MATHEMATICAL THOUGHT
AND PRACTICE
Chapter 8: The Mathematics of Scheduling
Chasing the Critical Path
The Basic Elements of Scheduling
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The Mathematics of Scheduling
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The tasks.
We define a task as an
indivisible unit of work and
thus always carried out by a
single
i l processor. W
We use
capital letters A, B, C…, to
represent the tasks.
At any time, a task can be :
The processors.
Every job requires workers.
N represents the number of
processors and P1, P2, P3, …
PN to denote the processors
themselves.
The Mathematics of Scheduling
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The processing times.
The processing time for a given task X is the
amount of time, without interruption, required by
one processor to execute X.
X Thus,
Th the
h notation
i X(5)
tells us that the task called X has a processing time
of 5 units.
Completed
In execution
† Ready
† Ineligible
†
†
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The Mathematics of Scheduling
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The Mathematics of Scheduling
The precedence relations.
Precedence relations are formal restrictions on the
order in which the tasks can be executed, much like
those course prerequisites that tell you that you
can’t take course Y until you have completed
course X.
A precedence relation can be conveniently
abbreviated by writing X→Y or described
graphically in (a).
The Mathematics of Scheduling
When a pair of tasks X and Y have no
precedence requirements, the tasks are
independent. Graphically, there are no
arrows connecting them (b).
The Mathematics of Scheduling
We cannot have a set of precedence relations
that form a cycle as shown in (d). Clearly,
this is logically impossible!
The Mathematics of Scheduling
Precedence relations are transitive: If X→Y
and Y→Z , then it must be true that X→Z as
shown in (c).
Repairing a Wreck
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Imagine you just wrecked
your car. The garage has
two processors, P1 and P2.
The repairs can be broken
i
into
four
f
tasks:
k (A) exterior
i
body (4 hours), (B) engine
repairs (5 hours), (C)
painting and exterior finish
(7 hours), and (D) repair
transmission (3 hours).
The only precedence
relation A→ C .
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Repairing a Wreck
(a) is very inefficient: All short tasks are assigned to P1 and long to P2.
The project finishing time (duration from the start to finish) is 12 hours. So
Fin = 12 hours.
(b) looks better but violates precedence: A→ C. But if we force P2 to be
idle for one hour, we get a perfectly good schedule as shown in (c). Fin =
11 hours.
Digraphs
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Repairing a Wreck
Can we do better? No! The precedence relation A(4)→ C(7)
implies that the 11 hours is a minimum barrier. So (c) is an
optimal schedule and Fin = Opt = 11 hours is the optimal
finishing time. Another optimal schedule is (d).
Romantic Love
A directed graph or digraph
for short, is a graph in
which the edges have a
direction associated with
them, typically shown with
arrows. Digraphs are
particularly useful when
we want to describe
asymmetric relationships.
We have four possible
scenarios: Neither loves the
other (a), X loves Y but Y
doesn’t love X (b), Y loves X
but X doesn’t love Y (c), and
they love each other (d).
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Terminology
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Example
If XY is an arc, X is incident to Y or Y is
incident from X.
The arc YZ is said to by adjacent to XY if YZ
starts where XY ends.
Ap
path from consists of a sequence
q
of arcs
such that each is adjacent to the one before
and no arc appears more than once.
When the path starts and ends at the same
vertex, we call it a cycle of the digraph.
The outdegree of vertex X is the number of
arcs that start at X (outgoing arcs); the
indegree of X is the number of arcs that end
at X (incoming arcs).
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Scheduling with Priority Lists
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Priority Lists
A priority list is a list of
all the tasks prioritized in
the order we prefer
them. If task X comes
before task Y, then X
gets priority over Y. So
X is executed ahead of
Y unless X is not ready
for execution, in which
case we skip X.
Priority List Timelines
To keep track on the priority list itself, ready
tasks are circled in red, (a). Executed task have
a single red slash, (b). Completed tasks have a
second slash, (c). Tasks that remain ineligible
remain unmarked as shown in (d).
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Decreasing-Time Algorithm
To find a good priority list:
(1) Do longer jobs first and
(2) Shorter jobs last.
