Next: Up: Previous:

Matrix Multiplication Approach

Consider a shortest path p from vertex i to vertex j containing at most m edges.

If i=j, then p has no edges
If i $\neq$ j, then p = $i \stackrel{p'}{\leadsto} k \rightarrow j$ , where p' contains m-1 edges

By Lemma 25.1 (subpaths of shortest paths are shortest paths), p' is a shortest path from i to k, and $\delta(i,j) \;=\; \delta(i,k) \;+\; w_{kj}$ .

Thus, the shortest path problem exhibits optimal substructure.

Next: Up: Previous: