multiprocessor scheduling

HASSO-PLATTNER-INSTITUTfür Softwaresystemtechnik GmbH an der Universität Potsdam

Multiprocessor Scheduling

Integrating List Heuristics into genetic Algorithms for

Multiprocessor Scheduling [1]

May, 22 2006Stefan 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 solvedHeuristics 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 processor

a. 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 processorc. 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 – AnimationPopulation 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 37661 large 2810 1168 111 181 219352 medium 360 326 11 116 8492 large 1681 1511 73 274 185053 medium 1452 933 31 166 92303 large 2038 1251 64 223 150684 medium 871 548 41 110 133384 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 ?

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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.

multiprocessor scheduling

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