All Pairs Shortest Path

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All Pairs Shortest Path
Instructor Yao-Ting Huang
Bioinformatics Laboratory, Department of Computer
Science Information Engineering, National Chung
Cheng University.
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All pair shortest Path Problem
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Another Dynamic Programming Idea
Let P(i, j) be the shortest path between i and j.
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Dynamic Programming Idea
Input weighted, directed graph D(V,E,w) of n
vertices, say V1,2, , n. Let P(i, j) be the
shortest path between i and j. DP-related
Question How can we decompose the problem into
some simpler ones?
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Dynamic Programming Idea
Input weighted, directed graph D(V,E,w) of n
vertices, say V1,2,,n. Let P(i, j) be the
shortest path between i and j. DP-related
Question How can we decompose the problem into
some simpler ones? A It should go through some
vertex k.
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Dynamic Programming Idea
Let P(i, j) be the shortest path between i and
j. Let Vk1, 2, , k. Sk(i, j) length of a
shortest path between i and j whose interior
vertices are all in Vk.
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Dynamic Programming Idea
Let P(i,j) be the shortest path between i and
j. Let Vk1,2, , k. Sk(i, j) length of the
shortest path between i and j whose interior
vertices are all in Vk. What is the length of the
shortest path between i and j?
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Dynamic Programming Idea
Let P(i,j) be the shortest path between i and
j. Let Vk1,2,k. Sk(i, j) length of the
shortest path between i and j whose interior
vertices are all in Vk. What is the length of the
shortest path between i and j? It is Sn(i, j).
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Dynamic Programming Idea
Let Vk1,2,k. Sk(i, j) length of the
shortest path between i and j whose interior
vertices are all in Vk. If k is on the shortest
path from i to j, Sk(i, j) ???
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Key Observation

• Sk(i, j) is determined by the shortest paths
  whose intermediate from 1, , k-1
• Case1 If k is not an intermediate vertex of P
• Path P is a shortest path from i to j with
  intermediates from 1,k-1
• Case2 If k is an intermediate vertex of path P
• Path P can be broken down into i ? p1? k ? p2 ? j
• P1 is the shortest path from i to k with all
  intermediate in the set 1,2,,k
• P2 is the shortest path from k to j with 1,2,,k

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Key Observation
P2All intermediate vertices in 1,2,..,k-1
P1All intermediate vertices in 1,2,..,k-1
Sk-1(i, k)
Sk-1(k, j)
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p1
p2
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P All intermediate vertices in 1,2,..,k
Sk (i, j)
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Dynamic Programming Idea
Let Vk1,2,k. Sk(i, j) length of the
shortest path between i and j whose interior
vertices are all in Vk. If k is on the shortest
path from i to j, Sk(i, j) Sk-1(i, k) Sk-1(k,
j)
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Dynamic Programming Idea

• Let Vk1,2,k.
• Sk(i, j) length of the shortest path between i
  and j whose interior vertices are all in Vk.
• If k is on the shortest path from i to j,
• Sk(i, j) Sk-1(i, k) Sk-1(k ,j).
• If k is not on the shortest path from i to j
• Sk(i, j) Sk-1(i, j).

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Dynamic Programming Formula

• Let Vk1,2,k.
• Sk(i, j) length of the shortest path between i
  and j whose interior vertices are all in Vk.
• S0(i, j) w(i, j).
• Sk(i, j) min Sk-1(i, j), Sk-1(i, k) Sk-1(k,
  j) .

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Floyd-Warshall Algorithm

• Negative edges is allowed
• But assume that no negative-weight cycle
• Dynamic Programming
• A recursive solution
• Computing from bottom-up
• Constructing a shortest path

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Dynamic Programming Algorithm
Floyd(W1n,1n,S1n,1n) For i 1 to n For
j 1 to n if lti, jgt in E then S(i, j) ?
W(i, j) else S(i, j) ? 8 For k 1 to n For i
1 to n For j 1 to n if Si,
j gt Si, k Sk, j then Si, j
? Si, k Sk, j
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Dynamic Programming
Sk (k1,2,3,4,5)
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A detailed example
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S0W
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A detailed example
For k 1 to n For i 1 to n For j 1
to n if Si, j gt Si, k Sk, j
then Si, j ? Si, k Sk, j
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A detailed example
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A detailed example
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A detailed example
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Extracting the Shortest Paths

• How can we retrieve the shortest paths?
• Maintain a predecessor pi, j.
• If the shortest path does not pass through any
  intermediate vertex, then the predecessor pi, j
  0.

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How do we retrieve the actual path?

• We maintain Pki, j as follows
• P0i, j 0 for all i, j in V
• Pki, j Pk-1i, j, if Ski, j Sk-1i, j
• Pki, j k, if Ski, j ? Sk-1i, j

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How do we retrieve the actual path?

• We maintain Pki,j as follows
• P0i, j 0 for all i,j in V
• Pki, j Pk-1i,j, if Ski,j Sk-1i,j
• Pki, j k, if Ski,j ? Sk-1i,j
• To reconstruct the shortest path in Pni, j
• If Pni, j0, edge lti, jgt is the shortest path.
• If Pni, jk, then k is an interior vertex on
  the path, other interior vertices can be obtained
  by checking Pni, k and Pnk, j.

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Floyds Algorithm (with matrix P)
Floyd(W1n,1n,S1n,1n,P1n,1n) For i 1
to n For j 1 to n Pi,j ? 0 if
lti,jgt in E then Si,j ? Wi,j else Si,j ?
8 For k 1 to n For i 1 to n For j
1 to n if Si,j gt Si,k Sk,j
then Si,j ? Si,k Sk,j
Pi,j ? k
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A detailed example
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How do we retrieve the shortest path from 3 to 5?
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A detailed example
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How do we retrieve the shortest path from 3 to
5? P3,52, so 325
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A detailed example
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How do we retrieve the shortest path from 3 to
5? P3,52, so 325 P3,21,so
3125 P3,1P2,5P1,20 So 3?1?2?5.
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Analysis

• Running time is clearly T(?)
• T(n3) -gt T(V3)
• Faster than previous algorithms.
  O(V4),O(V3lgV)