Lecture 7 All-Pairs Shortest Paths

All-Pairs Shortest Paths

Path Counting Problem

Adjacency Matrix

1 3 2 1 2 3 1 2 3

Theorem Proof. We prove it by induction on k.

k=1 True! 1 3 2 1 2 3 1 2 3

Induction Step

All-Pairs Shortest Paths with at most k edges

Recursive formula Proof.

Case 1. The path with length contains at most edges. Case 2. the path with length contains exactly edges.

Key Observation

Dynamic Program

Speed Up dynamic Program

Idea

Weighted Adjacency Matrix

1 4 5 6 3 2 1 2 3 1 2 3

A New Multiplication

Associative Law

Theorem Proof. We prove it by induction on k.

1 k=1 True! 4 5 6 3 2 1 2 3 1 2 3

Induction Step

All-Pairs Shortest Paths Theorem Proof

How to Compute

Lemma

Theorem

Theorem

Floyd-Warshall Algorithm

Observation

Dynamic Program

Theorem

Theorem

What we learnt in this lecture? The relationship between shortest path and matrix multiplication. Faster-All-Pairs-Shortest-Paths algorithm Floyd-Warshall algorithm.

Puzzle 1

Puzzle 2

Puzzle 3