Intermediate Structure Of Shortest Path

An intermediate vertex of a simple path p = <v₁, v₂, .., v_l>is any vertex of p other than v₁ and v_l.

The Floyd-Warshall (FW) algorithm works by successively reducing the number of intermediate vertices that can occur in a shortest path and its subpaths.

Let graph G = (V, E) have vertices V numbered 1 .. n, V = {1, 2, .., n}, and consider a subset {1, 2, .., k} for some k.

Let p be the minimum weight path from vertex i to vertex j whose intermediate vertices are drawn from {1, 2, .., k}. One of two situations then occur:

1.

k is not an intermediate vertex of p

contains vertices from {1, 2, .., k-1}

2.

k is an intermediate vertex of p

contains vertices from {1, 2, .., k-1}

p₁ is the shortest path from i to k

p₂ is the shortest path from k to j

(by Lemma 25.1)