
4
1
6
2
3
2
6
6
4
1
5
1
3
5
3
5
4
2
4
3
6
2
4
6
6
2
5
42
140
Soln.16
5
4
5
1
2
4
1
3
4
2
3
5
6
1
3
1
5
6
6
1
5
4
3
2
5
6
4
3
1
2
2
6
3
4
6
2
44
Soln.17
4
3
5
1
3
5
1
1
2
5
6
4
6
1
4
4
1
4
3
2
2
6
3
2
5
6
2
4
1
3
5
5
2
3
6
6
42
Soln.18
5
1
3
4
5
4
4
6
3
1
1
1
6
5
2
1
3
5
1
2
5
4
6
2
2
3
4
6
5
2
4
2
3
3
6
6
44
Soln.19
5
3
4
1
4
6
5
1
3
2
5
4
1
1
6
2
1
5
3
4
5
6
3
2
4
6
3
2
5
1
2
2
3
4
6
6
44
Soln.20
5
5
1
1
2
6
3
3
1
2
4
3
5
4
4
6
1
3
2
1
2
1
4
6
5
4
3
2
5
4
5
3
6
6
2
6
42
Soln.21
5
3
5
4
2
1
6
3
5
4
1
5
1
6
1
3
4
2
1
5
2
6
5
3
4
4
2
3
2
4
2
6
1
3
6
6
43
Soln.22
1
4
3
3
4
2
5
6
5
1
6
4
4
3
2
1
5
1
2
6
1
3
6
4
3
2
5
6
1
2
3
2
4
5
5
6
43
Soln.23
3
5
4
5
1
1
2
6
3
1
5
4
3
6
5
2
5
1
4
2
3
4
3
1
6
2
4
2
1
5
2
6
3
6
4
6
43
Soln.24
2
6
2
1
4
3
1
5
2
1
6
5
4
3
1
3
4
6
2
1
5
3
4
5
4
1
3
6
6
3
5
2
2
4
6
5
44
Soln.25
4
1
6
5
5
1
4
4
3
1
3
3
1
6
2
6
5
2
2
4
3
5
1
3
1
5
3
6
4
5
2
6
6
2
2
4
42
Soln.26
1
4
5
5
3
6
6
2
4
1
4
1
3
4
2
1
1
3
1
2
4
3
5
5
3
2
5
6
2
6
3
4
6
6
5
2
43
Soln.27
3
4
1
5
5
1
3
1
4
3
2
6
4
1
2
3
1
6
5
4
4
1
5
2
5
2
6
5
3
2
3
4
2
6
6
6
43
Soln.28
6
1
5
5
1
4
6
4
5
2
1
2
5
4
5
3
1
1
1
3
3
2
4
4
3
2
5
6
2
4
3
3
2
6
6
6
43
Soln.29
4
3
4
5
1
6
2
5
1
1
5
1
1
1
2
2
3
4
3
6
6
3
3
2
2
4
3
4
4
5
2
5
6
5
6
6
43
Soln.30
1
5
5
1
4
5
3
1
2
4
6
1
2
3
6
4
1
3
2
1
4
5
2
3
6
6
2
3
3
4
4
6
2
5
6
5
42
141
Soln.31
4
5
6
3
1
3
4
2
5
4
1
2
1
6
1
3
4
2
3
2
1
6
3
2
6
5
4
1
2
3
5
5
6
5
6
4
43
Soln.32
4
4
3
5
1
5
3
2
1
5
4
1
5
3
2
1
6
6
5
2
3
1
4
1
3
2
4
6
2
6
6
4
3
6
2
5
43
Soln.33
3
5
4
5
6
1
4
3
2
1
1
2
5
5
1
6
4
5
3
5
4
6
1
2
4
2
3
6
2
1
2
3
6
4
3
6
43
Soln.34
5
4
2
5
1
3
1
2
3
5
4
1
6
1
1
3
2
1
6
2
5
4
4
4
6
2
2
3
6
3
5
3
6
5
6
4
42
Soln.35
3
1
5
4
5
1
3
5
6
4
1
3
6
2
2
1
6
1
2
1
4
3
5
6
4
3
2
4
2
3
6
5
2
5
6
4
42
Soln.36
4
5
1
5
1
1
6
2
3
4
1
6
1
2
3
5
4
4
1
2
3
5
6
2
5
3
3
3
4
5
6
6
2
2
4
6
42
Soln.37
6
5
1
3
4
4
5
2
1
2
5
1
3
3
5
1
5
6
4
1
6
4
1
2
3
2
2
3
6
4
3
5
4
2
6
6
43
Soln.38
5
1
2
5
6
1
1
4
2
4
5
3
4
1
3
1
2
6
1
3
5
5
6
3
2
3
2
4
6
4
4
3
5
6
2
6
42
Soln.39
5
5
2
1
3
4
1
3
6
6
2
1
4
3
4
1
5
2
5
1
1
4
4
6
2
5
3
3
5
6
2
3
4
2
6
6
42
Soln.40
4
5
3
6
4
5
1
1
3
1
1
5
1
2
4
3
2
2
5
1
6
2
6
3
4
3
6
2
5
4
6
5
4
2
3
6
42
Soln.41
6
1
6
4
3
5
5
1
1
4
5
4
5
3
1
2
1
2
3
3
4
2
3
2
6
4
6
2
2
5
6
5
4
1
3
6
43
Soln.42
2
1
4
2
5
3
3
1
2
1
4
6
6
2
5
3
1
4
1
5
3
1
6
5
4
5
4
2
6
2
3
5
4
3
6
6
43
Soln.43
1
4
6
5
3
4
2
5
4
5
2
1
5
3
6
3
3
1
2
4
1
1
6
2
4
6
2
3
1
3
5
2
4
5
6
6
45
Soln.44
1
5
5
3
1
6
3
5
1
4
3
4
2
1
2
5
6
4
6
1
4
2
1
3
2
4
3
3
2
4
2
6
5
5
6
6
43
Soln.45
1
3
5
3
5
6
2
4
1
4
1
2
1
5
4
1
1
3
3
2
6
5
3
4
6
3
6
6
5
4
2
2
2
4
6
5
42
142
Soln.46
5
1
5
4
1
1
3
6
4
2
2
1
4
2
5
1
3
5
6
3
3
2
6
2
1
3
4
4
5
5
6
4
3
6
6
2
42
Soln.47
4
5
5
1
4
6
1
5
1
1
2
3
3
2
2
4
3
6
2
1
5
1
6
4
3
3
4
6
5
3
2
2
6
4
5
6
42
Soln.48
3
5
1
5
4
6
3
4
1
5
4
2
5
1
6
3
2
1
4
1
4
3
5
6
6
2
3
1
5
2
3
2
6
2
6
4
43
Soln.49
4
6
4
1
5
5
5
5
2
3
1
2
4
1
3
3
5
1
1
2
6
6
3
5
4
1
4
3
2
2
4
2
6
3
6
6
43
Soln.50
5
4
1
4
2
5
1
5
3
6
4
2
3
5
1
1
6
3
5
1
4
5
2
3
4
1
3
2
6
6
2
4
3
2
6
6
43
Soln.51
5
4
1
5
3
4
3
1
6
5
4
6
5
2
3
1
1
2
2
3
1
6
6
4
5
3
4
1
2
2
3
6
4
2
6
5
43
Soln.52
5
1
3
5
4
1
6
1
2
3
2
3
4
1
6
1
2
4
5
6
2
5
4
2
1
3
5
3
4
6
3
6
6
2
4
5
42
Soln.53
1
5
5
6
3
4
1
5
4
5
2
1
5
1
3
4
6
1
2
3
4
5
4
2
2
1
6
3
2
4
6
3
3
6
6
2
43
Soln.54
5
2
1
3
5
3
1
6
1
6
4
5
2
3
4
1
2
5
4
3
6
1
2
2
1
3
4
4
2
6
5
6
5
3
6
4
42
Soln.55
5
5
1
4
5
3
3
1
2
2
4
6
1
5
3
5
4
1
6
3
4
2
3
6
1
6
2
6
1
3
5
2
4
4
2
6
43
Soln.56
3
5
5
1
2
1
4
4
1
1
6
3
1
6
4
3
2
2
1
6
5
2
2
4
3
5
3
6
4
4
5
6
3
2
6
5
44
Soln.57
5
3
4
5
1
1
5
1
3
4
3
6
4
2
1
2
3
1
2
6
5
4
2
6
3
4
1
2
6
4
6
5
3
5
2
6
42
Soln.58
3
5
5
4
1
6
4
2
3
1
1
6
5
4
2
4
1
2
1
2
1
5
3
3
4
3
6
5
3
6
6
5
6
2
2
4
42
Soln.59
4
6
1
5
5
3
4
1
1
2
1
3
4
5
1
3
2
6
2
1
5
3
4
3
4
5
3
5
2
6
6
2
2
6
6
4
42
Soln.60
6
5
4
4
5
3
1
6
5
2
1
5
3
2
1
4
1
5
3
4
1
2
1
4
2
3
5
6
3
2
4
2
3
6
6
6
43
143
Soln.61
3
5
6
3
4
1
4
5
2
5
4
3
3
5
2
6
3
4
5
1
3
2
6
1
4
5
2
1
1
2
2
6
6
1
4
6
44
Soln.62
4
4
5
1
5
1
6
1
2
6
3
2
3
3
4
1
5
1
3
2
5
6
2
1
6
4
3
2
3
6
4
2
4
6
5
5
43
Soln.63
4
5
5
3
1
4
3
5
5
1
3
1
6
2
4
6
2
4
1
3
1
5
2
2
6
2
1
4
2
3
6
3
5
6
4
6
43
Soln.64
1
4
3
5
4
2
6
5
1
3
6
3
1
1
5
2
4
2
2
1
4
4
6
3
1
2
2
3
3
6
5
6
5
6
4
5
43
Soln.65
4
5
6
1
4
3
3
5
5
4
2
1
1
3
5
2
4
3
1
6
4
1
6
2
3
5
2
1
6
5
2
4
6
3
6
2
44
Soln.66
1
4
4
1
5
6
1
6
3
2
3
1
5
3
5
6
1
1
5
4
2
4
3
5
2
3
5
2
2
4
2
4
3
6
6
6
45
Soln.67
3
1
5
4
3
4
5
2
4
1
3
1
6
5
4
5
5
2
6
3
1
2
6
6
1
3
1
4
2
6
3
2
2
4
6
5
43
Soln.68
3
5
5
1
1
4
1
3
3
4
2
5
4
2
5
6
5
1
3
2
6
1
1
4
3
6
6
2
6
5
4
2
3
4
2
6
43
Soln.69
5
5
4
1
4
3
6
4
3
5
1
2
2
1
3
1
4
6
2
1
1
5
6
2
3
5
4
6
2
5
3
2
3
4
6
6
43
Soln.70
4
1
6
3
2
5
5
4
1
5
2
4
1
3
1
1
2
6
3
3
4
1
6
6
4
2
2
5
3
2
5
3
6
5
4
6
42
Soln.71
1
5
3
4
5
3
4
1
5
4
6
1
2
1
6
1
2
5
2
3
3
3
6
4
1
2
3
4
5
2
2
6
5
6
4
6
43
Soln.72
4
5
2
5
4
1
1
2
1
5
1
3
6
3
1
1
2
5
3
4
6
5
3
4
4
6
6
5
2
2
3
6
2
3
6
4
42
Soln.73
5
5
1
4
4
6
1
5
1
3
2
3
6
4
4
2
1
5
1
3
1
6
2
2
3
5
6
4
3
5
6
2
4
3
2
6
43
Soln.74
5
1
3
4
5
1
1
6
2
3
5
2
4
6
4
5
2
1
4
3
5
6
3
1
2
2
4
4
3
5
1
6
3
2
6
6
44
Soln.75
4
1
2
3
6
5
5
4
3
1
4
2
5
6
5
1
1
2
5
1
3
3
3
6
4
2
5
4
2
2
6
6
1
3
4
6
43
144
Soln.76
5
4
1
3
4
3
5
4
1
1
3
1
2
5
2
2
6
4
3
4
1
5
1
6
2
6
5
2
5
6
3
2
4
6
3
6
43
Soln.77
1
4
5
3
6
5
5
1
3
4
1
4
3
6
2
1
1
2
5
2
3
1
5
4
6
3
3
4
2
5
6
2
2
4
6
6
43
Soln.78
1
5
3
4
3
3
5
6
4
5
2
1
2
6
5
5
1
2
4
1
5
6
4
3
6
3
2
1
4
2
2
4
6
1
3
6
44
Soln.79
1
5
5
1
6
4
4
1
6
2
3
2
5
1
3
3
4
3
6
2
4
1
1
5
4
2
5
3
3
2
6
6
6
5
4
2
42
Soln.80
5
5
1
3
2
4
1
2
5
4
3
6
4
6
1
1
3
1
1
2
5
5
2
3
6
6
3
4
6
2
4
5
2
3
6
4
42
Soln.81
5
3
4
5
3
4
4
1
1
3
6
1
2
2
1
1
3
2
5
5
3
1
6
5
6
4
4
6
2
3
5
2
6
2
4
6
43
Soln.82
1
5
3
5
3
4
1
4
1
2
1
4
5
6
5
2
6
4
1
3
1
5
6
2
2
3
3
2
4
3
6
4
6
5
2
6
43
Soln.83
1
2
5
6
3
5
6
1
4
1
5
2
4
3
1
2
5
3
1
6
4
3
2
4
4
1
3
2
4
2
6
3
6
5
6
5
42
Soln.84
5
6
1
5
6
4
4
3
1
5
3
4
2
5
1
2
3
5
1
1
4
2
6
3
4
2
6
2
3
5
6
4
1
2
3
6
43
Soln.85
1
4
2
3
5
5
4
1
6
3
5
5
1
4
1
2
1
6
3
4
6
5
2
3
6
4
3
1
6
2
2
3
2
5
6
4
43
Soln.86
1
3
5
6
5
4
1
4
2
4
5
1
2
3
3
1
5
1
3
2
6
3
4
6
1
5
6
4
2
5
2
6
2
3
6
4
43
Soln.87
1
4
5
5
3
2
6
5
4
1
3
5
