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.

My T. Thaimythai@cise.ufl.edu

2

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:

My T. Thaimythai@cise.ufl.edu

3

Outline

Shortest paths and matrix multiplication

Floyd-Warshall algorithm

Johnson’s algorithm

My T. Thaimythai@cise.ufl.edu

4

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 edge

Time:

My T. Thaimythai@cise.ufl.edu

6

Time:

My T. Thaimythai@cise.ufl.edu

7

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

My T. Thaimythai@cise.ufl.edu

8

Time:

Example

My T. Thaimythai@cise.ufl.edu

9

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:

mythai@cise.ufl.edu

10

Floyd-Warshall algorithm

Recursive formula:

Time:

My T. Thaimythai@cise.ufl.edu

11

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}

My T. Thaimythai@cise.ufl.edu

12

(Use vertex k)

Example

My T. Thaimythai@cise.ufl.edu

13

My T. Thaimythai@cise.ufl.edu

14

My T. Thaimythai@cise.ufl.edu

15

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

My T. Thaimythai@cise.ufl.edu

16

Reweighting

My T. Thaimythai@cise.ufl.edu

17

Proof of lemma 25.1

First, we prove

With cycle ,

My T. Thaimythai@cise.ufl.edu

18

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

mythai@cise.ufl.edu

19

Johnson’s algorithmTime:

My T. Thaimythai@cise.ufl.edu

20

Example

My T. Thaimythai@cise.ufl.edu

21

My T. Thaimythai@cise.ufl.edu

22

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:

My T. Thaimythai@cise.ufl.edu

23

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)

My T. Thaimythai@cise.ufl.edu

24