A priority list in decreasing
order of processing times
is called a decreasingtime priority list, and
creating a schedule using
such a list is called
decreasing-time
algorithm.
Decreasing-Time Algorithm
Decreasing-time priority list: AD(8), AP(7), IW(7),
AW(6), FW(6), AF(5), IF(5), ID(5), IP(4), HU(4),
PU(3), PD(3), EU(2), IC (1)
Project Digraph
Decreasing-Time Algorithm
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Critical Paths and Critical Times
The critical path for a vertex
X is the path from X to
END with longest
processing time. When we
add the processing times
of all the tasks along the
critical path for a vertex X,
we get the critical time for
X. (By definition, the
critical time of END is 0.)
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Critical Paths and Critical Times
The critical path for a project
is the path with longest
processing time from
START to END. The total
processing time for this
path is the critical time for
the project.
The Backflow Algorithm
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Step 1 Find the critical time
for every vertex by starting
at END and working toward
START according to this rule:
critical time for X =
processing time of X +
largest critical time of vertices
incident from X.
Step 2 Once we have the
critical time for every vertex,
critical paths are found by
just following the path along
largest critical times.
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The Critical Path Algorithm
The Critical Path Algorithm
Critical paths can be used to create very good
schedules using critical times rather than
processing times to prioritize the tasks. The
priority
i it list
li t we obtain
bt i when
h we write
it the
th tasks
t k
in decreasing order of critical times (with ties
broken randomly) is called the critical-time
priority list, and the process of creating a
schedule using the critical-time priority list is
called the critical-path algorithm.
Critical-Path Algorithm
Backflow & Critical Path Algorithms
‰Step 1 Find
critical times
using the
backflow
algorithm.
algorithm
‰Step 2
Create priority
list.
‰Step 3
Create
schedule.
(b) What is the critical path?
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Scheduling with Independent Tasks
What happens to scheduling
problems in the special case
when there are no precedence
relations to worry about tasks that are all
independent? There are no
efficient optimal algorithms
known for scheduling, even
when the tasks are all
independent.
Scheduling with Independent Tasks
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We assign the tasks to the
processors as they become
free in exactly the order
given by the priority list.
The critical-path time of a
task equals its processing
time, meaning the
decreasing-time algorithm
and the critical-path
algorithm become one and
the same.
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Relative Error
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If we know the optimal
finishing time Opt, we can
measure how “close” a
particular schedule is
p
being optimal using the
concept of relative error.
For a project with
finishing time Fin, the
relative error is given by ε.
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ε=
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Fin − Opt
Opt
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MATHEMATICAL THOUGHT
AND PRACTICE
Mini-Excursion 2: A Touch of Color
Graph Coloring
Graph Coloring
Can’t We All Get Along?
Planning a wedding
reception for a contentious
family, we need to seat
people so that fights don’t
break out, unfortunately.
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Can’t We All Get Along?
Coloring
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Coloring Complete Graphs
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Coloring Complete Graphs
The only possible way
to color is to use a
different color for
each vertex, as in (b).
Thus, we can conclude
that X(K5) = 5.
We generalize so
X(Kn) = n.
X(K100) = 100.
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Greedy Algorithm for Coloring
Brook’s Theorem
Sudoku
Sudoku
Each row, column,
and block must
have 1, 2, 3, 4,
5, 6, 7, 8, and
9 appearing
exactly once.
(C6) = 2 but X(C5) = 3.
If n is odd and greater
than 3, X(Cn) = 3.
If n is even,
even X(Cn) = 2
2.
… X
Each vertex represents
a cell, and the cells
that cannot have the
same numbers are
connected with edges.
Each of the 81
vertices then has
degree 20.
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Sudoku
Four Color Theorem
To solve the puzzle,
we need only color
the graph (Greedy
Algorithm) and
then replace colors
with numbers.
Map Coloring
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Any map (in a
plane) can be
colored with four
colors or less so
that no two
adjacent
countries have
the same color.
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Each vertex is a country and edges show
adjacencies. A is adjacent to F, E, and B. Then,
color with the Greedy Algorithm.
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Homework
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Do online homework
Do paper homework
Do quiz
M t with
Meet
ith me
Prepare group projects
Do the final exam review
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