All Pairs Shortest Paths Problem (APSPP)

Consider a weighted complete graph G with vertex set G.V = {v0, v1, v2, …, vn-1}. The weight of the edge from vi and vj is denoted as G.w(i, j). It is assumed that the weights of the edges are non-negative. In other words, the weights satisfy the following constraints: G.w(i, j) > 0 if i ≠ j G.w(i, j) = 0 if i = j The All Pairs Shortest Paths Problem (APSPP) is, given G, to find the distance network D which is a weighted complete graph such that (i) D has the same vertex set as G.V. In other words, D.V=G.V= {v0, v1, v2, …, vn-1}; (ii) The weights of the edges in D represents the lengths of the shortest paths in G, In other words, D.w(i, j)=length of the shortest path from vi and vj APSPP problem can be solved by the following approaches: Approach A (Dijkstra’s algorithm): Repeatedly solving the Single Source Shortest Paths Problem (SSSPP) using Dijkstra’s algorithm which is a well known greedy algorithm. Approach B (Floyd Algorithm): This approach solves APSPP using Dynamic Programming. It finds all the constrained shortest paths in the graph that only go via intermediate nodes {v0, v1, v2, …, vk}, for k=0, 1,2,.. n-1. When k=n-1, there is no more constraint. Thus all-pairs shortest paths problem is solved when k=n-1.