hasso-plattner-institut für softwaresystemtechnik gmbh an der universität potsdam multiprocessor...

HASSO-PLATTNER-INSTITUTfür Softwaresystemtechnik GmbH an der Universität Potsdam

Multiprocessor Scheduling

Integrating List Heuristics into genetic Algorithms for

Multiprocessor Scheduling [1]

May, 22 2006

Stefan Hüttenrauch

Multiprocessor SchedulingProf. Dr. Lars Lundberg 2


May, 22nd 2006Stefan Hüttenrauch

Roadmap

Part I – Introduction The problem to be solved Heuristics in general

Genetic algorithms List heuristics

Part II – Integrating list heuristics into genetic algorithms One genetic algorithm in detail Integrating a list heuristic Results and comparisons


Part I – Introduction

The problem to be solved

Heuristics in general Genetic algorithms (GA) List heuristics (LH)




Introduction – The Problem to be solved

Computer with a set P of m processors Set T of n tasks with precedence order Schedule S assigns tasks to processors in a

specific order

Goal: find schedule with smallest makespan

T1

T2 T5

T4T3

T6

T7

T8

acyclic digraph

feasible schedule for m=2

T1 T2

T3 T4 T5

T6 T7

T8

Refs: [1][3]




Heuristics in general

Consistent algorithms for optimization problems

Has strategy of searching in set of all feasible solutions

No guarantee for optimal solution

Examples: Generic algorithms (meta heuristics) List heuristics

Refs: [4]




Heuristics in general – Genetic Algorithms

History Developed by John H. Holland in mid-1960’s Imitate nature’s evolution processes

Comparison to trad. optimization methods Explore greater range of possible solutions Use probabilistic transition rules

Disadvantages Finding the best solution is not necessarily given

Refs: [1][2][3]




Heuristics in general – Genetic Algorithms cond.

Genetic Algorithms consists of:

String representation (genes)

Initial population

Fitness function

Genetic operators and a stochastic assignment

Refs: [1][3]




Heuristics in general – List Heuristics

In contrast to GA:

Schedule is build step by step

Use knowledge about the problem and about what happened in the steps before to find better solution

One example is the “critical path/most immediate successors first” (CP/MISF) list heuristic

Refs: [1][5]


Part II – Integrating list heuristics into genetic algorithms

One genetic algorithm in detail

Integrating a list heuristic

Results and comparisons




Individual: set s of m strings for m processors String: ordered set of tasks scheduled to Pj

Precedence relations among tasks satisfied Every task is present && appears only ones

One GA Details – String Representation

T1

T2 T5

T4T3

T6

T7

T8

s1

s2

one individualT1 T2

T3 T4 T5

T6 T7

T8

0 0

1 1 1

2 2

3

Refs: [1][3]




One GA Details – initial Population

How to define feasible individuals Define height-ordering on task set T

in each string sj: height(th) ≤ height(ti)

with th and ti in sj && h < i

Only a necessary condition may not find optimal schedule

How to define that ordering height(ti) random with plp(ti) < height(ti) < pls(ti)

plp(ti)… max path length between t1 and immediate predecessor of ti

pls(ti)… max path length between t1 and immediate successor of ti

Refs: [1][3]




One GA Details – initial Population cond.

Algorithm to create initial population1. Compute height for every task in task graph TG2. Separate tasks according to their height (set G(h))3. For all m-1 processors do 4.

4. Form schedule for a processora. For every G(hi) in G(h) create random number r with

0 ≤ r ≤ |G(hi)|

b. Pick r tasks of G(hi) assigning them to current processor

c. Delete assigned tasks from G(hi)

5. Assign remaining tasks to the last processor

Refs: [1][3]




One GA Details – Fitness Function

For evaluation of search nodes (individuals)

Controls genetic operators

Here: finishing time FT of a schedule S FT(S) = max ftp(Pj)

for all j in {0, 1,…, m}ftp(Pj)… finishing time of last task in Pj

m… number of processors

Refs: [1][3]




Genetic Algorithms – genetic Operators

Reproduction Clone individuals and assign them to new population

Do it (number of individuals in population) times Clone individual with highest fitness value

Crossover Partition of strings Pair-wise exchange of parts of the strings

Do it (number of individuals in population) / 2 times

Mutation Random alternation of a string with small probability

Exchange two tasks with same height within one string Do it (number of individuals in population) times

Refs: [1][3]




Population size = 10, max. number of iterations = 1500 Reproduction One crossover (5 times, with probability propCros = 1) One mutation (10 times, with probability propMut = 0.05) Replace individual with smallest fitness value

One GA Details – genetic Operators – Animation

Population 1 Population 2Population 1Population 1




One GA Details – Problems

Initial population In average pi has more tasks than pi+1

No uniform distribution due to initial population generation scheme.

Crossover Some feasible solution cannot be generated Derives from height-ordering

Refs: [1]




Integrating a List Heuristic – initial Population

Basis also a task digraph

Iterative method Determine free tasks Randomly choose one and assign it to a free

processor

Tasks not necessarily in height-order

Refs: [1]




Integrating a List Heuristic – Crossover

Determine partitions V1 and V2 Remember the genetic algorithm decision based

on heights of tasks Now take a digraph with dependencies from original

tasks digraph and the two schedules and a nice algorithm

Perform the crossover V1 remains the same in s1 and s1’ as in GA For V2 use CP/MISF list heuristic

Choose task with smallest introduction date first (CP) If several possibilities choose task with more successors

(MISF)Refs: [1][5]




Integrating a List Heuristic – Mutation

Also choose one individual (as in GA)

Sort the tasks according to their precedence constraints Take task with smallest introduction date

Also use CP/MISF assign it to a randomly chosen processor

Refs: [1]




Results and Comparisons

Better initial population

Shorter makespan better quality of solutions

But: longer execution time

Refs: [1]




Results and Comparisons cond.

DigraphCP/MISFparallelsolution

Genetic Algorithm Integrated solution

parallelsolution

execution time

parallel solution

execution time

1 medium 988 332 24 69 3766

1 large 2810 1168 111 181 21935

2 medium 360 326 11 116 849

2 large 1681 1511 73 274 18505

3 medium 1452 933 31 166 9230

3 large 2038 1251 64 223 15068

4 medium 871 548 41 110 13338

4 large 1158 722 85 110 33191

Synthetic diagraphs generated with ANDES-Synth 1. Bellford 2. Diamond1 3. Diamond3 4 Diamond4

Refs: [1]

16 processors


Integrating List Heuristics intogenetic Algorithms for

Multiprocessor Scheduling [1]

I thank you for your attention

¿ Questions ?

[email protected]




References

[1] R.C. Corrêa, A. Ferreira, and P. Rebreyend. Integration list heuristics into genetic algorithms for multiprocessor scheduling. 8th IEEE Symposium on Parallel and Distributed Processing , 1996.

[2] John H. Holland. Genetic Algorithms. http://www.econ.iastate.edu/tesfatsi/holland.GAIntro.htm, April 2006.

[3] E. Hou, N. Ansari, and H. Ren. A genetic algorithm for multiprocessor Scheduling. IEEE Transactions on Parallel and Distributed Systems, 5(2):113-120, Feb. 1994.

[4] J. Hromkomvič. Algorithmics for Hard Problems. Springer Verlag, 2001.[5] A. Doboli, P. Eles. Scheduling under Data and Control Dependencies for

Heterogeneous Architectures. ICCD '98: Proceedings of the International Conference on Computer Design, 1998.

hasso-plattner-institut für softwaresystemtechnik gmbh an der universität potsdam multiprocessor...

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