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PARALLEL GRAPH PARTITIONING ON A HYPERCUBE

DISTRIBUTED GENERATION OF PAIRWISE COMBINATIONS

F. Ercal, P. Sadayappan, and J. Ramanujan

University of Missouri-Rolla and The Ohio State University

PROBLEM DEFINITION

- Given a graph G(V,E), |V|=N |E|=e
- Obtain a K partitions from G with the following constraints:
- Balanced: Each partition has equal size
- Minimum cut: number of edges across partition is minimized
- arises in: TasK Allocation, VLSI layout, File Placement etc.
- Intractable, no polynomial time algorithm is Known
- Heuristics needed
- Kernighan-Lin Mincut Heuristic (1970)
- Time complexity: O(N2logN)
- Extension by Fiduccia and Mattheyses (1982)
- Used Buckets and moves. Linear time algorithm: O(e)

P2

0

-2

v7

v1

+2

+1

v2

v6

-2

v3

-1

v8

+1

+1

v4

v5

-2

0

v7

v1

v2

-1

v6

-2

v3

-3

v8

+1

+1

v4

v5

MINCUT ALGORITHMCUT=5

IF V2 MOVES GAIN=2 and TOT_GAIN=2

CUT=3

IF V5 MOVES GAIN=1 and TOT_GAIN=3

TIME COMPLEXITY

Sequential Time Complexity for Recursive Bisection

N + 2*(N/2) + 4*(N/4) + …….2p*(N/2p) ===> O(N*logK)

Parallel Time Complexity for Recursive Bisection

N + N/2 + N/4 + ……. N/2p ===> O(N)

- COMMENT:
- speedup is very limited
- to increase speedup, bisection algorithm must be parallelized

P2

P3

P6

P4

P5

P7

P8

PAIRWISE MINCUTPAIRS TO BE CONSIDERED FOR MINCUT

(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)

(2,3) (2,4) ………….. (2,8)

…….

(7,8)

TIME COMPLEXITY

Sequential Time Complexity for Pairwise Mincut

Parallel Time Complexity for Recursive Bisection

(100% processor utilization)

- CONCLUSIONS
- Sequential Recursive Bisection (RB) has much lower time complexity than Pairwise Mincut (PM)
- but superior parallelizability of PM renders its parallel time complexity comparable to that of parallel RB

- Perform repeated bisection, each time doubling the number of partitions, until K partitions are obtained

Time Complexity

N+ 2*(N/2) + 4*(N/4)+….+2P*(N/2P) ==> O(N*logK)

2) PAIRWISE MINCUT

- Initially obtain K partitions. Try to reduce the cut-size between each pair of partitions. K(K-1)/2 pairs (each of size 2N/K) must be considered

Time Complexity

3) Any combination of

RECURSIVE BISECTION+PAIRWISE MINCUT

DISTRIBUTED GENERATION OF PAIRWISE COMBINATIONS ON A HYPERCUBE

Problem

- Given 2P disjoint items, P*(2P-1) distinct pairs can be formed.
- How would you efficiently generate these pairs on the processors of a hypercube ?
- Similar to the problem of distributed scheduling of a round-robin tournament between 2C players using C courts, where the paths between courts form a hypercube topology
- maximum utilization of courts (processor utilization)
- +
- minimum walking between courts (min. comm. overhead)

C1

C2

C1

B00

B01

B10

B11

P00

P01

A00

A01

A10

A11

P10

P11

C1

C2

C1

C2

C1

C2

C1

C2

A00

A01

P01

A10

A11

P10

B00

B01

P11

B10

B11

C1

C2

d=0

d=1

d=2

P00

P01

P10

P11

A00

A01

A10

A11

B00

B01

B10

B11

P00

Distributed PC Algorithm on a 2d Hypercube (4 Processors)

A2

A3

:

:

AK/2

AK/2+1

:

:

AK

B1

B2

B3

:

:

BK/2

BK/2+1

:

:

BK

1

CYCLIC-TOUR

RING-FRAGMENTATION

2

A1

A2

:

AK/4

AK/4+1

:

AK/2

AK/2

AK/2+1

:

A3K/4

A3K/4+1

:

AK

B1

B2

:

BK/4

BK/4+1

:

BK/2

BK/2

BK/2+1

:

B3K/4

B3K/4+1

:

BK

CYCLIC-TOUR

CYCLIC-TOUR

RING-FRAGMENTATION

Ring Communication in different phases of Distributed PC algorithm

0110

0111

1110

1111

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1100

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(a) d=0 1 ring of size 16

1110

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0110

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0001

(b) d=1 2 rings of size 8

Ring Communication in different phases of Distributed PC algorithm (Contd..)

1110

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0110

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0100

1110

0110

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1011

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(c) d=2 4 rings of size 4

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0110

0111

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0100

1110

0110

1010

1011

0010

0011

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0000

1001

0001

(d) d=3 8 rings of size 2

Ring Communication in different phases of Distributed PC algorithm (Contd..)

1110

1111

0110

0111

1100

0100

1110

0110

1010

1011

0010

0011

1000

0000

1001

0001

(e) d=4 16 rings of size 1

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