All-Pairs Shortest Paths

Problem:

Given a directed graph G = (V,E) with weight function w: E $\rightarrow$ R mapping edges to real-valued weights, find the shortest path between every pair of vertices u,v $\in$ V that minimizes the sum of the weights along the edges of the path.

Solution 1:

Run Single-Source Shortest Path on every vertex.

Dijkstra, O(VE lg V) using binary heap and priority queue

Bellman-Ford, O(V²E)

Note that E = O(V²) in the worst case.

Solution 2:

Dynamic Programming approach

Matrix Multiplication Approach with repeated squaring,

Floyd-Warshall,

Application: Mileage chart for a road atlas.