I t d ti t Al ithIntroduction to Algorithms

All-Pairs Shortest Paths

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Single-Source Shortest Paths ProblemSingle-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 ( j) g pfrom i to j.

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

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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 vertexDijkstra s algorithm once from each vertex Time:

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OutlineOutline Shortest paths and matrix multiplication

Floyd-Warshall algorithm Floyd Warshall algorithm

Johnson’s algorithm Johnson’s algorithm

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Recursive solutionRecursive solution

Optimal substructure: subpaths of shortest paths are shortest paths

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

WhWhere wij:My T. Thai

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Computing the shortest-path weights bottom upbottom up

All simple shortest paths contain ≤ n − 1 edges

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

iTime:

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iTime:

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

lik t i d t L(i+1) L(i) Wlikes matrix product: L(i+1)= L(i).W Compute L(1), L(2), L(4) . . . , L(r) with

Time:Time:

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ExampleExample

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Floyd-Warshall algorithmFloyd 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 i i {1 2 k}vertices in {1, 2, . . . , k}:

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

If k is an intermediate vertex:

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Floyd-Warshall algorithmFloyd Warshall algorithm Recursive formula:

Time: N t i h t t Time: Note: since we have at most n vertices, return

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Constructing a shortest pathConstructing 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}

(Use vertex k)

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ExampleExample

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Johnson’s algorithmJohnson 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 vertexj g

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ReweightingReweighting

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Proof of lemma 25 1Proof of lemma 25.1 First, we prove

i h l With cycle ,

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Producing nonnegative weightsProducing nonnegative weights Construct

Since no edges enter s, has the same set of cycles Gas G

has a negative-weight cycle if and only if G does

Define: Define: Claim: Proof: Triangle inequality of shortest paths Proof: Triangle inequality of shortest paths

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Johnson’s algorithmJohnson s algorithmTime:

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ExampleExample

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SummarySummary Dynamic-programming algorithm based on matrix

multiplication Define sub-optimal solutions based on the length of

thpaths

Use the technique of “repeated squaring” Time: Time:

Floyd-Warshall algorithm Define sub optimal solutions based on the set of 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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