Observation%20on%20Parallel%20Computation%20of%20Transitive%20and%20Max-closure%20Problems - PowerPoint PPT Presentation

About This Presentation
Title:

Observation%20on%20Parallel%20Computation%20of%20Transitive%20and%20Max-closure%20Problems

Description:

TC problem has numerous applications in many areas of computer science. ... Max-closure is a main ingredient of the TC closure. Slide 18. Max-Closure -- TC ... – PowerPoint PPT presentation

Number of Views:39
Avg rating:3.0/5.0
Slides: 25
Provided by: IgorPo
Category:

less

Transcript and Presenter's Notes

Title: Observation%20on%20Parallel%20Computation%20of%20Transitive%20and%20Max-closure%20Problems


1
Observation on Parallel Computation of Transitive
and Max-closure Problems
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.

3
What is transitive closure?
GENERIC TRANSITIVE CLOSURE PROBLEM (TC) Input a
matrix A with elements from a semiring S lt ?,?
gt Output the matrix A, A(i,j) is the sum of
all simple paths from i to j lt ? , ? gt
TC lt or , and gt boolean closure - TC of a
directed graph lt MIN, gt all pairs shortest
path ltMIN, MAXgt minimum spanning tree all(i,j)
A(i,j)A(i,j)
4
Finegrain 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)

5
Warshall algorithm
k
k1
k2
  • for k1 to n
  • for all 1?i,j?n parallel do
  • Operation(i, k, j)
  • ----------------------------------
  • Operation(i, k, j) a(i,j)a(i,j) ? a(i,k) ?
    a(k,j)
  • ----------------------------------

k
k1
k2
Warshall algorithm
6
(No Transcript)
7
Coarse-Grained computations
A11
A24
n
A32
n
8
Naïve Course Grained Algorithms
9
II
I
10
Course-grained Warshall algorithm
  • Algorithm 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 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)
  • --------------------------------------------------
    --------------------

11
Implementation of Warshall TC Algorithm
k
k
k
k
k
The implementation in terms of multiplication of
submatrices
12
II
I
13
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)

14
(No Transcript)
15
KGT algorithm
16
An example graph
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

18
Max-Closure --gt TC
  • Max-to-Transitive
  • performs 1/3 of the total operations

Max-closure algorithm
  • Max-closure computation performs 2/3 of total
    operations

The algorithm Max-to-Transitive reduces TC to
matrix multiplication once the Max-Closure is
computed
19
A Fine Grained Parallel Algorithm
Algorithm Max-Closure for k 1 to n do for all
1 ? i,j ? n, max(i,j) gt k, i?j parallel do
Operation(i,k,j)
  • Algorithm Max-to-Transitive
  • Input matrix A, such that Amax A
  • Output transitive closure of A
  • For all k ? n parallel do
  • For all i,j max(i,j) ltk, i?j
  • Parallel do Operation(i,k,j)

20
Coarse-grained Max-closure Algorithm
  • Algorithm CG-Max-Closure Partial
    Blocks-Warshall
  • for K1 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) gt 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)

21
Implementation of Max-ClosureAlgorithm
k
k
k
k
k
The implementation in terms of multiplication of
submatrices
22
Experimental results
3.5 h
23
Increase / Decrease of overall time
  • While computation time decreases when adding
    processes the communication time increase
  • gt there is an ideal number of processors
  • All experiments were carried out on cluster of 20
    workstations
  • gt some processes were running more than one
    worker-process.

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
Write a Comment
User Comments (0)
About PowerShow.com