APPLYING LEARNING CLASSIFIER SYSTEMS for
MULTIPROCESSOR SCHEDULING PROBLEM
Franciszek Seredynski, Damian Kurdej
Polish Academy of Sciences and
Polish-Japanese Institute of Information Technology
Motivations
• New scheduling algorithms are proposed near every day
• In the light of
– NP-hard compliteness of the scheduling problem, and
– No free lunch theorem concerning metaheuristics
this situation may last forever, at least till the moment of
appearing quantum computers
• Can we use the knowledge gained from the experience
with already known scheduling algorithms
(hypeheuristics approach) ?
• We will use GA-based Learning classifier systems (LCS)
to extract some knowledge and use it in the scheduling
process





Multiprocessor Scheduling Problem
The idea of LCS
The concept of LCS-based scheduling
Experimental results
Conclusions
MUTIPROCESSOR SCHEDULING PROBLEM
t1
5
t2
5
3
t3
1
t4
2
P1
P2
P3
P4
3
Examples of a precedence task
graph (a) and a system graph (b).
5
a)
b)
Multiprocessor system: undirected, unweighted graph Gs=(Vs,Es), called a system graph.
Parallel program: weighted, directed, acyclic graph GP=<VP,EP>, called a precedence task
graph or a program graph.
The purpose of the scheduling is to distribute the tasks among the processors in such
a way that the precedence constraints are preserved and the response time T (the
total execution time) is minimized.
T = f (allocation, scheduling_policy = const)
4
Problem formulation
• Given a set of program graph instances
• Given a multiprocessor system
• Given a number of scheduling algorithms (heuristics)
solving instances of the scheduling problem with
some efficiency
• Is it possible to train LCS system to match a given
instance of the scheduling problem with the best for
it scheduling algorithm (to minimize the total
exectution time) from a set of scheduling algorithms
?
Idea of GA-based Learning
Classifier System (LCS)
Learning Classifier System
Evaluation
system
Decision
system
Environment
System for
discovery
of new
rules
Idea of Learning Classifier
System (LCS)
Learning Classifier System
System for
Evaluation
system
Decision
system
discovery
of new rules
Environment state
or message
e.g.10100
Environment
Idea of Learning Classifier
System (LCS)
Learning Classifier System
System for
Evaluation
system
Decision
system
action
e.g. Turn right
discovery
of new rules
Environment state
e.g.10100
Environment
Idea of Learning Classifier
System (LCS)
Learning Classifier System
System for
Evaluation
system
reward
e.g. 120
Decision
system
action
e.g. Turn right
discovery
of new rules
Environment state
e.g.10100
Environment
Classifier (rule) in classical LCS
• The structure of a classifier
– Condtition part C
– Action A
– Strength S
• Strength S
A
#011: 01 : 43
C
– Used when a classifier is selected from a set of
classifiers to perform an action
– Used when GA creates new rules
S
Classifier in XCS
ε
a
exp
as
010##0#####:0 1000 2,504 0,77 499 19924 146,76 109
C
•
•
•
•
•
•
•
p
f
ts
num
C: condition part
a: action
p: expected reward
e: prediction error
f: fitness
exp: experience of classifier
ts: remembers recent time when GA was applied to this
classifier
• as: expected population size [A], in which appears
classifier
• num: numerosity of classifier
XCS
Environment
5.
1.
Efector
Detector
Match set [M]
0011
σ
C : a : p: e: f
#011:01:43:.01:99
#0##:11:11:.13:9
001#:01:27:.05:52
#0#1:11:18:.24:3
2.
Population [P]
C : a : p: e: f
#011:01:43:.01:99
#0##:11:11:.13:9
001#:01:27:.05:52
#0#1:11:18:.24:3
11##:00:32:.02:92
1#01:10:24:.17:15
01
a
4.
Action set [A]
C : a : p: e: f
#011:01:43:.01:99
001#:01:27:.05:52
3.
9.
Prediction array PA
00
-
01
37.49
10
-
GA
11
12,75
8.
Subsumption
...
Action set [A]-1
7.
cover
C : a : p: e: f
11##:00:32:.02:92
Enforcement
ρ
6.