1
3
6
4
5
6
5
4
1
2
3
1
4
2
2
3
6
6
2
1
3
4
2
6
43
Soln.88
5
1
5
4
3
1
3
5
4
2
4
6
1
3
2
3
1
1
1
2
6
6
5
4
3
5
4
2
2
6
2
5
6
4
3
6
42
Soln.89
5
1
4
3
3
5
4
1
6
1
5
2
5
1
4
2
3
6
2
6
4
3
2
5
1
1
5
2
4
4
3
3
6
6
2
6
43
Soln.90
6
1
6
3
5
5
1
4
1
4
1
5
4
5
3
2
4
3
1
6
2
3
2
1
5
3
4
3
6
2
5
6
2
6
4
2
43
145
Soln.91
4
5
5
2
1
4
1
3
4
6
3
1
5
6
5
1
2
2
1
4
2
4
3
6
2
3
5
4
3
1
6
3
6
6
2
5
43
Soln.92
1
6
4
4
5
3
6
5
5
1
5
1
4
2
1
1
3
3
5
2
4
2
2
4
1
5
3
6
3
6
2
4
6
3
2
6
43
Soln.93
3
5
1
3
5
1
1
6
2
4
3
4
2
4
1
3
1
6
6
2
5
4
2
5
2
1
3
4
3
4
5
2
6
5
6
6
42
Soln.94
1
4
5
2
5
6
1
2
1
4
1
3
2
1
4
6
3
3
5
5
3
4
2
3
6
2
4
1
4
2
6
5
6
5
3
6
42
Soln.95
1
5
4
3
5
1
3
1
4
4
2
6
3
5
1
2
6
5
2
1
1
4
2
3
6
5
3
4
2
3
5
4
2
6
6
6
43
Soln.96
1
6
3
4
5
5
2
1
3
6
1
3
2
4
4
1
3
1
5
2
2
1
6
4
5
4
2
6
2
3
4
3
6
5
6
5
42
Soln.97
4
5
3
1
6
5
1
3
2
1
4
4
5
6
2
5
5
4
3
1
1
2
2
6
3
3
5
1
4
6
6
3
2
6
4
2
43
Soln.98
5
1
6
5
3
4
6
4
1
4
1
5
5
2
3
1
3
2
4
5
2
6
3
2
3
1
6
6
1
2
5
3
6
4
4
2
43
146
PROBLEM 10
Soln.1
4
2
5
2
5
3
1
2
6
1
2
3
2
5
2
4
4
1
1
6
6
6
4
3
5
1
4
6
5
3
5
4
1
6
3
3
54
Soln.2
4
5
2
2
5
3
2
4
1
4
6
2
5
1
1
2
1
2
4
5
4
3
6
6
1
4
1
5
3
3
6
6
5
3
3
6
55
Soln.3
2
5
2
2
5
3
1
4
4
1
4
1
3
2
3
2
4
6
5
6
5
6
1
5
1
2
6
4
3
1
4
3
3
5
6
6
55
Soln.4
2
2
2
5
4
4
5
4
3
6
2
5
1
1
3
4
1
4
2
6
1
5
4
2
3
5
6
3
3
5
1
6
6
6
1
3
55
Soln.5
2
4
2
2
6
5
3
1
5
1
2
3
4
4
4
1
5
6
4
5
6
2
3
5
1
3
2
5
1
6
1
6
3
3
6
4
56
Soln.6
2
4
2
5
1
5
6
3
2
2
1
4
4
3
2
5
1
5
4
1
5
6
3
4
3
1
3
6
6
2
5
6
4
6
1
3
55
Soln.7
5
4
2
2
5
2
4
3
2
5
1
3
6
3
1
4
4
1
5
5
6
6
4
6
1
3
2
5
1
3
2
4
3
1
6
6
55
Soln.8
5
1
4
3
4
5
4
2
6
3
1
2
5
1
4
2
4
6
5
2
4
1
5
5
3
3
2
6
1
1
6
2
6
6
3
3
56
Soln.9
4
2
2
2
1
6
5
4
2
1
5
4
2
1
5
4
3
1
5
2
4
6
3
5
6
3
6
4
3
1
5
6
6
1
3
3
54
Soln.10
5
3
1
5
4
4
5
4
2
6
3
2
4
4
2
2
4
6
1
1
1
6
1
3
6
5
5
1
3
6
6
5
2
2
3
3
56
Soln.11
2
4
2
5
2
6
5
4
2
3
5
1
3
3
4
1
2
6
2
1
1
1
5
6
4
4
5
1
5
6
3
6
4
3
3
6
54
Soln.12
5
5
1
4
2
3
4
1
1
4
4
5
5
2
4
2
6
5
3
1
6
6
2
6
3
2
3
5
1
4
2
3
6
6
1
3
56
Soln.13
4
5
2
5
5
1
2
2
3
4
1
3
4
4
6
2
1
2
2
4
5
1
4
3
6
6
3
5
1
5
1
3
6
6
3
6
56
Soln.14
4
4
5
5
3
3
2
4
2
2
1
4
5
5
5
6
2
1
5
4
6
1
1
1
4
2
3
1
3
3
6
3
6
2
6
6
56
Soln.15
5
2
2
5
4
2
4
3
2
1
6
4
5
1
4
3
4
1
4
6
2
3
5
6
6
5
2
5
6
1
3
1
1
6
3
3
55
147
Soln.16
1
5
4
5
2
4
1
1
3
5
5
2
6
4
4
1
3
2
1
6
4
2
1
4
3
3
6
2
5
6
2
3
5
6
3
6
56
Soln.17
2
4
2
5
5
2
4
1
2
1
5
3
4
1
4
1
4
6
2
3
6
5
1
2
5
3
6
6
3
5
6
3
6
1
3
4
54
Soln.18
2
2
4
5
1
2
4
3
1
5
4
2
6
5
2
4
1
5
1
3
3
5
6
4
1
6
2
4
5
3
1
6
6
3
3
6
55
Soln.19
4
4
2
1
5
3
2
2
5
4
1
4
3
5
2
6
3
6
2
4
1
6
1
6
5
6
5
3
1
5
4
6
1
3
3
2
56
Soln.20
5
4
2
1
5
4
5
2
1
2
6
5
1
3
4
6
2
4
3
6
5
3
4
1
4
3
2
2
1
6
3
1
5
6
3
6
56
Soln.21
5
2
2
4
2
5
3
2
4
6
1
5
4
3
1
1
4
4
3
5
2
1
2
4
3
5
6
6
5
3
1
6
6
1
6
3
55
Soln.22
4
6
2
2
5
5
3
4
5
2
1
4
3
6
4
3
4
1
1
5
6
2
2
6
2
4
1
6
3
5
1
5
1
6
3
3
56
Soln.23
2
2
4
3
1
5
6
1
3
5
3
2
3
4
2
5
1
5
4
6
4
4
2
6
1
2
6
5
1
4
1
3
3
6
5
6
55
Soln.24
2
4
5
5
2
2
1
4
4
6
1
6
1
3
3
2
5
6
4
4
4
2
6
5
5
3
2
5
1
3
6
3
6
3
1
1
55
Soln.25
5
4
1
2
4
6
5
1
1
5
3
4
4
2
3
3
5
2
5
6
4
1
6
5
2
3
2
4
2
6
3
6
3
1
6
1
56
Soln.26
5
5
1
4
4
1
4
5
2
4
1
4
6
3
4
2
3
1
2
2
6
6
5
3
2
5
6
6
3
5
6
2
3
3
1
1
56
Soln.27
3
2
3
1
2
3
4
2
1
5
3
5
2
4
3
1
5
5
6
4
2
4
3
5
4
1
6
6
2
5
1
6
4
6
1
6
55
Soln.28
5
2
3
4
2
2
6
2
5
1
1
4
1
2
5
4
5
3
2
3
4
1
5
4
4
3
5
3
6
6
3
1
6
1
6
6
54
Soln.29
1
5
5
4
6
1
4
2
4
4
1
3
5
3
4
1
2
2
2
6
6
3
1
5
4
6
6
2
3
5
2
5
1
6
3
3
56
Soln.30
2
6
5
2
1
2
5
4
4
1
4
1
2
3
3
3
5
1
6
4
4
2
2
6
3
1
4
5
5
5
3
3
6
1
6
6
55
148
Soln.31
1
2
4
5
4
3
6
5
4
2
3
3
5
1
6
1
4
2
6
4
5
4
1
5
1
1
2
6
6
5
2
3
6
3
2
3
56
Soln.32
6
3
2
3
2
3
3
2
1
4
5
4
5
2
3
1
4
2
1
5
6
4
6
4
1
2
1
4
5
5
6
3
5
1
6
6
55
Soln.33
2
1
2
4
5
3
5
2
2
4
6
3
1
1
4
3
6
5
4
6
5
4
4
1
6
2
1
5
2
1
3
5
3
6
6
3
55
Soln.34
1
3
2
5
3
5
4
4
3
4
3
1
2
1
1
3
5
4
6
2
5
6
4
6
5
5
1
4
6
2
1
6
2
2
3
6
56
Soln.35
5
2
1
2
4
2
5
4
6
2
3
4
5
3
2
4
1
1
4
2
5
3
6
1
3
5
4
6
5
3
1
3
6
6
1
6
55
Soln.36
2
5
4
2
2
4
1
6
5
3
2
1
4
2
4
3
1
5
2
6
5
1
6
4
3
1
5
3
1
5
4
3
6
6
3
6
55
Soln.37
4
5
6
4
5
1
2
5
1
3
2
4
6
1
3
4
2
5
3
1
5
4
5
2
1
3
6
3
6
3
2
6
1
6
4
2
56
Soln.38
3
6
5
3
3
5
2
3
2
2
4
4
1
1
4
1
4
2
2
5
4
2
6
4
5
5
1
1
3
6
6
1
3
6
6
5
55
Soln.39
2
5
3
1
2
2
4
1
2
5
3
5
4
6
1
4
4
1
2
4
3
5
6
5
2
3
1
6
6
1
4
3
6
5
6
3
54
Soln.40
2
1
2
2
4
5
5
2
4
4
1
5
2
1
4
3
5
2
6
3
6
3
1
4
1
5
4
3
6
3
5
6
6
1
3
6
54
Soln.41
2
5
4
2
1
2
3
6
4
5
2
1
4
3
1
4
6
2
5
5
2
3
4
1
5
6
3
6
3
4
5
3
1
1
6
6
55
Soln.42
2
1
2
2
4
5
5
4
6
4
3
1
2
3
5
1
4
2
5
5
5
4
6
3
1
4
3
2
1
1
3
6
6
6
3
6
55
Soln.43
4
5
3
5
2
2
2
4
1
2
5
6
1
4
2
6
1
2
3
3
4
5
5
4
6
1
4
6
5
3
1
3
6
1
6
3
55
Soln.44
5
5
5
4
3
4
2
1
4
6
4
4
3
1
2
2
1
3
2
5
5
5
4
6
6
1
2
1
3
3
6
3
2
1
6
6
56
Soln.45
5
1
2
4
1
2
3
5
4
4
3
1
3
2
6
6
6
5
1
2
4
2
6
6
4
1
4
3
5
1
5
5
3
2
6
3
56
149
Soln.46
3
5
3
3
2
5
4
2
1
2
3
4
2
1
5
6
1
4
4
3
2
2
6
4
4
5
6
6
5
6
1
6
1
5
1
3
55
Soln.47
2
4
5
1
2
2
5
5
1
4
3
5
3
1
6
4
2
4
6
4
2
3
2
5
1
6
6
1
5
6
3
4
1
3
6
3
56
Soln.48
4
5
2
5
6
4
2
1
4
3
5
6
2
2
4
1
1
3
4
1
2
4
1
3
5
2
3
5
6
3
6
5
6
3
6
1
56
Soln.49
2
4
5
2
2
4
5
2
2
3
1
1
4
3
1
2
5
3
4
1
5
4
5
3
3
6
6
4
6
6
1
6
5
1
3
6
54
Soln.50
2
5
5
1
4
4
3
3
4
5
6
3
4
2
2
6
5
2
4
4
1
1
5
5
1
3
6
2
1
1
3
3
6
2
6
6
56
Soln.51
5
2
4
3
1
2
2
5
2
4
3
2
1
5
1
6
4
6
4
3
5
1
4
2
6
6
1
4
3
5
6
3
6
3
1
5
55
Soln.52
4
1
3
1
5
3
4
4
3
6
3
1
5
1
2
4
6
2
3
2
5
5
4
5
6
5
2
1
1
4
3
2
6
6
2
6
56
Soln.53
4
2
5
5
5
1
4
3
4
6
2
1
4
3
2
1
3
4
6
1
6
1
2
5
5
2
4
3
6
2
1
6
5
3
6
3
56
Soln.54
5
5
4
1
6
4
3
2
4
1
5
4
4
1
5
1
2
4
5
3
2
6
3
5
6
1
2
6
3
1
3
6
2
6
2
3
56
Soln.55
2
4
5
2
3
2
4
5
2
1
4
4
5
6
4
1
1
6
2
3
5
5
4
1
3
3
5
3
2
6
6
3
1
1
6
6
55
Soln.56
5
5
3
4
4
1
2
3
4
5
2
1
2
4
6
2
4
1
5
4
3
2
5
6
2
1
5
3
6
1
6
1
3
6
6
3
56
Soln.57
1
4
2
5
1
3
6
4
5
5
1
2
3
4
1
3
6
2
1
4
6
2
2
4
1
5
6
6
2
5
3
4
6
3
5
3
56
Soln.58
4
2
1
5
2
2
5
2
4
5
6
1
2
4
6
3
1
3
3
5
1
4
4
2
6
5
1
6
3
4
3
5
1
6
6
3
54
Soln.59
5
6
2
5
2
2
4
1
1
5
2
4
4
1
6
3
4
5
3
3
1
1
4
6
3
5
5
6
1
2
4
3
6
6
3
2
56
Soln.60
1
5
5
4
2
1
5
6
3
4
4
6
1
4
5
3
1
2
1
4
5
4
5
2
6
3
2
6
3
3
2
3
2
6
1
6
56
150
Soln.61
5
4
1
4
1
6
5
4
1
2
5
2
