DISTRIBUTED GENERATION OF PAIRWISE COMBINATIONS

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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
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PROBLEM DEFINITION

• Given a graph G(V,E), VN Ee
• 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)

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MINCUT ALGORITHM
CUT5
IF V2 MOVES GAIN2 and TOT_GAIN2
CUT3
IF V5 MOVES GAIN1 and TOT_GAIN3
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MINCUT ALGORITHM (Contd..)
-2
0
v7
v1
v2
-1
v6
v5
-2
v3
-3
v8
v4
-1
CUT2
IF V1 MOVES GAIN0 and TOT_GAIN3
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RECURSIVE BISECTION
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TIME COMPLEXITY
Sequential Time Complexity for Recursive Bisection
N 2(N/2) 4(N/4) .2p(N/2p) gt
O(NlogK)
Parallel Time Complexity for Recursive Bisection
N N/2 N/4 . N/2p gt
O(N)

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

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PAIRWISE MINCUT
PAIRS 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)
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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

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1) RECURSIVE BISECTION

• 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) gt O(NlogK)
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 BISECTIONPAIRWISE MINCUT
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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)

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C1
C2
A00 A01 A10 A11
B00 B01 B10 B11
P00 P01 P10 P11
d0 d1 d2
P00
Distributed PC Algorithm on a 2d Hypercube (4
Processors)
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A1 A2 A3 AK/2 AK/21 AK
B1 B2 B3 BK/2 BK/21 BK
1
CYCLIC-TOUR
RING-FRAGMENTATION
2
A1 A2 AK/4 AK/41 AK/2
AK/2 AK/21 A3K/4 A3K/41 AK
B1 B2 BK/4 BK/41 BK/2
BK/2 BK/21 B3K/4 B3K/41 BK
CYCLIC-TOUR
CYCLIC-TOUR
RING-FRAGMENTATION
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Ring Communication in different phases of
Distributed PC algorithm
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Ring Communication in different phases of
Distributed PC algorithm (Contd..)
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Ring Communication in different phases of
Distributed PC algorithm (Contd..)
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