Features of XCS
• Creates population of classifiers
• Processes messages received from
environment
• Applies GA to evolve classifiers
• Sends action to environment
• Learns, generalizes and modifies the set
of classifiers
Our problem
• Given 200 program graph instances created on the
base of the 15-tree graph: training set
• Each instance is a tree with different random task
and communication weights
• Two processor system is considered
• Given 5 scheduling heuristics
• We want to train LCS system to select in the best way
the scheduling heuristic to solve given set of
instance of the scheduling problem to provide the
best possible solutions ?
Set of list algorithms
•
•
•
•
•
ISH (Insert Scheduling Heuristic)
MCP (Modified Critical Path)
STF (Shortest Time First)
LTF (Longest Time First)
own list algorithm: priority of a task
depends on a size of the subgraph
• We know how works each algorithm
(response time) on the set of scheduling
instances
XCS-based scheduling system
Program
graph
+
System graph
1.
XCS
1. XCS receives
information about an
instance of the
scheduling problem
XCS-based scheduling system
Program
graph
+
System graph
1.
XCS
2.
scheduling
algorithm
1. XCS receives
information about an
instance of the
scheduling problem
2. XCS selects the best
available heuristic
XCS-based scheduling system
Program
graph
+
System graph
3.
1.
XCS
2.
scheduling
algorithm
1. XCS receives information
about an instance of the
scheduling problem
2. XCS selects the best
heuristic from the set of
available heuristics
3. Program graph and a
system graph become
input data of scheduling
algorithm
XCS-based scheduling system
Program
graph
+
System graph
3.
1.
XCS
2.
scheduling
algorithm
4.
Gantt diagram
1. XCS receives information
about an instance of the
scheduling problem
2. XCS selects the best
heuristic from the set of
available heuristics
3. Program graph and a
system graph become input
data of scheduling
algorithm
4. Scheduling algorithm
delivers a solution
Program graph signature: the basic
information concerning program graph
• LCS receives from environment a signature of program graph
• The signature codes some static properties of program graph
– comm/comp – the averaged communication to computation time for
a program graph (3 bits)
– information about distribution of tasks with a given computational
requirements (12 bits)
– information about distribution of communication time requirements
to communicate between tasks (12 bits)
– Information about critical path based on evaluation of comp/comm
(16bits)
• The length of the signature: 43 bits
Distribution of tasks with a given computational requirements/
distribution of communication time requirements
Coding information concerning critical path based on
evaluation of comp/comm
• Computing ratios on critical
path:
ratios[0] = 1/4, ratios[1] = 5/3,
ratios[2] = 1/3
• Normalization:
ratios[0] = 3/27, coding as 01,
ratios[1] = 20/27,coding as 11,
ratios[2] = 4/27, coding as 01.
• Coding signal concerning
critical path:
0111010000000000
Training LCS: number of correct matching
scheduling algorithms to instances as function of
number of training cycles
Training LCS: population size of rules as
function of a number of cycles
Training: summary of experiments
• Nontrained system correctly matched heuristics with
scheduling instances in 40-50% cases
• The system was able to learn to match correctly in 100%
heuristics to instances
• It means that information about the matching process was
extracted during the learning process
• Classifiers contain this information and during the learning
process the process of generalization of rules was observed
• Learning process is a costly process, but the gained
information can be used in the scheduling
LCS-based scheduling system:
normal operation mode
• Modification of instances (program graphs)
from training set: testing set
• All computation and communication weights
were scailed by 10
• Next, weights of k tasks or communications
were changed by constant d
Experiment: k=1, d=1
Experiment: k=2, d=2
Experiment: k=3, d=3
Normal operation mode: summary
of experiments
Number of correct
matching heuristics to
scheduling instances
Number k of modified
weights (tasks or
communications)
Difference d between
initial weight value and
the value after
modification
90%
1
1
80%
2
2
75%
3
3
Conclusions
• LCS has been proposed to learn optimal matching scheduling algorithms
to instances
• Instances were represented by specially signatures
• During the learning process the knowledge about matching was extracted
in the shape of LCS rules, and next generalized
• Creating signatures is one of the most crucial issues in the proposed
approach
• Performance of the system depends also on many parameters of LCS
• We believe that encouraging results of experiments open new possibilities
in developing hyperheuristics