4
2
6
1
5
6
3
1
1
2
3
3
4
4
6
5
3
3
5
3
2
6
2
6
56
Soln.62
5
2
4
2
2
4
5
3
2
6
5
1
3
2
1
4
6
4
2
1
1
3
4
6
3
4
5
5
6
6
3
5
6
3
1
1
54
Soln.63
3
3
3
2
4
2
2
5
3
5
1
2
1
6
5
4
4
4
6
4
2
2
1
3
6
5
1
1
5
3
6
6
6
4
1
5
55
Soln.64
2
4
2
6
5
5
2
4
4
1
1
1
6
3
2
5
6
2
2
1
5
4
3
4
6
6
3
1
4
3
5
1
3
5
3
6
54
Soln.65
5
2
3
1
2
4
1
5
2
4
2
1
2
5
1
4
6
3
4
5
2
6
3
6
4
3
6
1
5
4
5
6
1
6
3
3
54
Soln.66
2
4
2
5
2
3
1
5
4
4
6
1
1
3
4
3
2
5
6
5
1
2
6
1
5
6
4
2
3
6
3
5
6
3
1
4
54
Soln.67
2
2
2
5
2
5
5
4
1
3
4
6
3
1
4
6
3
2
4
1
2
4
5
1
6
5
5
6
3
4
6
3
1
6
3
1
55
Soln.68
5
4
2
4
5
4
1
3
2
2
4
3
5
6
3
1
4
1
2
5
4
1
6
3
5
6
1
5
6
3
2
6
6
1
3
2
56
Soln.69
6
4
2
5
1
2
1
4
5
3
5
4
4
2
2
5
4
4
6
1
3
2
1
6
5
3
3
6
5
1
2
6
3
6
1
3
56
Soln.70
1
2
4
4
6
5
1
2
1
5
5
5
2
1
3
4
3
4
6
4
5
4
6
5
6
1
6
3
2
3
2
1
3
2
6
3
56
Soln.71
2
5
5
6
2
4
6
2
2
5
4
1
1
3
4
1
3
5
1
6
6
4
1
4
5
3
2
3
6
5
3
6
4
2
3
1
55
Soln.72
4
2
5
2
2
3
4
6
1
5
2
1
4
3
6
1
5
4
3
4
5
4
5
5
2
1
3
1
3
2
3
1
6
6
6
6
55
Soln.73
4
3
2
3
3
5
5
2
2
4
3
1
4
1
2
2
4
1
5
4
6
5
6
2
6
5
5
1
6
6
6
3
1
4
3
1
55
Soln.74
3
2
3
6
3
2
4
1
5
5
2
3
2
4
3
1
5
4
6
3
4
1
4
1
5
2
2
6
4
6
5
1
1
6
5
6
55
Soln.75
1
4
2
5
1
3
5
4
4
2
5
6
4
1
1
3
4
5
4
2
6
5
5
1
3
6
6
2
6
3
2
1
2
6
3
3
56
151
Soln.76
4
2
1
5
2
5
1
2
4
4
3
2
1
2
5
4
4
5
3
2
5
4
3
6
6
1
1
1
6
3
6
5
3
6
6
3
55
Soln.77
2
5
5
3
4
3
2
2
2
5
1
4
6
4
6
2
1
3
1
2
4
4
5
6
6
1
1
5
3
6
5
3
3
4
6
1
55
Soln.78
2
2
1
5
3
2
4
2
1
5
4
4
2
5
3
1
3
4
6
1
6
5
2
6
5
5
1
4
3
4
3
3
6
6
1
6
54
Soln.79
2
4
5
2
5
3
2
3
2
1
6
4
1
1
4
2
6
5
5
4
1
3
4
5
3
3
3
6
1
1
6
5
2
4
6
6
55
Soln.80
5
2
4
5
3
4
1
2
4
5
3
4
4
4
1
2
6
6
5
1
3
5
5
6
2
2
2
1
6
1
6
3
3
3
1
6
56
Soln.81
2
4
5
2
1
6
2
3
1
5
2
3
4
4
2
5
1
4
5
4
3
1
4
5
5
6
6
6
1
6
3
3
1
2
6
3
55
Soln.82
5
3
4
5
3
2
2
1
2
4
5
6
4
1
6
2
4
1
5
6
1
2
3
1
4
6
5
3
6
1
2
3
4
5
3
6
56
Soln.83
3
2
1
4
3
4
3
5
2
2
3
5
1
4
4
1
2
6
6
5
5
6
2
2
3
5
4
6
3
5
1
1
6
6
4
1
56
Soln.84
2
5
6
2
4
2
1
5
1
4
4
3
2
5
1
6
6
4
1
6
3
2
4
3
4
1
6
5
1
5
3
6
2
5
3
3
55
Soln.85
5
4
2
2
6
2
4
1
5
3
4
2
1
3
1
5
5
6
4
1
6
5
2
2
3
4
6
3
6
6
5
1
3
4
1
3
54
Soln.86
2
4
2
5
1
2
5
4
3
2
6
5
1
4
3
1
6
1
2
5
4
6
1
3
2
5
4
6
4
6
1
3
3
3
6
5
55
Soln.87
2
5
2
2
4
5
6
3
1
2
1
4
4
5
3
1
6
3
2
4
5
4
1
2
6
6
3
1
5
6
6
4
3
5
1
3
55
Soln.88
1
2
4
5
4
6
5
1
1
2
4
3
5
5
4
6
4
3
2
6
6
5
6
1
1
5
2
1
3
3
6
3
2
4
2
3
56
Soln.89
5
2
2
4
2
6
1
5
3
3
4
4
5
5
2
1
2
4
3
6
2
4
1
5
1
5
3
4
6
1
3
3
1
6
6
6
56
Soln.90
2
4
5
2
3
1
2
5
4
6
4
2
1
1
5
1
2
4
4
6
3
1
3
5
6
5
4
3
1
5
3
6
6
3
2
6
55
152
Soln.91
2
5
4
2
5
6
3
2
3
2
1
4
4
1
2
5
1
1
4
4
5
3
6
1
3
5
1
6
2
6
5
6
6
3
4
3
55
Soln.92
5
2
2
4
2
5
2
4
4
5
1
1
3
3
1
4
4
3
6
6
1
4
3
5
6
6
5
3
5
3
1
1
6
2
6
2
55
Soln.93
5
5
4
2
2
1
2
1
4
4
1
3
3
2
4
5
1
4
5
2
2
6
6
3
3
5
6
3
6
6
5
6
4
1
1
3
54
Soln.94
4
5
1
2
4
5
1
2
4
3
3
1
6
5
4
1
5
6
2
4
2
3
1
4
6
6
6
5
3
1
6
3
5
2
3
2
56
Soln.95
2
5
2
4
2
5
2
4
1
5
1
4
4
3
4
1
5
3
1
2
5
6
6
3
6
3
4
6
1
5
6
2
1
3
6
3
55
Soln.96
4
2
2
5
6
5
1
2
2
1
2
5
4
4
3
4
1
3
1
3
4
6
5
1
6
6
3
2
6
5
5
3
1
4
3
6
54
Soln.97
2
2
2
2
5
1
4
5
5
4
1
4
1
5
4
5
3
4
1
3
3
2
6
2
6
1
5
3
3
4
6
3
1
6
6
6
55
Soln.98
4
2
5
2
5
1
3
6
2
3
4
2
1
1
5
4
3
6
4
1
5
6
5
1
4
6
4
3
2
3
1
6
6
3
2
5
55
Soln.99
2
2
1
5
2
4
3
5
4
2
3
2
5
4
1
3
4
1
1
5
1
2
4
5
4
6
5
6
3
3
1
6
6
6
3
6
56
Soln.100
2
5
4
5
2
2
2
1
3
4
6
4
1
2
1
4
6
5
3
2
1
3
5
1
4
5
3
4
5
6
6
3
3
6
1
6
54
153
PROBLEM S1
Soln.1
2
4
1
5
4
2
3
5
6
3
1
4
6
5
4
1
1
6
5
5
4
6
3
2
6
3
2
5
2
3
3
1
4
1
6
2
46
Soln.2
4
4
2
6
6
3
4
1
1
5
2
4
5
5
3
6
5
1
3
4
1
5
6
2
6
3
1
2
3
5
3
2
6
1
4
2
46
Soln.3
4
1
1
3
5
2
6
2
5
1
4
3
4
6
1
6
4
2
5
5
4
6
3
4
6
5
5
3
3
1
2
3
1
2
2
6
46
Soln.4
4
3
6
3
1
5
2
6
1
6
5
1
4
4
2
4
5
1
1
3
6
4
5
5
6
2
3
4
2
2
5
2
3
1
3
6
46
Soln.5
6
1
4
5
6
2
4
1
4
5
2
4
1
1
5
4
3
6
3
6
3
2
4
6
5
5
1
3
2
2
5
2
1
3
6
3
46
Soln.6
3
4
3
1
6
1
4
5
2
5
6
6
4
4
1
6
5
5
1
6
2
2
5
3
3
5
4
2
2
6
1
3
1
3
2
4
46
Soln.7
4
6
1
6
5
1
2
5
4
4
5
4
2
6
3
1
4
5
3
3
1
1
5
2
6
3
1
2
2
4
6
5
3
3
6
2
46
Soln.8
4
1
2
5
3
3
4
1
2
5
6
4
6
4
6
1
5
1
5
4
6
5
3
3
4
2
2
1
6
5
3
2
6
3
2
1
46
Soln.9
4
2
3
1
5
1
5
1
2
6
3
4
4
4
4
6
5
1
6
5
5
1
3
3
2
4
6
1
2
3
6
5
2
2
6
3
46
Soln.10
3
2
1
5
2
4
6
6
4
5
4
1
5
6
4
3
1
4
1
5
5
3
6
3
1
2
2
6
4
2
1
5
3
2
6
3
46
154
Soln.11
2
1
5
4
3
3
4
5
1
1
1
6
4
6
2
4
6
5
3
5
5
2
1
3
4
1
2
4
6
3
6
2
5
2
6
3
46
Soln.12
4
2
6
1
4
3
5
6
6
4
1
5
5
1
3
6
4
1
5
2
6
3
2
5
3
2
4
5
1
1
6
2
4
3
2
3
46
Soln.13
4
3
2
1
5
4
3
5
6
1
6
4
2
6
1
4
1
4
5
3
5
5
2
6
6
3
5
3
1
2
1
4
2
6
2
3
46
Soln.14
4
3
4
3
1
2
4
6
5
6
2
5
1
4
5
1
5
6
1
4
5
2
3
4
6
3
1
6
2
2
5
2
3
1
3
6
46
Soln.15
4
3
2
5
5
4
1
4
2
6
6
1
1
5
6
5
3
6
2
4
2
1
5
3
1
2
6
4
3
1
5
3
3
2
6
4
47
Soln.16
5
4
4
4
1
2
5
6
6
1
2
3
4
6
1
2
5
2
1
3
5
5
4
2
1
6
6
3
3
6
3
5
2
3
1
4
47
Soln.17
2
3
5
6
1
4
4
1
4
2
6
5
4
5
3
2
5
2
6
3
1
4
1
6
4
1
2
5
3
6
1
2
5
3
6
3
47
Soln.18
1
4
5
2
4
2
6
6
3
5
4
5
5
1
4
3
2
5
1
2
6
1
2
1
6
3
4
3
3
1
2
6
4
5
3
6
47
Soln.19
4
2
1
5
3
6
5
1
4
2
3
6
6
4
2
5
5
1
5
6
3
2
4
1
3
4
1
6
2
1
6
4
3
5
2
3
47
Soln.20
5
1
2
3
3
2
4
4
4
6
5
1
2
1
4
1
4
6
5
5
1
1
2
2
6
6
3
5
6
3
4
2
5
3
3
6
47
155
PROBLEM S2
Soln.1
1
5
6
6
3
5
4
1
2
3
5
3
4
2
3
6
5
4
3
2
6
5
1
1
2
4
6
5
2
1
3
4
4
6
1
2
53
Soln.2
5
6
3
3
2
4
1
5
4
2
4
5
6
6
3
6
1
5
3
2
6
1
3
2
1
3
4
2
5
4
5
6
4
2
1
1
53
Soln.3
6
5
4
2
1
5
3
2
6
5
3
1
3
6
4
3
3
2
4
3
1
6
2
6
1
5
5
5
6
4
2
1
2
4
1
4
53
Soln.4
5
1
6
3
5
4
2
3
1
2
4
6
2
6
4
3
2
6
3
6
5
3
1
3
5
4
5
5
1
2
6
4
2
1
1
4
53
Soln.5
6
3
5
2
1
6
5
6
3
4
3
2
1
6
2
4
3
5
4
6
2
5
3
4
5
3
5
4
1
2
1
1
2
4
1
6
53
Soln.6
1
6
5
5
6
2
4
3
3
3
5
2
4
3
4
1
5
1
6
2
5
6
2
5
3
4
1
6
2
3
4
1
1
2
4
6
53
Soln.7
5
3
6
1
5
4
3
4
6
4
2
2
3
6
1
2
3
6
2
4
5
2
1
3
3
1
5
6
1
4
5
6
1
5
2
4
53
Soln.8
3
5
6
4
2
2
3
6
5
4
4
1
2
5
3
3
5
1
5
6
2
6
1
3
3
1
4
6
6
5
4
4
2
1
1
2
53
Soln.9
4
6
5
5
5
1
3
6
4
3
4
5
1
2
2
1
3
3
6
6
3
3
1
4
2
2
2
1
5
6
5
4
1
2
6
4
53
Soln.10
1
4
6
6
5
3
2
2
3
5
4
4
6
3
5
2
3
2
3
5
1
6
6
1
4
5
2
3
1
6
5
1
4
4
1
2
53
156
Soln.11
6
1
3
4
3
5
6
4
5
3
2
2
5
6
3
4
5
6
2
4
1
3
3
2
1
4
6
5
5
2
1
6
2
4
1
1
53
Soln.12
5
3
2
5
1
4
2
4
6
3
6
1
3
3
1
5
6
3
1
6
4
2
4
6
5
5
3
4
2
2
1
6
5
4
1
2
53
Soln.13
3
1
5
6
3
4
1
4
2
5
5
