introduction to algorithms all-pairs shortest paths my t. uf

Introduction to Algorithms

All-Pairs Shortest Paths

My T. Thai @ UF

Single-Source Shortest Paths Problem

Input: A weighted, directed graph G = (V, E) Output: An n × n matrix of shortest-path

distances δ. δ(i, j) is the weight of a shortest path from i to j.

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1 2 3 4 5

1 0 1 -3 2 -4

2 3 0 -4 1 -1

3 7 4 0 5 3

4 2 -1 -5 0 -2

5 8 5 1 6 0

Can use algorithms for Single-Source Shortest Paths

Run BELLMAN-FORD once from each vertex Time:

If there are no negative-weight edges, could run Dijkstra’s algorithm once from each vertex Time:

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Outline Shortest paths and matrix multiplication

Floyd-Warshall algorithm

Johnson’s algorithm

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Recursive solution

Optimal substructure: subpaths of shortest paths are shortest paths

Recursive solution: Let = weight of shortest path from i to j that contains ≤ m edges.

Where wij:My T. Thai

mythai@cise.ufl.edu

Computing the shortest-path weights bottom up

All simple shortest paths contain ≤ n − 1 edges

Compute from bottom up: L(1), L(2), . . . , L(n-1). Compute L(i+1) from L(i) by extending one more edgeTime:

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Time:

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Shortest paths and matrix multiplication Extending shortest paths by one more edge

likes matrix product: L(i+1)= L(i).W Compute L(1), L(2), L(4) . . . , L(r) with

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Time:

Example

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Floyd-Warshall algorithm For path p = <v1, v2, . . . , vl> , v2 … vl-1 are

intermediate vertices from v1 to vl Define = shortest-path weight of any path from i

to j with all intermediate vertices in {1, 2, . . . , k} Consider a shortest path with all intermediate

vertices in {1, 2, . . . , k}: If k is not an intermediate vertex, all intermediate vertices

in {1, 2, . . . , k -1} If k is an intermediate vertex:

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Floyd-Warshall algorithm Recursive formula:

Time:

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Note: since we have at most n vertices, return

Constructing a shortest path is the predecessor of vertex j on a shortest

path from vertex i with all intermediate vertices in the set {1, 2, . . . , k}

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(Use vertex k)

Example

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Johnson’s algorithm Reweighting edges to get non-negative weight

edges: For all u, v V∈ , p is a shortest path using

w if and only if p is a shortest path using

For all (u, v) E, ∈ Run Dijkstra’s algorithm once from each vertex

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Reweighting

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Proof of lemma 25.1 First, we prove

With cycle ,

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Producing nonnegative weights Construct

Since no edges enter s, has the same set of cycles as G has a negative-weight cycle if and only if G does

Define: Claim:

Proof: Triangle inequality of shortest pathsMy T. Thai

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Johnson’s algorithmTime:

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Example

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Summary Dynamic-programming algorithm based on matrix

multiplication Define sub-optimal solutions based on the length of paths Use the technique of “repeated squaring” Time: Floyd-Warshall algorithm Define sub-optimal solutions based on the set of allowed

intermediate vertices Time:

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Johnson’s algorithm Reweight edges to non-negative weight edges Run Dijkstra’s algorithm once from each vertex Time: Faster than Floyd-Warshall algorithm when the graph

is dense E = o(V2)

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