Uniformly Distributed Sampling: An Exact Algorithm for GA’s Initial Population in A Tree Graph

1. Uniformly Distributed Sampling:An Exact Algorithm for GA’s Initial Population in A Tree Graph H. S. Shahhoseini, PhD Assistant Professor at Iran University of Science & Technology Director of Talent Student Affairs of the University IEEE TFCC Coordinator in Middle East Region Countries IEEE TFCC Executive Committee Member email: h_s_shahhoseini@hotmail.com h_s_shahhoseini@iust.ac.ir http://h_s_shahhoseini.tripod.com/papers/ASC2003UDS.ppt 1h_s_shahhoseini@hotmail.com

2. Overview of Presentation • Task Graph Scheduling Problems and Issues • Uniform Initial Population • Previous Works • Uniformly Distributed Sampling (UDS) • How the Algorithm works • Future Works 2h_s_shahhoseini@hotmail.com

3. Task Graph Scheduling Task Scheduling Problem: Finding the best sequence of the task to the processors in a parallel system. Task Scheduling is an NP-Hard optimization problem which means the time of operation is a non-polynomial function of the size of the problem. 3h_s_shahhoseini@hotmail.com

4. Problems and Issues Two main solution: • Heuristic algorithms : Usually restricts the search space. • Search algorithms : Globally investigate the search space for finding the best solution. Search algorithms are very sensitive to the start point. 4h_s_shahhoseini@hotmail.com

5. Heuristic Usually heuristics are list-based algorithm. • Assigning a property to any node on basis of the weight of the graph’s links and nodes. • Constructing a list of nodes according their properties in descending or ascending manner. • Selecting the nodes from head of the list. • Assigning to the processor who can start their job earlier. Examples: HLFET (by t_level Property), PDEFT (by b_level Property) and MCP (by ALAP Property) 5h_s_shahhoseini@hotmail.com

6. The Structure of the Heuristic 6h_s_shahhoseini@hotmail.com

7. Search Algorithm • The space of valid permutations was searched for finding the best permutation. • Examples: Genetic Algorithm and Tabu Search. 7h_s_shahhoseini@hotmail.com

8. Genetic Algorithm • A group of the individuals are selected as initial population, named chromosome. • The population is regenerated from them by fitness, mutation functions. • The most fitted chromosomes are selected as a next generation by selection functions. The initial population affects on the speed of reaching the optimum schedule. 8h_s_shahhoseini@hotmail.com

9. Example of a graph Valid Permutation 9h_s_shahhoseini@hotmail.com

10. Previous Methods 10h_s_shahhoseini@hotmail.com

11. Example of a graph Valid Permutation • In previous algorithm b and c are similarly selected from set F as second node which is incorrect. • To have a uniformly distributed initial population, the selection probability must be non-uniform. • The selection probability must be according to remaining selection subspace size, Nrss, which produced by selecting the previous node in the permutation. 11h_s_shahhoseini@hotmail.com

12. Uniformly Distributed Sampling To describe Uniformly Distributed Sampling, UDS: • Defining ordered-combinationof permutation with variable lengths. • Proving a lemma for determining the number of ordered-combination of two permutation, R(m,n). • Defining the node’s Valid Permutation’s Attributes, VPA 12h_s_shahhoseini@hotmail.com

13. ordered-combination • Consider two arbitrary permutation A1 and A2 with lengths of L1 and L2. • The ordered combination of and is a new permutation with length of L1+L2 whose element consist of the elements of A1 and A2, with their order in A1 and A2. • There are many ordered combinations for two permutations 1234 and abc. For example 12a3b4c and a1b2c34 are two ordered combination of and . 13h_s_shahhoseini@hotmail.com

14. Lemma : Number of ordered-combination Equations (1) and (2) can be simply proved, so they are accepted and the last equation can be prove by inductive proof. 14h_s_shahhoseini@hotmail.com

15. Lemma : Number of ordered-combination Equation (3), can be extended in the same manner for more than two permutations as follows: where p , m , n are the lengths of three different permutation. 15h_s_shahhoseini@hotmail.com

16. Valid Permutation’s Attributes • Valid Permutation’s Attributes, or VPA is defined as an ordered pair for any node, which is shown by (lk , pk). • lk : is the number of valid permutations, which contain node k and its entire successor nodes. • pk : is the length of these permutations. 16h_s_shahhoseini@hotmail.com

17. Computation of VPA for node k • In the Tree graph the hierarchical computing can be used for finding VPA of nodes in the graph. 17h_s_shahhoseini@hotmail.com

18. Computation of VPA • To assign VPA to the nodes, UDS starts from the exit nodes of the graph and assign the (1,1) to them. • Then it can recursively compute VPA for the parent nodes VPA. • The selection probability are proportional to remaining selection subspace size, Nrss, which produced by selecting the previous node in the permutation. 18h_s_shahhoseini@hotmail.com

19. Selection probability • For node nj for selecting k-th element of permutation. So the selection probability of j-th node of set F , when the k-th element of permutation to be selected will be: 19h_s_shahhoseini@hotmail.com

20. UDS Summary 20h_s_shahhoseini@hotmail.com

21. Example For second node F ={b,c,d} and In the same manner: 21h_s_shahhoseini@hotmail.com

22. Conclusion • A sampling algorithm, UDS, was proposed for making uniformly selected initial population of GA in the domain for the task graph scheduling . • The validity of UDS is mathematically investigated. 22h_s_shahhoseini@hotmail.com

23. Future Works • showing how this initial selection reduces the run time of GA for finding the best schedule of the task graph in different applications. • Uniformly Distributed Sampling, UDS, is introducedfor graph with Tree structure. Another area for future work is to extend this approach for the other topologies of the graph. 23h_s_shahhoseini@hotmail.com

24. Thank You. 24h_s_shahhoseini@hotmail.com