6
3
1
4
2
5
2
3
5
1
2
6
3
6
6
6
3
4
5
2
1
4
2
4
1
53
Soln.14
5
1
6
6
4
3
2
4
1
3
4
5
2
2
5
3
5
3
6
6
5
2
6
1
3
1
4
2
3
1
4
5
6
2
4
1
53
Soln.15
1
5
6
3
6
4
5
4
3
2
1
6
2
3
3
4
2
5
2
3
1
1
6
4
6
6
5
5
3
2
1
4
4
5
1
2
53
Soln.16
4
6
5
3
5
1
1
3
2
5
2
5
6
3
6
4
6
3
3
4
3
4
5
2
2
1
1
4
6
5
6
2
4
2
1
1
53
Soln.17
1
5
6
2
4
3
6
4
5
2
6
3
3
5
2
3
4
1
6
5
2
5
4
5
3
3
4
6
2
1
4
1
6
1
2
1
53
Soln.18
1
6
5
3
6
5
4
1
5
2
3
4
3
2
3
3
5
4
1
6
2
5
6
4
3
4
2
1
2
5
1
6
2
6
4
1
53
Soln.19
4
5
6
3
1
5
2
4
6
6
5
3
3
3
4
6
4
5
1
2
1
5
3
2
2
2
6
1
3
6
5
2
4
1
4
1
53
Soln.20
5
1
2
4
6
3
2
6
2
6
5
4
3
3
3
5
4
6
4
2
1
3
5
1
5
2
3
1
6
5
6
4
2
1
4
1
53
157
PROBLEM S3
Soln.1
1
1
1
4
2
6
6
3
2
3
4
4
2
5
1
6
4
2
2
5
2
4
6
3
5
3
5
3
6
4
1
3
5
6
1
5
60
Soln.2
4
1
2
3
2
1
4
3
2
5
2
6
4
4
6
3
5
2
5
1
2
4
5
6
3
3
1
5
4
3
1
6
1
6
6
5
60
Soln.3
1
1
1
2
4
3
2
6
4
1
5
4
2
5
3
6
6
3
4
2
2
5
1
2
4
5
3
5
1
4
6
3
5
3
6
6
60
Soln.4
3
4
2
1
4
6
3
1
2
1
2
4
2
5
6
4
5
3
1
6
2
6
4
5
2
5
3
3
6
4
5
1
5
3
6
1
60
Soln.5
4
1
3
2
1
6
4
2
2
1
1
5
5
4
3
2
4
3
2
5
6
1
5
6
2
4
6
3
3
4
5
3
1
6
5
6
60
Soln.6
1
4
1
2
3
1
2
2
6
4
5
2
3
3
4
5
6
4
2
3
5
2
1
4
5
6
5
6
3
1
4
6
5
3
6
1
60
Soln.7
1
2
4
2
4
3
5
6
4
2
6
1
3
5
4
6
2
3
1
2
5
3
4
5
4
1
6
1
2
6
3
5
6
3
5
1
60
Soln.8
3
6
1
2
4
2
1
4
3
4
2
3
2
6
5
4
5
2
1
2
6
5
6
5
3
1
4
4
3
5
1
3
6
1
6
5
60
Soln.9
2
3
4
1
2
6
3
1
6
6
4
5
2
4
2
4
5
1
3
3
2
6
1
4
3
5
2
5
5
6
4
3
5
1
6
1
60
Soln.10
1
2
4
3
4
3
6
1
3
4
5
2
1
6
2
2
4
2
1
5
3
4
1
5
6
6
2
5
3
5
5
4
3
1
6
6
60
158
Soln.11
1
2
1
2
3
6
4
6
1
4
4
6
2
5
2
3
4
5
3
4
2
1
3
5
1
6
5
2
5
3
4
3
6
5
1
6
60
Soln.12
6
1
6
2
2
4
1
4
3
1
5
2
3
2
1
5
4
4
2
6
3
1
5
5
3
2
5
4
3
4
3
6
1
6
5
6
60
Soln.13
5
1
4
3
3
6
2
5
1
4
4
6
1
3
5
4
1
2
5
3
5
6
1
6
2
2
2
6
3
3
4
1
2
5
6
4
61
Soln.14
3
5
1
2
1
1
3
5
5
3
6
4
6
3
2
3
6
5
6
1
2
4
1
4
2
5
3
4
2
6
1
4
4
6
2
5
61
Soln.15
1
1
4
3
3
1
2
5
6
3
2
4
5
4
5
3
6
5
3
6
1
1
5
6
2
3
6
4
6
1
2
2
4
5
2
4
61
Soln.16
3
2
1
1
4
5
1
2
3
6
6
6
1
5
3
6
3
2
4
4
5
5
6
1
3
5
2
4
2
1
4
3
6
4
5
2
61
Soln.17
4
2
5
3
6
4
5
1
3
6
1
6
1
1
3
5
4
3
2
5
5
4
1
2
3
4
5
6
1
2
3
6
2
2
4
6
61
Soln.18
1
5
2
3
1
6
6
5
1
3
1
5
2
6
3
4
4
4
3
5
2
3
6
5
2
4
6
3
1
2
4
1
5
6
4
2
61
Soln.19
2
1
4
3
3
6
5
5
2
6
5
3
6
1
2
5
2
4
3
2
4
1
5
6
3
4
1
1
4
5
2
3
1
4
6
6
61
Soln.20
2
5
3
1
6
3
4
5
4
1
1
5
4
2
2
3
2
5
3
2
5
6
6
1
2
4
3
3
6
1
1
4
4
6
6
5
61
159
PROBLEM S4
Soln.1
6
3
3
4
6
6
1
6
6
3
4
4
1
5
4
3
3
2
5
3
1
1
4
6
2
2
1
5
4
1
2
2
5
5
2
5
48
Soln.2
4
4
1
4
5
1
5
4
1
3
1
6
4
2
2
2
6
6
3
5
5
6
3
2
4
5
3
1
1
3
2
6
6
2
5
3
50
Soln.3
6
1
4
4
6
4
1
1
1
3
5
2
3
6
1
6
4
5
4
2
4
5
2
3
5
6
3
2
5
3
6
3
2
2
5
1
50
Soln.4
6
1
4
1
2
3
5
4
6
4
5
1
4
3
2
6
3
4
1
2
2
4
5
1
3
5
5
1
6
3
6
2
3
6
5
2
50
Soln.5
3
1
4
4
1
6
5
4
6
1
6
3
1
4
2
6
5
2
6
4
1
3
2
3
4
5
2
5
2
5
1
6
5
3
2
3
50
Soln.6
1
6
4
4
4
1
3
1
5
5
4
2
4
1
3
6
2
5
4
1
6
6
2
5
2
5
1
6
2
3
3
2
3
5
6
3
50
Soln.7
4
1
5
4
5
1
3
6
4
4
2
6
1
5
6
3
4
1
1
6
4
2
2
2
6
5
3
3
2
5
5
1
3
3
2
6
50
Soln.8
4
1
6
1
4
5
6
4
1
2
4
6
3
4
2
5
2
1
1
3
1
5
2
6
2
4
5
3
5
5
6
3
3
6
3
2
50
Soln.9
4
3
4
6
1
5
4
1
2
1
6
5
4
4
1
2
5
6
2
4
2
3
6
5
5
6
3
5
3
1
2
3
1
2
6
3
50
Soln.10
1
6
4
4
4
3
6
4
1
2
1
5
3
5
6
5
2
4
5
6
1
4
1
3
2
1
6
3
2
5
5
2
3
3
2
6
50
160
Soln.11
6
1
4
4
5
1
4
5
3
1
1
6
6
4
1
6
4
2
5
2
2
2
3
3
2
4
5
6
5
3
2
3
3
1
6
5
50
Soln.12
3
4
1
1
4
5
4
6
6
5
6
4
1
2
3
3
6
2
1
4
5
6
2
4
5
3
1
2
1
5
2
6
2
5
3
3
50
Soln.13
4
4
5
1
6
3
4
1
1
6
6
1
2
4
4
5
2
6
3
4
6
2
3
1
3
1
5
2
3
5
2
5
6
3
5
2
50
Soln.14
5
4
6
4
1
4
4
1
5
1
3
5
1
2
6
4
2
1
4
1
6
3
5
3
3
6
6
3
2
2
5
3
2
2
6
5
50
Soln.15
1
4
4
4
6
5
1
1
2
6
3
4
2
4
6
5
3
2
5
3
1
6
5
1
2
4
5
3
6
3
2
6
1
5
3
2
50
Soln.16
3
6
6
4
1
6
3
2
5
2
6
4
4
2
5
6
2
1
4
1
3
1
4
3
5
4
5
5
6
5
3
2
2
1
1
3
50
Soln.17
6
4
5
6
1
4
6
2
6
4
3
1
2
1
2
4
6
3
5
5
1
3
4
2
3
6
2
5
1
1
3
2
5
4
5
3
50
Soln.18
4
1
4
6
5
6
4
1
5
4
3
1
1
2
2
5
4
2
2
4
3
6
3
6
1
5
3
6
5
2
3
6
5
2
3
1
50
Soln.19
4
3
3
6
1
4
5
5
4
4
4
1
1
6
2
6
5
1
2
2
6
5
2
4
1
2
3
6
3
3
5
1
5
3
2
6
50
Soln.20
3
5
4
4
5
1
1
3
6
6
4
1
4
6
1
6
2
4
2
3
5
1
6
5
4
3
3
2
2
6
5
1
2
5
3
2
50
161
PROBLEM S5
Soln.1
3
4
3
3
6
4
1
1
5
2
3
3
4
2
2
5
6
6
4
5
6
4
5
5
4
3
6
1
6
1
5
1
2
2
1
2
55
Soln.2
4
4
3
3
2
5
6
3
3
2
1
1
5
1
3
4
5
2
3
6
2
6
6
2
4
5
5
1
6
1
2
4
5
4
1
6
55
Soln.3
3
3
3
2
2
3
2
1
5
5
4
1
6
4
4
1
6
5
3
3
1
5
6
2
6
2
4
5
2
6
5
4
4
1
1
6
55
Soln.4
2
3
3
6
5
3
1
1
4
2
4
3
5
4
6
1
2
3
3
6
5
5
2
6
1
5
2
6
4
4
4
1
1
2
6
5
55
Soln.5
2
3
3
4
3
1
6
3
2
5
1
2
3
5
4
4
5
6
5
1
2
5
2
6
4
1
6
1
1
3
4
6
4
2
6
5
55
Soln.6
3
4
1
3
4
1
5
5
3
4
3
2
6
2
3
6
2
5
1
2
3
6
5
4
6
5
2
1
1
5
1
4
2
6
6
4
55
Soln.7
3
4
2
1
3
5
5
1
4
3
2
6
3
4
5
3
2
1
2
3
6
6
4
1
5
1
4
2
5
6
6
1
5
6
2
4
55
Soln.8
2
3
1
3
4
3
1
3
5
3
2
6
4
1
2
5
6
4
6
2
5
3
6
4
5
1
5
2
1
4
6
5
2
6
1
4
55
Soln.9
4
3
1
2
4
3
3
3
5
2
2
4
6
1
1
6
3
5
2
4
5
6
3
1
4
5
6
2
2
1
5
6
6
1
4
5
55
Soln.10
3
3
4
3
4
3
1
5
2
6
3
2
5
1
4
1
5
6
2
6
1
3
2
6
5
4
4
5
2
6
1
1
6
5
2
4
55
162
Soln.11
4
3
3
1
3
1
2
4
2
5
6
1
5
3
4
2
3
2
5
6
5
4
5
1
2
6
3
1
6
6
1
4
2
5
4
6
55
Soln.12
4
3
3
6
4
3
2
1
4
3
2
5
6
1
2
3
1
6
5
5
6
2
5
4
2
1
4
5
3
2
1
6
6
1
4
5
55
Soln.13
2
3
3
3
6
4
5
3
1
3
2
4
1
4
5
2
1
3
6
6
2
1
2
1
5
5
1
6
4
6
4
5
4
6
2
5
55
Soln.14
3
1
1
3
2
6
4
3
3
5
5
1
2
2
2
4
3
6
6
4
5
2
4
5
1
6
6
1
5
1
4
4
3
6
2
5
55
Soln.15
3
4
2
4
3
6
3
3
1
4
5
2
3
1
5
6
2
5
4
1
6
4
6
1
5
4
2
5
5
6
2
1
3
2
6
1
55
Soln.16
3
2
1
3
4
5
3
5
2
4
3
1
6
2
4
3
6
6
1
5
5
2
5
1
3
6
1
2
4
6
2
5
1
4
4
6
55
Soln.17
5
2
1
3
1
2
4
3
4
5
2
3
6
6
3
5
6
3
1
4
6
2
5
1
2
3
5
6
6
4
1
4
1
2
4
5
58
Soln.18
2
3
2
3
3
2
1
3
4
5
6
5
1
1
5
2
4
2
6
4
4
5
6
6
3
5
3
1
1
4
6
5
4
2
1
6
58
Soln.19
1
5
2
3
3
2
6
1
3
4
5
2
1
3
4
3
4
2
6
5
5
6
6
1
1
2
6
3
4
4
5
6
1
2
4
5
58
Soln.20
5
3
1
3
2
4
2
3
2
1
5
6
3
4
5
3
4
1
6
5
6
2
1
5
4
6
4
5
2
6
1
3
4
2
1
6
58
163
PROBLEM S6
Soln.1
2
6
3
6
5
2
4
3
1
1
2
6
3
1
6
2
4
3
4
5
4
2
2
6
4
1
3
5
1
5
6
4
3
1
5
5
54
Soln.2
6
3
2
4
6
5
1
6
4
1
3
5
2
2
1
3
4
3
2
4
6
2
6
1
4
3
1
5
2
6
4
5
5
3
1
5
54
Soln.3
2
6
4
1
5
3
2
6
2
6
4
2
1
4
3
1
5
3
6
2
3
6
4
3
1
4
6
1
2
4
5
1
5
5
3
5
54
Soln.4
3
6
2
1
6
1
4
6
5
2
2
4
4
3
3
1
3
2
6
5
6
4
2
3
2
5
1
6
3
4
1
5
5
4
1
5
54
Soln.5
6
5
2
2
3
