asc2003 (july 15,2003)1 [email protected] uniformly distributed sampling: an exact...
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ASC2003 (July 15,2003) 1 [email protected]
Uniformly Distributed Sampling:
An Exact Algorithm for GA’s Initial Population in A Tree
GraphH. 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: [email protected]
[email protected]://h_s_shahhoseini.tripod.com/papers/ASC2003UDS.ppt
ASC2003 (July 15,2003) 2 [email protected]
Overview of Presentation
Task Graph Scheduling Problems and IssuesUniform Initial PopulationPrevious WorksUniformly Distributed Sampling (UDS)How the Algorithm worksFuture Works
ASC2003 (July 15,2003) 3 [email protected]
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 [email protected]
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 [email protected]
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) 7 [email protected]
Search Algorithm
The space of valid permutations was searched for finding the best permutation.
Examples: Genetic Algorithm and Tabu Search.
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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) 11 [email protected]
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.
ASC2003 (July 15,2003) 12 [email protected]
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
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ordered-combinationConsider 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 .
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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 [email protected]
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.
),(),(
),(),(),,(
mpRnmpR
nmRnmpRnmpR
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Valid Permutation’s AttributesValid Permutation’s Attributes, or VPAis 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.
ASC2003 (July 15,2003) 17 [email protected]
Computation of VPA for node k
In the Tree graph the hierarchical computing can be used for finding VPA of nodes in the graph.
321321 ),,( ppplllRp 1321 llll
ASC2003 (July 15,2003) 18 [email protected]
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.
ASC2003 (July 15,2003) 19 [email protected]
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:
s
innnnnnnRSS isjjj
pllllllRN1
),,,1,,,,(1121
k
ki
kkn
kj
kj
kj
kkkj
n
innnnnnnn
pllllllRN1
),,,1,,,,(1121
k
ki
kj
n
in
nkj
N
Nn
1
)Pr(
ASC2003 (July 15,2003) 21 [email protected]
Example
For second node F ={b,c,d} and
In the same manner:
8
3
)3,1,3()4,0,3()4,1,2(
)4,1,2()Pr(
RRR
Rb
8
1)Pr( c
2
1
8
4)Pr( d
ASC2003 (July 15,2003) 22 [email protected]
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.
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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.
