ASC2003 (July 15,2003)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

ASC2003 (July 15,2003)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

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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)

ASC2003 (July 15,2003)6 The Structure of the Heuristic

ASC2003 (July 15,2003)7 Search Algorithm  The space of valid permutations was searched for finding the best permutation.  Examples: Genetic Algorithm and Tabu Search.

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)9 Example of a graph Valid Permutation

ASC2003 (July 15,2003)10 Previous Methods

ASC2003 (July 15,2003)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, N rss, which produced by selecting the previous node in the permutation.

ASC2003 (July 15,2003)12 Uniformly Distributed Sampling To describe Uniformly Distributed Sampling, UDS:  Defining ordered-combination of 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

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)16 Valid Permutation’s Attributes  Valid Permutation’s Attributes, or VPA is defined as an ordered pair for any node, which is shown by (l k, p k ). l k : is the number of valid permutations, which contain node k and its entire successor nodes. p k : is the length of these permutations.

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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, N rss, which produced by selecting the previous node in the permutation.

ASC2003 (July 15,2003)19 Selection probability  For node n j 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:

ASC2003 (July 15,2003)20 UDS Summary

ASC2003 (July 15,2003)21 Example For second node F ={b,c,d} and In the same manner:

ASC2003 (July 15,2003)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.

ASC2003 (July 15,2003)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 introduced for graph with Tree structure. Another area for future work is to extend this approach for the other topologies of the graph.

ASC2003 (July 15,2003)24 Thank You.