4
1
6
3
4
6
2
3
4
3
1
6
2
5
6
1
3
6
5
2
2
1
1
4
4
5
4
3
1
5
5
54
Soln.6
6
2
3
4
1
2
6
5
4
3
6
2
1
3
1
6
5
4
6
2
3
2
1
3
4
4
1
6
2
1
5
3
5
5
4
5
54
Soln.7
6
2
4
2
3
5
3
6
2
5
6
1
1
2
3
1
6
2
4
4
4
6
3
3
1
4
5
2
6
1
5
3
4
5
5
1
54
Soln.8
5
2
3
6
4
2
1
6
3
2
6
5
2
1
3
4
2
4
6
2
1
4
3
6
5
1
4
6
3
1
1
3
4
5
5
5
54
Soln.9
6
2
3
1
6
2
4
6
3
5
4
2
1
3
4
6
2
3
5
1
2
2
1
6
4
3
5
4
6
3
1
1
5
5
4
5
54
Soln.10
2
6
4
3
6
2
1
3
5
2
6
2
1
1
4
3
5
2
4
6
3
2
6
4
3
1
6
5
4
1
3
5
4
5
5
1
54
164
Soln.11
3
6
5
2
2
4
6
1
2
3
3
6
2
5
1
1
4
6
4
2
1
4
5
3
6
4
3
6
1
2
5
3
4
5
5
1
54
Soln.12
3
2
6
6
2
1
6
5
4
4
2
1
3
2
4
5
1
2
3
6
6
2
3
5
1
3
4
3
1
5
6
4
1
5
4
5
54
Soln.13
6
2
1
3
2
5
4
6
6
2
3
3
6
4
4
6
3
2
1
5
3
2
6
1
5
2
4
1
1
4
5
4
5
3
1
5
54
Soln.14
6
4
3
2
6
1
6
2
5
3
4
2
1
4
5
3
1
2
6
4
2
1
6
2
3
4
4
3
5
1
6
5
5
1
3
5
54
Soln.15
6
3
6
5
2
4
1
3
2
4
4
2
1
5
3
6
1
3
2
2
4
1
6
5
6
3
4
1
2
6
1
4
5
3
5
5
54
Soln.16
6
2
2
5
3
6
4
1
2
1
3
1
4
6
3
4
3
5
6
2
2
1
4
4
2
5
6
4
3
3
1
5
6
1
5
5
54
Soln.17
6
2
3
4
5
2
2
3
6
3
1
4
5
6
1
2
1
5
2
6
2
4
6
1
5
3
1
5
4
4
6
1
4
5
3
3
56
Soln.18
1
5
3
6
5
2
4
1
2
1
5
2
4
6
5
3
3
4
6
5
2
1
2
6
4
5
2
1
4
6
3
6
1
3
4
3
58
Soln.19
1
5
1
6
5
3
5
4
2
1
2
5
3
6
4
2
5
3
4
6
3
2
5
6
1
4
1
6
4
1
3
2
3
2
4
6
58
Soln.20
4
3
1
5
1
2
5
2
6
4
5
1
2
5
3
1
4
5
6
5
1
6
4
3
3
2
6
6
2
4
2
3
6
3
4
1
58
165
PROBLEM S7
Soln.1
1
6
1
1
2
3
4
5
5
1
6
3
6
1
6
3
2
4
4
5
5
5
6
2
1
6
3
2
4
2
5
4
3
3
2
4
51
Soln.2
3
6
1
1
1
5
5
6
3
1
1
5
2
6
3
4
3
5
6
2
6
2
4
4
5
4
2
3
2
5
6
3
1
4
4
2
51
Soln.3
2
6
1
5
1
3
1
3
1
6
5
6
3
2
4
6
2
1
5
4
5
3
4
6
4
1
2
3
4
3
2
2
5
5
6
4
51
Soln.4
6
2
3
3
1
1
1
5
5
6
4
1
1
6
6
3
2
5
2
5
5
5
3
3
6
4
2
4
1
4
4
3
2
4
6
2
51
Soln.5
1
1
6
1
5
2
3
5
1
6
1
6
2
5
4
6
3
3
4
2
5
1
2
4
5
3
3
5
6
4
4
6
3
4
2
2
51
Soln.6
5
1
3
2
6
1
1
6
5
3
2
4
1
6
1
5
5
6
2
3
2
1
4
3
6
4
4
4
5
2
2
5
3
6
4
3
51
Soln.7
3
2
1
5
1
1
5
2
6
1
3
4
6
2
6
1
3
4
5
6
5
4
3
5
6
2
4
1
3
3
5
4
2
2
4
6
51
Soln.8
1
2
6
3
1
5
1
4
3
6
5
1
1
5
2
2
6
3
5
1
4
6
2
3
6
4
5
4
2
4
5
6
3
3
2
4
51
Soln.9
3
5
1
2
6
3
1
1
5
6
2
1
4
1
6
2
4
3
6
5
3
3
4
6
4
2
4
2
1
5
3
6
2
5
5
4
51
Soln.10
1
3
2
6
6
1
1
5
1
6
5
4
4
1
3
2
2
6
3
5
5
6
4
5
1
5
3
3
2
4
4
2
2
3
4
6
51
166
Soln.11
1
5
6
1
2
4
1
5
3
2
1
6
6
6
3
2
4
3
1
5
5
3
5
2
3
6
4
3
5
1
4
2
6
4
4
2
51
Soln.12
1
1
3
1
5
6
2
1
6
5
3
4
1
6
4
3
5
6
2
6
4
2
1
2
5
3
6
4
5
5
3
2
4
3
2
4
51
Soln.13
6
3
1
2
6
3
1
5
1
5
1
4
1
6
2
4
6
3
2
5
3
6
4
3
6
5
5
1
2
5
4
3
4
4
2
2
51
Soln.14
1
2
6
1
1
6
5
3
1
3
1
5
6
4
3
6
5
6
5
4
2
2
5
4
3
4
5
2
6
3
3
1
4
2
4
2
51
Soln.15
6
1
1
3
1
5
2
6
5
1
6
4
4
2
3
1
4
5
6
5
3
6
5
3
4
6
3
2
5
2
4
2
1
4
2
3
51
Soln.16
1
6
5
6
1
1
1
3
3
5
6
1
2
5
4
3
2
6
4
2
4
5
6
3
5
6
1
3
5
2
4
2
4
2
3
4
51
Soln.17
3
1
2
1
6
1
4
6
5
1
4
5
1
2
6
3
2
5
1
3
6
4
5
3
2
3
6
3
4
5
6
2
5
4
4
2
51
Soln.18
1
1
1
5
1
3
6
6
3
5
2
1
6
4
2
5
4
6
2
6
5
3
3
2
4
1
4
2
5
3
4
3
5
4
6
2
51
Soln.19
1
5
1
3
6
6
3
2
1
5
1
6
1
4
6
5
3
3
2
2
5
5
4
4
6
4
2
1
3
4
2
6
5
2
3
4
51
Soln.20
1
5
6
3
6
1
1
3
1
6
1
2
3
5
5
4
6
3
6
2
4
3
2
4
5
1
4
6
2
3
4
2
2
5
5
4
51
167
PROBLEM S8
Soln.1
2
4
3
2
1
5
2
6
3
5
1
3
4
4
6
5
1
3
6
3
2
1
4
1
6
5
6
3
2
4
5
5
6
1
4
2
67
Soln.2
3
6
3
2
6
3
4
1
1
5
2
2
2
4
2
6
5
3
6
5
4
1
1
5
6
1
3
4
5
4
6
5
2
1
4
3
67
Soln.3
2
2
6
2
3
1
5
4
5
6
1
4
3
3
3
5
4
4
2
1
6
6
3
5
2
1
1
4
1
5
5
3
6
6
2
4
67
Soln.4
1
2
6
4
2
1
3
2
3
5
5
5
3
4
2
6
2
4
1
2
3
4
1
5
4
6
3
6
1
6
1
5
4
6
5
3
67
Soln.5
1
2
4
1
2
2
3
5
6
6
5
3
1
3
2
4
5
4
3
2
5
4
6
1
2
3
6
5
4
1
6
3
4
5
6
1
67
Soln.6
2
4
6
3
1
5
5
2
6
3
1
4
2
2
2
3
5
4
5
2
3
6
1
4
3
1
1
3
6
4
6
6
4
1
5
5
67
Soln.7
6
3
2
6
1
5
2
5
1
4
5
3
3
5
3
2
4
2
6
6
1
4
1
1
4
3
5
2
3
6
1
5
2
6
4
4
67
Soln.8
2
6
2
1
3
5
2
3
1
2
4
4
5
3
6
5
4
5
3
1
1
6
4
3
4
2
1
6
3
5
1
6
2
5
6
4
67
Soln.9
6
6
3
1
5
1
3
2
2
5
4
5
3
6
3
5
1
2
4
2
6
4
6
1
3
4
2
4
3
1
5
1
2
6
4
5
67
Soln.10
3
6
2
1
1
3
6
2
3
3
5
5
4
5
2
4
5
6
6
4
1
2
4
1
6
2
5
1
6
5
2
3
4
3
1
4
67
168
Soln.11
6
4
1
2
4
3
5
1
5
2
4
3
3
5
4
6
2
2
3
6
1
3
6
1
6
1
5
4
4
5
2
6
3
2
1
5
67
Soln.12
6
2
3
1
5
1
3
2
6
3
5
3
2
4
2
2
6
6
4
5
1
5
4
4
5
6
1
4
1
1
5
3
6
2
4
3
67
Soln.13
3
2
4
6
1
5
4
5
3
3
1
6
3
2
2
4
4
2
6
5
6
5
5
4
2
1
3
1
1
6
3
1
5
6
2
4
67
Soln.14
6
1
4
1
4
2
5
5
4
3
3
4
2
2
3
5
6
6
3
4
5
1
2
1
4
3
6
2
6
3
1
5
2
5
1
6
67
Soln.15
3
2
2
1
3
6
1
6
4
5
5
2
2
5
2
1
3
4
6
4
3
6
4
6
3
1
1
5
4
6
3
5
1
2
5
4
67
Soln.16
3
4
2
3
1
3
5
6
2
5
5
1
4
2
5
6
3
4
6
1
2
1
4
6
1
6
2
5
3
4
1
2
5
3
6
4
67
Soln.17
3
1
2
6
2
5
4
6
3
4
1
5
3
2
5
2
4
6
1
3
2
4
6
5
1
5
1
3
4
3
6
5
6
1
4
2
67
Soln.18
3
3
1
2
1
2
4
5
4
5
6
3
6
3
5
4
4
1
3
2
5
1
6
4
2
2
6
1
6
4
6
2
5
3
5
1
67
Soln.19
3
2
6
1
4
5
2
1
2
2
3
3
5
6
4
4
1
3
4
6
5
1
3
3
5
2
4
6
1
6
5
1
6
2
4
5
67
Soln.20
2
3
4
1
6
2
2
3
5
1
5
3
4
4
3
4
6
1
5
5
2
6
2
6
1
6
3
5
3
1
4
5
2
6
4
1
67
169
PROBLEM S9
Soln.1
1
4
2
3
6
5
5
4
2
3
1
6
6
4
1
5
2
1
4
3
2
4
6
5
5
4
1
6
2
3
6
3
5
2
1
3
54
Soln.2
2
5
1
4
2
5
6
6
5
1
2
4
4
4
1
4
4
6
3
1
3
5
6
1
3
2
6
1
6
3
5
5
2
3
2
3
54
Soln.3
1
4
3
3
2
6
4
5
2
1
5
6
4
1
5
5
2
3
1
4
3
4
6
6
6
4
1
2
6
5
2
3
5
2
1
3
54
Soln.4
1
2
6
5
3
4
5
3
6
4
1
2
1
5
2
5
4
3
1
4
6
1
4
6
3
6
5
4
2
2
6
3
1
3
5
2
54
Soln.5
1
3
2
4
4
5
6
6
1
6
4
1
2
5
4
3
5
3
2
4
5
1
6
1
2
6
3
5
6
4
1
3
2
5
2
3
54
Soln.6
1
2
6
5
6
3
4
6
4
1
4
3
3
2
1
2
4
5
5
4
1
1
6
4
5
3
5
6
6
1
5
2
3
2
3
2
54
Soln.7
3
6
1
6
2
1
4
5
5
3
2
6
4
3
1
5
1
5
4
4
2
6
3
4
5
4
6
1
6
5
2
1
3
2
2
3
54
Soln.8
2
1
3
4
3
6
2
1
4
6
4
5
1
5
5
6
5
1
4
6
2
4
3
2
1
2
6
6
3
5
5
4
1
3
2
3
54
Soln.9
3
4
5
1
4
3
1
4
3
1
6
2
2
6
6
5
5
2
4
3
1
5
4
4
6
1
5
3
6
6
5
1
3
2
2
2
54
Soln.10
3
5
1
2
6
4
4
2
6
1
6
5
4
2
5
1
3
4
1
5
6
4
2
3
4
6
5
3
6
1
1
5
2
3
2
3
54
170
Soln.11
1
3
2
1
5
6
5
4
2
4
4
6
1
2
3
6
5
5
4
1
2
4
3
6
5
1
2
6
4
3
1
6
5
3
2
3
54
Soln.12
4
5
2
2
5
1
6
4
1
2
1
5
5
4
6
2
4
1
5
3
6
6
6
1
3
4
3
5
4
1
2
3
3
6
2
3
54
Soln.13
1
4
4
1
5
4
2
2
5
6
5
2
6
1
5
4
1
6
3
5
6
1
3
6
3
4
2
6
5
2
4
3
2
3
1
3
54
Soln.14
5
2
6
5
1
4
2
1
5
6
4
3
3
2
1
6
5
4
4
1
6
2
3
1
5
4
3
3
5
6
4
3
6
1
2
2
54
Soln.15
2
1
4
5
1
5
2
6
4
1
4
5
5
6
1
2
2
3
4
5
6
6
3
5
6
1
1
3
4
3
3
2
4
6
2
3
54
Soln.16
2
3
4
1
2
5
6
5
4
3
1
1
5
6
4
5
1
2
6
4
4
3
5
2
4
6
1
6
3
3
1
6
3
2
5
2
54
Soln.17
2
3
5
