All%20Pairs%20Shortest%20Path%20Algorithms

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All Pairs Shortest Path Algorithms

• Aditya Sehgal
• Amlan Bhattacharya

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Floyd Warshal Algorithm

• All-pair shortest path on a graph
• Negative edges may be present
• Negative weight cycles not allowed
• Uses dynamic programming
• Therefore bottom up solution

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Problem Description

• A graph G(V,E,w) be a weighted graph with
  vertices v1, v2, vn.
• Find the shortest path between each pair of
  vertices vi , vj, where i,j lt n and find out
  the all pairs shortest distance matrix D

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Principle on which it is based..

• For 2 vertices vi, vj in V, out of all paths
  between them, consider the ones which their
  intermediate vertices in v1, v2, ..vk
• If pki,j be the minimum cost path among them,
  the cost being d(k)i,j
• If vk is not in the shortest path then
• pki,j pk-1i,j

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Principles cont.

• If vk is in pki,j , then we can pki,j break
  into 2 paths - from vi to vk and
• from vk to vj
• Each of them uses vertices in the set
• v1, v2, .vk-1

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Recurrence equation

• d(k)i,j w(vi, vj) if k 0
• min d(k-1)i,j, d(k-1)i,k d(k-1)k,j if
  kgt0
• The output of the algorithm is the matrix
• D(n) (d(n)i,j )

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Algorithm

• procedure FLOYD_ALL_PAIRS (A)
• begin
• D(0) A
• for k 1 to n do
• for i 1 to n do
• for j 1 to n do
• d(k)i,j min d(k-1)i,j,
  d(k-1)i,k d(k-1)k,j
• end FLOYD_ALL_PAIRS

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Complexity of the sequential algorithm..

• Running time O (n3)
• Space complexity O (n2)

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An example graph..
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4
3
1
3
8
2
7
1
-5
-4
5
4
6
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How it works .

• 0 3 8 m -4
• m 0 m 1 7
• D(0) m 4 0 m m
• 2 m -5 0 m
• m m m 6 0

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How it works .

• 0 3 8 m -4
• m 0 m 1 7
• D(1) m 4 0 m m
• 2 5 -5 0 m
• m m m 6 0

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How it works .

• 0 3 8 4 -4
• m 0 m 1 7
• D(2) m 4 0 5 11
• 2 5 -5 0 -2
• m m m 6 0

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How it works .

• 0 3 8 4 -4
• m 0 m 1 7
• D(3) m 4 0 5 11
• 2 -1 -5 0 -2
• m m m 6 0

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How it works .

• 0 3 -1 4 -4
• 3 0 -4 1 -1
• D(4) 7 4 0 5 3
• 2 -1 -5 0 -2
• 8 5 1 6 0

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How it works .

• 0 1 -3 2 -4
• 3 0 -4 1 -1
• D(5) 7 4 0 5 3
• 2 -1 -5 0 -2
• 8 5 1 6 0

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Parallel formulation

• If p is the number of processors available,
• the matrix D(k) is partitioned into p parts
• For computation, each process needs some elements
  from the kth row and column in the matrix D(k-1)
• Use 2-D or 1-D block mapping

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2-D block mapping
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2-D Block mapping cont

• Each block size is (n / p0.5 ) by (n / p0.5 )
• kth iteration , Pi,j needs some part of kth
  row and column of the Dk-1 matrix
• Thus at this time p0.5 processors containing a
  part of the kth row sends it to the p0.5 - 1
  processes in the same column
• Similarly , p0.5 processors containing a part
  of the kth column sends it to the
• p0.5 -1 processes in the same row

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Communication patterns ..
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Algorithm .
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Analysis(computation)..

• Each process is assigned n / p0.5 n / p0.5
• i.e n2 / p numbers. In each iteration , the
  computation time is O (n2 / p )
• For n iterations the computation time is
• O (n3 / p )

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Analysis(communication)..

• After a broadcast, a synchronization time of O(n
  log p) is required
• Each process broadcasts n / p0.5 elements
• Thus total communication time takes
• O((n log p) / p0.5 )
• Thus for n iterations time
• O((n2log p) / p0.5 )

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Total parallel run time

• Tp O((n2log p) / p0.5 ) O (n3 / p )
• S O( n3 ) / Tp
• E 1 / (1 O((p0.5 log p)/n) )
• Number of processors used efficiently is
• O(n2 log2n )
• Isoeffiency function O (p1.5 log3 p )

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How to improve on this .

• Waiting for the k th row elements by each
  processor after all the processors complete the
  k-1 th iteration
• Start as soon as it finishes the k-1 th iteration
  and it has got the relevant parts of the D(k-1)
  matrix
• Known as Pipelined 2-D Block mapping

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How it is carried out ..

• Pi,j which has elements of the k th row after
  the k-1th iteration sends the part of the D(k-1)
  matrix to processors Pi,j1, Pi,j-1 .
• Similarly Pi,j which has elements of the k th
  column after the k-1th iteration sends the part
  of the D(k-1) matrix to processors Pi-1,i,
  Pi1,j
• Each of these values are stored and forwarded to
  the all the processors in the same row and
  column.
• The forwarding is stopped when mesh boundaries
  are reached

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Pipelined 2 D Block mapping
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Total parallel run time

• Tp O(n3/ p ) O (n )
• S O( n3 ) / O(n3/ p ) O (n )
• E 1/ (1 O(p/n2 )
• Number of processors used efficiently O(n2 )
• Isoeffiency function O (p1.5 )

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Comparison.
Algorithm Max Processes for E O(1) Parallel Run time Isoefficiency function

Djikstra source- partitioned O(n) O(n2) O(p3)
Djikstra source- parallel O(n2/ log n) O(n log n) O(p1.5 log1.5 p)
Floyd 1 D Block O(n/ log n) O(n2 log n) O(p3 log3 p)
Floyd 2-D Block O(n2/ log2 n) O(n log2 n) O(p1.5 log3 p)
Floyd pipelined 2-D Block O(n2) O(n) O(p1.5)