Euro PVM/MPI 2003 1/22 Venezia, Italia Efficient Parallel - - PowerPoint PPT Presentation

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• Euro PVM/MPI 2003

Venezia, Italia

Efficient Parallel Implementation of Transitive Closure of Digraphs

• C. E. R. Alves

Univsidade S˜ ao Judas Tadeu

• E. N. C´

aceres

Universidade Federal de Mato Grosso do Sul

• A. A. Castro Jr.

Universidade Cat´

• lica Dom Bosco
• S. W. Song

Universidade de S˜ ao Paulo

• J. L. Szwarcfiter

Universidade Federal do Rio de Janeiro

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• The Transitive Closure Problem
• Used in many areas such as

– Network Planning – Distributed Systems Design

• Used in problems such as

– All Shortest Paths in a Directed Graph – Breadth-First Spanning Trees

• Directed graph D(V, E) with |V | = n, |E| = m
• We present a parallel algorithm to compute its transi-

tive closure using – p processors – each with O(n2

p ) local memory

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• Example

5 3 2 4 6 1

A directed graph.

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• Example

5 3 2 4 6 1

Its transitive closure: green edges joining i to j if j can be reached from i.

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• BSP/CGM Model

CGM (Coarse Grained Multicomputer) model: p of pro- cessors, each with its own local memory, communicating through a network. The algorithm alternates between

• Computation round: each processor computes inde-

pendently.

• Communication round: each processor sends/receives

data to/from other processors. Goals:

• Obtain a linear speed-up on p.
• Minimize the number of rounds.

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• The CGM Model

Local computation Synchronization Barrier Global Communication Computation round Communication round P0 P1 P2 Pp−1

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• Previous Parallel Algorithms
• 1. PRAM:
• Karp et al.: CREW: O(log2 n) time with O(M(n))1

processors.

• J´

aJ´ a: CRCW: O(log n) time with O(n3) processors.

• 2. C´

aceres et al.: Acyclic digraph with linear extension labeling O(logp) rounds with O(n3/p) local time

• 3. Dependency Graph Approach:
• Pagourtzis et al.:

O(p) rounds with O(n3/p) local time

1M(n) is the best known sequential bound for multiplying two n × n matrices over a ring

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• Warshall’s Algorithm

Algorithm 1: Warshall’s Algorithm Input: Adjacency matrix Mn×n of graph G Output: Transitive closure of graph G

1: for k ← 1 until n do 2:

for i ← 1 until n do

3:

for j ← 1 until n do

4:

M[i, j] ← M[i, j] or (M[i, k] and M[k, j])

5:

end for

6:

end for

7: end for

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• Partitioning the Adjacency Matrix

1 1 2 2 3 3 4 4 t t t i k k j

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• The Parallel Algorithm

Algorithm 2: Parallel Warshall Input: Adjacency matrix M stored in the p processors: each processor q (1 ≤ q ≤ p) stores submatrices M[(q − 1)n

p + 1..q n p][1..n]

and M[1..n][(q − 1)n

p + 1..q n p].

Output: Transitive closure of graph G represented by the trans- formed matrix M.

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• Algorithm 3: Parallel Warshall

Each processor q (1 ≤ q ≤ p) does the following.

1: repeat 2:

for k = (q − 1)n

p + 1 until q n p do

3:

for i = 0 until n − 1 do

4:

for j = 0 until n − 1 do

5:

if M[i][k] = 1 and M[k][j] = 1 then

6:

M[i][j] = 1 (if M[i][j] belongs to processor different from q then store it for subsequent transmission to the corresponding processor.)

7:

end if

8:

Send stored data to the corresponding processors.

9:

Receive data that belong to processor q from other pro- cessors.

10:

end for

11:

end for

12:

end for

13: until no new matrix entry updates are done

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• The Main Idea
• Make a partition of V (D).
• In each partition, using the edges of D construct a

digraph formed by the edges of D that have at least

• ne of its extremes in the partition.
• Compute the Transitive Closure in each partition.
• Send the computed transitive edges to the proper par-

tition.

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• Example

1 5 3 2 8 4 6 7

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• Example

1 5 3 2 4 6 7

Processor 0

5 3 2 8 6 7

Processor 1

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• Example

1 5 3 2 4 6 7

Processor 0

5 3 2 8 6 7

Processor 1

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• Example

1 5 3 2 8 4 6 7

Processor 0

1 5 3 2 8 4 6 7

Processor 1

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• Implementation
• 64-node Beowulf cluster - low cost microcomputers

with 256MB RAM, 256MB swap memory, CPU In- tel Pentium III 448.956 MHz, 512KB cache.

• 100 Mb fast-Ethernet switch.
• Code in standard ANSI C and LAM-MPI Version 6.5.6.
• Tests on randomly generated digraphs with 20 % prob-

ability of an edge between two vertices.

• In all the tests, the number of communication rounds

required are less than log p.

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• Implementation Results
• 480x480
• 512x512

10 20 30 40 50 60 5 10 15 20 25

• No. Processors

Seconds

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• Implementation Results
• ◦ ◦
• 960x960
• • •
• 1024x1024

⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ 1920x1920 10 20 30 40 50 60 500 1000 1500

• No. Processors

Seconds

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• Implementation Results
• 480x480
• 512x512

10 20 30 40 50 60 5 10 15

• No. Processors

Speedup

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• Implementation Results
• 960x960
• 1024x1024

⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ 1920x1920 10 20 30 40 50 60 10 20 30

• No. Processors

Speedup

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• Conclusion

A BSP/CGM algorithm for the Transitive Closure problem.

• Digraph with n vertices and m edges.
• The number of communication rounds measured: O(log p).
• Local computation time: O(mn/p).