5
4
4
1
1
3
2
4
5
1
1
3
6
6
4
2
5
4
6
6
3
4
1
6
5
3
2
6
2
1
3
5
2
54
Soln.18
1
3
4
2
3
5
1
5
6
6
3
2
4
1
5
2
6
5
4
1
3
6
6
6
4
1
4
5
4
2
3
2
5
1
2
3
54
Soln.19
4
2
5
1
1
5
4
2
6
4
5
6
2
3
1
4
4
6
1
1
5
3
6
2
4
3
5
1
6
6
2
5
3
3
3
2
54
Soln.20
1
6
5
2
4
3
6
3
2
5
3
6
4
1
4
2
5
1
4
5
1
4
3
6
5
6
3
6
5
4
1
2
1
3
2
2
54
171
PROBLEM S10
Soln.1
1
4
6
4
5
3
1
4
5
6
5
3
5
1
2
1
3
3
4
1
2
6
2
4
2
3
6
4
2
2
5
1
3
5
6
6
60
Soln.2
3
5
1
4
6
1
6
4
5
5
2
6
4
2
3
3
1
1
4
5
2
4
1
2
2
6
3
3
5
5
4
1
6
2
3
6
60
Soln.3
3
4
6
4
6
5
2
6
3
1
4
2
5
1
1
4
2
2
4
1
5
5
6
6
4
6
3
3
1
3
5
1
2
5
3
2
60
Soln.4
6
4
4
1
3
5
4
1
2
3
6
6
1
4
1
2
4
3
5
6
3
5
3
5
2
2
6
4
2
5
5
6
1
1
2
3
60
Soln.5
6
3
4
4
2
5
6
4
1
6
5
4
3
2
1
1
4
1
5
6
3
5
2
3
1
2
1
3
5
4
6
2
2
3
5
6
60
Soln.6
1
4
5
1
6
5
3
4
2
1
5
3
4
6
1
5
3
1
4
2
3
4
5
2
6
6
6
3
3
6
2
1
4
2
2
5
60
Soln.7
4
1
5
1
6
4
5
1
3
3
5
6
4
1
6
2
1
2
4
2
6
3
6
5
1
2
3
4
5
2
5
2
6
3
4
3
60
Soln.8
6
3
4
5
4
1
6
2
2
4
5
2
3
4
4
1
3
1
6
2
1
5
6
3
1
6
5
3
6
2
5
5
4
2
3
1
60
Soln.9
3
5
6
4
1
4
5
5
3
1
3
6
4
2
5
6
3
1
1
1
1
4
2
3
2
6
4
3
2
4
5
6
2
2
5
6
60
Soln.10
4
6
1
6
5
4
4
2
1
3
3
6
2
1
5
4
1
6
2
3
4
5
3
2
5
2
1
6
3
5
1
4
2
3
6
5
60
172
Soln.11
4
3
6
5
6
4
4
2
1
4
3
5
1
3
6
1
5
1
2
1
2
3
4
6
5
2
2
5
3
6
6
2
5
3
1
4
60
Soln.12
4
3
6
5
4
6
1
4
3
5
5
2
3
4
5
1
1
2
1
6
5
3
1
2
6
6
3
4
2
2
3
5
6
4
1
2
60
Soln.13
4
6
5
5
6
1
4
1
3
5
4
3
2
5
1
6
3
1
2
1
1
3
2
4
4
2
5
3
6
6
4
2
5
2
3
6
60
Soln.14
5
6
1
4
5
4
3
2
3
6
3
1
1
2
1
5
4
2
2
1
6
1
5
3
4
3
6
5
6
2
4
4
2
3
5
6
60
Soln.15
3
1
5
4
6
4
4
1
4
3
6
2
1
3
2
6
3
5
6
5
2
1
4
3
2
2
6
4
5
1
5
5
6
1
3
2
60
Soln.16
1
4
6
4
2
3
6
5
2
4
3
5
4
6
1
4
3
2
5
2
6
2
6
1
1
5
2
1
1
6
3
5
5
3
4
3
60
Soln.17
6
4
5
3
6
4
1
3
2
2
4
1
4
2
1
6
5
5
1
6
3
2
4
1
2
3
5
3
4
6
3
5
6
2
5
1
60
Soln.18
6
4
6
4
3
2
5
5
3
3
6
1
2
4
4
1
4
6
6
1
5
1
1
4
5
1
2
2
3
6
2
3
2
3
5
5
60
Soln.19
3
4
6
1
5
4
6
4
1
2
3
6
1
5
5
3
2
4
6
2
4
3
1
1
6
2
1
5
3
5
3
4
2
5
2
6
60
Soln.20
3
4
1
6
5
4
6
4
2
4
6
6
2
2
5
1
1
3
5
1
4
3
2
6
3
2
5
1
5
4
3
6
3
1
2
5
60
173
APPENDIX C REGRESSION ANALYSIS
Job
6
5
2
1
3
4
6
5
2
1
3
4
6
3
5
2
1
4
5
1
3
2
6
4
2
1
5
4
6
3
1
6
2
4
5
3
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
2
PT
1
4
2
9
3
4
7
9
9
6
5
4
10
1
6
1
1
3
5
6
7
7
7
1
1
3
8
6
6
8
6
2
7
2
2
1
Score Val
2.7809
3.3741
5.5528
5.8895
5.9939
7.9127
8.7158
9.8054
11.0757
11.1487
11.873
13.8055
14.8286
15.5257
15.5638
15.8796
16.3372
18.1762
20.5591
20.7916
21.0446
21.3985
21.3985
21.9605
25.8764
26.2787
26.5049
27.2469
27.2885
27.3164
31.5583
31.7111
31.8113
32.1271
32.1687
32.2901
Job
2
6
3
1
4
5
2
6
3
1
4
5
4
2
6
1
3
5
4
6
1
3
2
5
3
6
1
2
4
5
3
1
6
5
2
4
Problem 3
Op
PT
1
6
1
7
1
9
1
1
1
2
1
3
2
8
2
8
2
10
2
1
2
1
2
8
3
1
3
10
3
5
3
9
3
8
3
4
4
9
4
6
4
7
4
2
4
5
4
2
5
4
5
6
5
3
5
4
5
6
5
1
6
7
6
6
6
6
6
4
6
6
6
5
Score Val
1.9032
2.448
2.4548
5.4035
6.5313
7.5548
7.9704
8.8862
9.268
9.4442
10.527
13.4658
14.4429
14.8286
15.0395
15.0953
16.32
18.8874
20.2188
20.3539
20.9198
21.027
21.0999
23.1504
25.6786
25.7065
26.2787
26.5733
26.6012
27.3125
31.3121
31.3919
31.4751
31.7179
31.7247
31.7629
Job
3
4
6
5
1
2
3
4
5
6
2
1
5
3
4
6
2
1
5
3
4
1
6
2
5
4
1
3
2
6
5
4
2
3
6
1
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
4
PT
5
1
4
6
2
5
5
6
1
4
2
9
1
9
9
9
8
3
7
3
3
2
4
4
6
6
4
9
1
5
7
4
4
2
1
4
Score Val
3.7941
4.3213
5.6434
5.7129
6.3649
7.4784
9.084
9.2333
10.6102
10.6626
11.6548
12.0126
14.4013
15.3455
16.0738
16.7611
17.0284
17.7391
19.3794
20.7392
21.1763
21.9639
22.4289
22.7412
25.8102
26.4975
26.5739
27.0907
27.3125
28.0966
31.0625
31.6347
31.6763
31.8775
32.0405
32.1755
174
Job
6
5
1
2
3
4
6
5
1
4
2
3
6
5
4
1
3
2
6
5
4
1
3
2
6
1
4
5
3
2
1
6
4
5
3
2
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
5
PT
4
5
1
3
7
4
6
6
10
2
9
7
2
7
5
2
2
6
8
3
2
3
7
5
8
4
6
7
4
1
7
7
5
4
1
2
Score Val
2.8954
4.148
5.1955
5.848
6.1534
6.7883
8.5876
9.712
10.7879
11.7169
11.9915
12.3175
13.2182
15.8795
16.2682
16.5697
17.0695
17.8747
18.5502
20.9472
21.1102
21.1552
22.3354
23.4524
25.1098
25.5954
26.2268
26.4177
27.8852
27.8955
31.3121
31.3953
31.4717
31.6347
32.1237
32.2103
Job
1
3
6
2
1
5
4
6
3
5
2
4
1
3
4
5
6
2
5
1
3
4
6
2
5
4
3
1
6
2
5
1
3
6
4
2
Problem
Op
1
1
1
1
2
1
1
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
PT
10
7
7
5
2
8
5
6
7
1
9
2
10
2
2
5
6
9
2
10
9
4
8
7
6
6
6
7
5
2
10
7
8
4
8
1
Score Val
1.1463
2.3021
3.7382
4.0437
5.3057
5.7191
6.5831
8.5466
9.0902
10.0272
10.5132
11.3003
12.1843
12.7606
14.5301
14.7067
15.3136
17.6347
18.7999
18.8693
19.0323
19.8473
21.7136
23.6672
24.9149
25.2067
25.5811
25.772
26.7853
27.6704
31.3223
31.4369
31.6899
31.9259
32.0227
32.4149
Job
5
2
4
3
1
6
4
5
1
2
6
3
4
6
5
3
1
2
5
4
6
3
1
2
5
2
4
3
6
1
5
3
2
6
4
1
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
7
PT
9
10
7
9
1
1
1
7
7
9
4
9
9
1
8
2
8
9
2
6
9
8
3
2
9
3
6
1
5
10
7
8
7
7
7
1
Score Val
2.1214
2.8121
3.4886
3.6413
4.841
5.2787
8.1329
8.2787
9.1951
9.3062
9.6841
10.638
13.4928
14.4218
14.4673
15.3416
15.7581
16.6773
19.2575
20.1248
20.2393
21.256
21.3216
21.7137
25.1964
25.8416
25.8518
25.8969
25.9732
26.9072
31.3121
31.3987
31.6865
31.7281
31.9361
32.7061
175
Job
2
5
4
3
1
2
5
4
6
3
1
2
4
6
5
3
1
6
2
5
4
3
6
1
2
6
5
3
4
1
4
5
3
1
6
2
Problem
Op
1
1
1
1
1
2
2
2
1
2
2
3
3
2
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
8
PT
8
1
1
1
6
7
9
6
4
3
1
5
3
1
4
8
6
1
7
6
10
4
2
6
8
1
10
6
6
2
2
1
1
1
4
4
Score Val
1.8274
3.4471
4.384
6.8191
8.1275
8.6537
9.3894
9.712
10.2652
11.0752
12.6504
14.1448
14.9911
15.1277
15.182
16.487
17.6872
19.2516
19.5253
20.375
20.9927
22.283
23.3168
23.3726
26.2547
26.8965
27.032
27.1432
27.2885
27.7952
31.7111
31.7493
31.7493
31.7493
31.8843
31.9259
Job
3
6
4
2
5
1
3
6
4
2
5
1
3
4
6
2
1
5
4
3
2
1
5
6
3
4
1
2
6
5
3
1
2
4
5
6
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
9
PT
3
7
6
5
3
8
2
3
4
7
8
2
6
1
10
1
1
1
6
8
2
5
9
8
2
6
2
6
2
1
8
6
4
6
1
1
Score Val
4.7031
4.7589
5.4422
6.6669
7.2009
7.3855
9.0943
9.993
10.5794
12.6714
12.8623
13.1325
14.5437
15.4419
16.369
17.1077
17.1282
17.42
19.8131
20.6941
21.1729
21.4333
22.9047
23.1714
25.4843
25.5401
