Observation on Parallel Computation of Transitive and Max-closure Problems

Slide 2

Motivation

TC problem has numerous applications in many areas of computer science.

Lack of course-grained algorithms for distributed environments with slow communication.

Decreasing the number of dependences in a solution could improve a performance of the algorithm.

Slide 3

What is transitive closure?

GENERIC TRANSITIVE CLOSURE PROBLEM (TC)

Input: a matrix A with elements from a semiring S= < , >

Output: the matrix A*, A*(i,j) is the sum of all simple paths

from i to j

< , > TC

< or , and > boolean closure - TC of a directed graph

< MIN, + > all pairs shortest path

<MIN, MAX> minimum spanning tree {all(i,j): A(i,j)=A*(i,j)}

Slide 4

Fine–grain and Coarse-grained algorithms for TC problem

Warshall algorithm (1 stage)

Leighton algorithm (2 stages)

Guibas-Kung-Thompson (GKT) algorithm (2 or 3 stages)

Partial Warshall algorithm (2 stages)

Slide 5

Warshall algorithm

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Warshall algorithm

k+1

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k+2

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for k:=1 to n

for all 1i,jn parallel do

Operation(i, k, j)

----------------------------------Operation(i, k, j): a(i,j):=a(i,j) a(i,k) a(k,j)----------------------------------

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Slide 7

Coarse-Grained computations

n

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A32

A11

A24

Slide 8

Naïve Course Grained Algorithms

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011001

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Actual =

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5II

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011001

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Actual =

Slide 10

Course-grained Warshall algorithm

Algorithm Blocks-Warshallfor k :=1 to N do

A(K,K):=A*(K,K)

for all 1 I,J N, I K J parallel do

Block-Operation(K,K,J) and Block-Operation(I,K,K)

for all 1 I,J N parallel do Block-Operation(I,K,J)

----------------------------------------------------------------------Block-Operation(I, K, J): A(I,J):=A(I,J) A(I,K) A(K,K) A(K,J)

----------------------------------------------------------------------

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Implementation of Warshall TC Algorithm

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k k

k

The implementation in terms of multiplication of submatrices

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NEW

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Decomposition properties In order to package elementary operations

into computationally independent groups we consider the following decomposition properties:

A min-path from i to j is a path whose intermediate nodes have numbers smaller than min (i,j)

A max-path from i to j is a path whose intermediate nodes have numbers smaller than max(i,j)

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Slide 15

KGT algorithm

KGT algorithm

AA’B

B’

C

C’

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An example graph

a max-path

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an initial path

It is transitive closure closure of the graph

Slide 17

What is Max-closure problem?

Max-closure problem is a problem of computing all max-paths in a graph

Max-closure is a main ingredient of the TC closure

Slide 18

Max-Closure --> TC

Max-closure computation performs 2/3 of total operations

Max-closure

algorithm

Max-to-Transitiveperforms 1/3 of the

total operations

The algorithm ”Max-to-Transitive ” reduces TC to matrix multiplication once the Max-Closure is computed

Slide 19

A Fine Grained Parallel Algorithm

Algorithm Max-to-TransitiveInput: matrix A, such that Amax = AOutput: transitive closure of AFor all k n parallel do For all i,j max(i,j) <k, ij Parallel do Operation(i,k,j)

Algorithm Max-Closurefor k :=1 to n do for all 1 i,j n, max(i,j) > k,

ij parallel do Operation(i,k,j)

Slide 20

Coarse-grained Max-closure Algorithm

Algorithm CG-Max-Closure {Partial Blocks-Warshall}

for K:=1 to N do

A(K,K):= A*(K,K)

for all 1 I,J N, I K J parallel do

Block-Operation(K,K,J) and Block-Operation(I,K,K)

for all 1 I,J N, max(I,J) > K MIN(I,J) parallel do

Block-Operation(I,K,J)

--------------------------------------------------------------------------------Blocks-Operation(I, K, J): A(I,J):=A(I,J) A(I,K) A(K,J)

Slide 21

Implementation of Max-Closure

Algorithmk

k

k k

k

The implementation in terms of multiplication of submatrices

Slide 22

Experimental results~3.5 h

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Increase / Decrease of overall time

While computation time decreases when adding processes the communication time increase => there is an ”ideal” number of processors

All experiments were carried out on cluster of 20 workstations=> some processes were running more than one worker-process.

Slide 24

Conclusion The major advantage of the algorithm is

the reduction of communication cost at the expense of small communication cost

This fact makes algorithm useful for systems with slow communication