26.1921
26.6223
27.9616
28.291
31.0659
31.3919
31.5515
31.5999
32.0405
32.2069
Job
5
4
1
2
3
6
4
5
2
1
3
4
6
2
5
1
3
4
6
2
5
1
3
6
4
2
1
5
3
6
4
2
3
5
1
6
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
3
2
3
3
3
3
4
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
10
PT
7
5
9
8
9
9
8
10
2
4
3
1
9
5
10
8
1
9
3
7
4
3
9
1
6
2
6
6
2
1
7
5
2
1
1
1
Score Val
2.5728
2.6076
5.6399
6.1568
7.2014
7.618
8.7614
9.2264
10.9054
10.9333
12.8447
13.9015
14.6979
15.9975
16.5354
16.9241
17.0034
19.511
20.2164
22.002
22.1998
22.654
22.7377
24.4999
25.727
26.8788
27.6424
27.8088
28.1069
28.6238
31.3537
31.6797
31.8775
32.0405
32.3317
32.5813
176
Job
6
4
2
1
3
5
6
4
5
3
2
1
6
4
5
3
1
2
4
6
5
3
1
2
6
4
5
3
2
1
6
2
5
3
1
4
Problem S1
Op
PT
1
7
1
6
1
9
1
9
1
1
1
2
2
1
2
4
2
4
2
6
2
7
2
7
3
9
3
5
3
7
3
3
3
1
3
7
4
6
4
4
4
2
4
6
4
8
4
1
5
4
5
7
5
6
5
4
5
4
5
4
6
8
6
4
6
5
6
5
6
2
6
4
Score Val
4.0095
4.6717
5.0364
5.5983
5.6953
6.0315
9.1114
9.9337
10.4751
10.7737
11.6097
11.9636
14.8457
14.8526
15.9422
16.1776
16.6495
17.6074
20.167
20.4303
20.819
21.374
22.0681
22.5429
25.5743
26.3345
26.3516
26.4696
26.5323
27.4481
31.3571
31.7179
31.8045
31.8045
31.8359
32.0091
Job
6
4
3
5
1
2
4
6
3
5
2
1
6
4
3
2
5
1
6
3
4
5
2
1
3
6
4
5
2
1
3
5
6
2
4
1
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S2
PT
10
10
8
7
9
4
1
4
3
4
5
10
4
9
10
7
9
3
9
1
10
3
7
1
6
5
5
2
1
8
10
7
10
5
6
1
Score Val
1.9379
2.5414
3.0965
4.7378
4.9532
5.2268
7.3419
7.5607
8.4104
10.662
10.8113
12.3488
12.7047
13.1178
14.6616
16.3998
16.5525
17.9887
19.0944
19.4205
20.347
21.9046
22.3143
22.4802
24.8733
24.9115
26.0359
26.4622
27.4994
27.774
30.9063
31.2289
31.3223
31.5549
31.8495
32.3733
Job
4
3
2
5
1
4
6
3
1
5
2
6
4
3
1
5
2
6
4
1
5
3
6
2
4
1
5
2
6
3
4
5
2
3
1
6
Problem
Op
1
1
1
1
1
2
1
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S3
PT
2
4
10
7
4
5
10
9
1
4
10
1
9
5
8
7
6
3
8
1
1
9
1
5
6
10
7
1
10
9
10
10
10
4
8
4
Score Val
0.9937
2.0411
2.1875
3.7382
4.5395
6.2106
7.0794
7.0996
8.4036
8.8304
9.2469
12.1295
12.5969
13.6655
13.8222
14.1727
15.2515
16.0528
18.2795
18.9207
19.233
19.3862
20.1283
20.7044
24.8317
24.9291
25.0015
25.1059
25.9703
26.4661
30.9063
31.2807
31.6135
31.8843
31.9395
32.3003
177
Job
1
4
6
2
3
5
4
1
6
3
2
5
4
6
1
3
5
2
4
6
1
3
2
5
4
1
6
3
5
2
4
5
6
1
3
2
Problem S4
Op
PT
1
6
1
4
1
3
1
8
1
2
1
8
2
1
2
7
2
1
2
1
2
5
2
4
3
10
3
3
3
4
3
8
3
5
3
5
4
2
4
6
4
8
4
3
4
4
4
2
5
8
5
1
5
8
5
4
5
2
5
1
6
7
6
4
6
3
6
6
6
3
6
3
Score Val
4.0893
4.2067
6.8886
7.0937
7.2186
7.4892
7.9876
10.4648
10.5897
11.5887
12.415
12.4532
14.0376
15.3866
15.7234
17.0489
18.0377
18.2873
19.3202
20.375
21.631
22.5292
22.7623
23.0045
25.2762
26.1055
26.6086
26.6782
27.2327
27.5832
31.3953
31.4267
31.4649
31.6415
32.0473
32.0473
Job
2
3
6
1
4
5
2
3
4
6
1
3
5
2
4
6
1
3
5
2
4
6
5
1
3
4
2
5
6
1
3
4
6
5
1
2
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
3
2
3
3
3
3
4
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S5
PT
10
2
8
5
3
7
5
4
2
10
5
5
10
10
6
4
7
7
1
10
7
10
4
7
10
5
8
4
1
3
10
5
4
3
2
2
Score Val
0.793
1.6599
3.6584
5.4177
5.473
6.2987
6.7731
6.8107
9.7394
9.851
10.874
11.3968
12.4115
13.3503
14.9808
16.0978
16.9617
17.4013
17.9192
20.7425
21.2315
22.1376
22.1998
22.7098
23.8885
26.5978
26.9625
27.3438
27.3957
27.6943
31.0311
31.7213
31.9675
32.2553
32.2935
32.3351
Job
3
2
6
4
1
5
2
6
3
1
4
5
1
6
3
2
4
5
2
6
1
3
4
5
6
2
1
3
4
5
2
6
1
3
4
5
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S6
PT
10
10
9
9
7
2
8
5
8
1
8
7
5
9
8
8
7
9
2
4
10
7
2
1
1
2
3
5
5
1
10
10
5
3
3
2
Score Val
2.1254
2.9785
3.5786
4.9942
5.5909
7.3639
8.9489
8.9797
9.427
9.9856
11.4672
13.129
14.6235
15.0332
15.2794
15.6333
16.9207
18.9261
20.7979
20.909
21.5335
21.9393
22.2135
23.938
25.0227
25.6917
26.4662
26.848
27.1392
28.6027
30.9063
31.4055
31.7629
32.0057
32.0473
32.2103
178
Job
4
2
3
1
6
5
4
2
6
1
3
5
4
1
2
6
3
5
1
6
4
5
2
3
1
4
3
6
5
2
1
2
3
4
6
5
Problem S7
Op
PT
1
10
1
3
1
3
1
2
1
9
1
10
2
1
2
8
2
3
2
2
2
10
2
5
3
10
3
3
3
6
3
3
3
3
3
5
4
4
4
4
4
10
4
1
4
10
4
2
5
3
5
1
5
6
5
10
5
7
5
4
6
10
6
2
6
3
6
6
6
1
6
1
Score Val
3.2287
3.6619
4.9533
5.8241
6.285
6.3722
7.7796
9.1569
10.6387
10.9259
12.1613
12.4771
14.1624
15.3034
15.6675
16.0528
17.4684
17.975
20.1596
20.909
21.4298
21.6277
21.6589
22.026
25.0711
26.6879
26.9762
27.1152
27.521
27.7393
31.3639
31.4199
32.0057
32.0159
32.3317
32.8309
Job
1
5
6
4
3
2
6
5
4
1
3
2
5
6
4
1
2
3
5
2
6
4
1
3
6
5
2
1
4
3
2
5
6
1
4
3
Problem
Op
1
1
1
1
1
1
2
2
2
2
2
2
3
3
3
3
3
3
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S8
PT
9
6
1
7
1
7
9
4
1
9
9
3
10
10
9
9
4
3
7
3
9
10
10
9
4
8
9
2
8
9
7
9
9
7
4
1
Score Val
0.0402
0.1753
0.4694
2.8429
3.4676
4.9663
6.0789
6.1861
7.0296
7.6193
9.4515
9.9092
11.9763
12.6841
12.8471
14.2416
14.7238
14.8868
18.671
19.9681
20.0518
20.2427
20.8468
21.1135
25.0124
25.1924
25.446
26.4206
26.6707
27.7358
31.2705
31.3189
31.3189
31.8113
32.1339
32.6229
Job
1
5
2
1
4
3
6
5
2
4
1
3
6
5
4
2
3
1
6
5
4
2
6
3
1
5
4
6
3
2
1
5
4
6
3
2
Problem
Op
1
1
1
2
1
1
1
2
2
2
3
2
2
3
3
3
3
4
3
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
S9
PT
1
6
1
9
5
6
9
6
10
1
10
6
1
5
7
9
7
9
4
5
7
8
3
5
9
8
5
1
3
3
8
7
4
5
1
1
Score Val
-1.5286
2.9854
3.5092
4.2473
5.7089
6.1289
8.2215
8.3796
9.5176
9.944
10.9773
11.9391
13.3582
14.2485
15.2133
15.9701
17.7322
18.2202
18.2212
19.2472
21.2942
22.6511
23.0911
23.1401
24.8841
25.0266
26.5773
27.0002
27.965
28.3394
31.1491
31.5617
31.8011
31.8877
32.2901
32.5397
179
Job
5
6
4
1
3
2
5
6
4
3
1
2
5
4
6
3
1
2
5
4
6
2
1
3
5
6
2
1
4
3
5
2
6
4
1
3
Problem S10
Op
PT
1
7
1
9
1
6
1
8
1
7
1
4
2
9
2
8
2
3
2
1
2
7
2
5
3
4
3
9
3
10
3
6
3
3
3
2
4
7
4
7
4
1
4
1
4
2
4
8
5
8
5
5
5
5
5
6
5
7
5
2
6
10
6
9
6
10
6
6
6
6
6
6
Score Val
0.4283
1.4763
2.2571
5.4279
5.4872
6.5381
6.8078
7.6165
7.7858
10.1731
11.4222
11.9562
12.6004
13.5965
14.8075
15.4595
16.2403
16.507
18.4003
19.7122
19.9824
20.1283
20.8811
21.5683
23.9649
24.9115
25.0158
26.1225
26.2091
26.7745
30.9479
31.3189
31.4055
31.8079
31.8079
32.0575
180
APPENDIX D SCHEDULES GENERATED BY SPT PROCEDURE
Seq. Posn.
17
18
19
20
21
22
2
23
24
25
26
27
3
9
10
13
15
16
4
6
7
8
11
12
5
28
29
30
31
32
1
14
33
34
35
36
Problem 2
Machine
4
2
5
3
6
1
1
6
2
5
3
4
3
4
2
5
1
6
2
4
1
3
5
6
6
4
1
2
3
5
3
4
6
5
1
2
ProcTime
9
6
1
6
3
6
2
9
1
7
1
7
3
5
1
7
8
1
4
4
3
1
6
2
4
9
6
5
8
2
1
7
10
7
6
2
Seq. Posn.
1
2
21
22
23
24
7
20
33
34
35
36
19
28
29
30
31
32
3
4
5
25
26
27
6
14
15
16
17
18
8
9
10
11
12
13
Problem 3
Machine ProcTime
3
1
1
1
2
9
4
7
6
3
5
6
5
6
4
8
6
10
3
5
2
4
1
6
1
9
3
10
2
8
4
2
5
4
6
7
6
2
4
1
5
1
1
9
2
6
3
5
3
3
2
8
1
4
4
2
6
1
5
4
4
7
3
8
2
5
1
6
5
6
6
6
Problem 4
Seq. Posn. Machine ProcTime
2
3
2
20
1
9
21
2
3
22
4
2
23
6
4
24
5
4
5
5
5
6
4
2
16
6
8
17
3
4
18
2
1
19
1
4
7
1
5
8
3
5
25
2
9
26
4
3
35
5
9
36
6
2
1
6
1
12
4
6
31
5
9
32
1
3
33
2
6
34
3
4
9
3
6
10
2
1
11
1
1
13
4
7
14
5
6
15
6
7
3
4
4
4
3
4
27
2
9
28
1
4
29
5
5
30
6
1
Seq. Posn.
1
32
33
34
35
36
2
27
28
29
30
31
14
15
16
19
20
21
3
4
7
8
12
13
6
11
17
18
22
23
5
9
10
24
25
26
Problem 5
Machine ProcTime
4
1
2
10
1
2
3
3
6
4
5
7
6
3
3
9
4
6
5
5
2
1
1
2
1
7
3
7
2
2
4
7
5
4
6
1
2
4
3
2
4
5
5
2
1
6
6
5
2
5
4
6
3
7
5
3
1
7
6
4
2
4
3
6
4
2
6
8
5
8
1
7
181
Seq. Posn.
30
31
33
34
35
36
1
22
23
24
25
26
7
11
12
27
28
29
2
3
4
5
6
13
14
15
16
17
18
32
8
9
10
19
20
21
Problem 6
Machine ProcTime
2
10
3
2
5
10
4
10
6
7
1
7
1
5
6
9
2
9
5
7
3
2
4
1
4
7
2
7
3
2
1
9
6
6
5
8
5
5
6
2
3
2
4
4
1
6
2
8
2
8
1
1
4
5
3
2
5
6
6
10
6
7
3
6
5
6
2
8
4
5
1
4
Problem 7
Seq. Posn. Machine ProcTime
1
2
1
7
3
7
8
5
8
9
4
3
35
6
10
36
1
1
29
1
10
30
6
9
31
2
9
32
5
2
33
3
3
34
4
7
10
4
9
15
2
9
16
3
2
17
1
8
18
6
1
19
5
8
5
5
7
6
6
1
20
3
9
21
4
6
22
1
6
23
2
7
11
2
9
12
1
7
13
4
8
14
3
2
27
5
9
28
6
7
2
6
1
3
3
4
4
5
1
24
2
9
25
4
5
26
1
7
Problem 8
Seq. Posn. Machine ProcTime
11
3
6
12
4
1
15
2
6
16
5
6
17
1
2
18
6
1
19
6
8
20
5
7
21
4
5
22
1
7
27
2
8
28
3
4
1
3
1
4
2
3
23
4
8
24
5
4
25
1
6
26
6
1
2
6
1
13
4
6
14
5
3
32
3
10
33
2
6
34
1
2
3
1
1
29
5
9
30
2
4
31
4
6
35
3
10
36
6
1
5
6
4
6
4
1
7
3
1
8
5
2
9
1
1
10
2
4
Problem 9
Seq. Posn. Machine ProcTime
19
3
8
20
4
2
21
2
1
22
5
5
23
1
2
24
6
6
4
6
5
14
4
7
15
5
1
16
3
2
17
2
6
18
1
4
1
1
3
2
3
2
11
2
6
25
4
8
26
5
2
29
6
8
5
6
6
6
5
4
7
4
1
8
3
6
9
1
6
10
2
6
3
2
3
27
4
8
28
3
1
30
5
9
31
6
1
32
1
1
12
1
7
13
3
3
33
2
10
34
4
8
35
5
2
36
6
1
182
Seq. Posn.
11
12
13
14
15
16
3
4
5
6
7
8
17
18
19
26
27
28
1
9
10
29
30
31
2
32
33
34
35
36
20
21
22
23
24
25
Problem 10
Machine ProcTime
6
9
1
4
2
8
5
3
3
6
4
1
6
8
1
2
2
5
5
7
3
2
4
5
4
9
3
3
2
1
5
9
6
2
1
2
1
5
2
8
3
1
6
9
5
6
4
7
4
7
3
10
5
10
2
4
6
6
1
1
1
9
3
9
2
3
4
1
6
1
5
1
Problem S1
Seq. Posn. Machine ProcTime
21
2
9
22
3
7
23
4
1
24
6
8
25
1
4
26
5
2
27
4
9
28
2
7
29
6
7
30
3
1
31
5
4
32
1
4
1
5
1
7
1
6
8
2
3
9
3
6
10
4
4
11
6
5
4
3
6
5
6
4
6
1
5
12
5
6
19
4
7
20
2
4
2
2
2
3
4
4
15
1
7
16
5
2
17
3
6
18
6
5
13
5
7
14
1
1
33
2
9
34
4
4
35
6
4
36
3
8
Problem S2
Seq. Posn. Machine ProcTime
11
2
9
24
5
10
25
1
3
26
6
1
27
3
8
28
4
1
1
4
4
2
6
5
5
5
7
6
1
7
7
3
1
8
2
5
9
5
8
10
1
3
29
3
10
30
2
1
31
6
6
36
4
10
21
6
10
22
2
1
23
1
9
32
3
10
33
4
5
34
5
6
3
4
7
4
3
4
12
6
9
13
1
3
14
5
2
15
2
7
16
1
10
17
2
4
18
4
4
19
3
9
20
5
5
35
6
10
Seq. Posn.
3
4
11
12
30
31
21
26
27
28
29
35
2
13
14
18
19
20
1
5
15
16
17
36
6
7
8
9
10
34
22
23
24
25
32
33
Problem S3
Machine ProcTime
5
4
4
1
6
8
3
1
2
10
1
8
2
10
5
10
3
6
6
5
4
1
1
10
1
4
4
9
2
5
3
9
5
9
6
4
3
2
2
5
1
9
4
8
5
6
6
10
2
7
3
4
4
7
6
1
1
7
5
10
1
10
5
1
6
3
4
1
3
10
2
4
183
Seq. Posn.
8
10
11
28
29
30
12
13
14
15
16
17
1
2
24
25
26
27
6
7
33
34
35
36
18
19
20
21
22
23
3
4
5
9
31
32
Problem S4
Machine ProcTime
4
6
5
7
6
4
3
8
1
1
2
6
5
8
4
5
2
5
1
4
6
1
3
3
4
2
6
1
5
8
2
3
1
4
3
3
5
4
1
1
6
10
4
2
2
8
3
7
2
8
1
4
6
5
5
2
3
2
4
4
2
3
4
1
5
3
6
6
3
8
1
3
Problem S5
Seq. Posn. Machine ProcTime
5
6
5
6
2
5
10
3
7
15
4
7
16
5
3
17
1
2
19
1
10
20
5
5
28
4
10
29
3
10
30
6
8
31
2
2
1
6
2
4
3
4
7
5
5
11
4
7
35
1
10
36
2
10
2
1
3
3
5
2
8
2
6
12
3
7
13
4
5
14
6
5
9
3
7
21
6
10
22
2
1
23
5
4
24
1
4
25
4
3
18
3
8
26
5
10
27
4
4
32
2
10
33
6
1
34
1
4
Problem S6
Seq. Posn. Machine ProcTime
2
6
7
3
4
1
4
3
5
32
2
10
33
5
3
34
1
5
21
6
10
22
5
8
23
1
8
24
3
2
25
2
2
35
4
10
26
4
10
27
2
8
28
1
8
29
6
7
30
5
5
31
3
3
6
2
9
7
6
8
8
5
7
9
3
2
10
1
5
11
4
3
1
1
2
5
2
7
14
3
9
15
5
1
16
6
1
17
4
2
12
2
9
13
4
5
18
3
9
19
1
4
20
5
1
36
6
10
Seq. Posn.
1
2
5
6
7
36
3
8
9
31
32
33
4
22
23
24
25
26
20
21
27
28
29
30
14
15
16
17
18
19
10
11
12
13
34
35
Problem S7
Machine ProcTime
6
2
5
2
2
3
3
4
1
3
4
10
5
3
2
8
4
6
3
10
1
4
6
2
6
3
4
10
3
3
1
2
5
6
2
3
5
10
6
1
2
10
1
10
3
1
4
6
3
10
2
5
1
5
6
1
5
7
4
1
4
9
6
3
3
3
5
4
1
10
2
1
184
Seq. Posn.
11
13
17
34
35
36
7
8
9
10
19
20
1
14
15
18
21
22
3
4
16
31
32
33
5
6
23
24
25
26
2
12
27
28
29
30
Problem S8
Machine ProcTime
4
9
1
9
2
9
5
10
6
2
3
7
2
7
5
3
6
4
1
3
3
9
4
7
4
1
3
9
6
3
1
9
2
9
5
1
2
7
4
1
6
9
1
10
3
8
5
4
4
6
2
4
5
10
3
7
1
8
6
9
3
1
2
9
6
10
1
9
4
4
5
9
Problem S9
Seq. Posn. Machine ProcTime
1
4
1
27
3
9
33
5
10
34
1
9
35
2
9
36
6
8
2
6
1
28
1
10
29
4
9
30
5
8
31
2
3
32
3
1
5
3
6
7
1
6
11
2
7
12
5
5
13
4
3
14
6
1
3
1
5
4
5
1
15
3
7
16
2
7
17
4
5
18
6
4
6
2
6
8
6
6
9
1
5
10
5
5
19
4
8
20
3
7
21
6
9
22
4
1
23
2
4
24
1
3
25
5
1
26
3
5
Problem S10
Seq. Posn. Machine ProcTime
12
2
8
13
5
7
14
1
3
15
4
2
16
6
6
17
3
6
1
3
4
2
2
5
3
5
2
4
1
1
5
4
5
31
6
9
8
4
7
9
1
1
10
3
6
18
5
8
19
6
2
20
2
6
6
1
6
7
3
3
27
5
9
28
4
7
29
2
7
30
6
6
11
5
7
23
3
9
24
6
4
25
2
7
26
1
8
35
4
10
21
3
9
22
1
8
32
6
10
33
4
1
34
5
5
36
2
10
185
APPENDIX E NORMAL DISTRIBUTION PLOTS OF MAKESPANS
GENETIC ALGORITHM
Probability Plot of GA
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
Percent
80
70
60
50
40
30
20
10
5
1
30
40
50
60
GA
70
52
6.307
19
0.222
0.801
186
REGRESSION ANALYSIS
Probability Plot of Regression
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
Percent
80
70
60
50
40
30
20
10
5
1
30
40
50
60
Regression
70
80
58.05
7.051
19
0.470
0.219
187
INDUCTION RULES
Probability Plot of Rules
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
Percent
80
70
60
50
40
30
20
10
5
1
30
40
50
60
Rules
70
80
90
59.37
8.139
19
0.526
0.157
188
SHORTEST PROCESSING TIME (SPT)
Probability Plot of SPT
Normal - 95% CI
99
Mean
StDev
N
AD
P-Value
95
90
Percent
80
70
60
50
40
30
20
10
5
1
30
40
50
60
70
80
SPT
90
100
110
120
71.58
12.34
19
0.401
0.327