Transitive Closure

Transitive closure asks: can you reach vertex $j$ from vertex $i$?

You can adapt Floyd-Warshall: instead of minimizing distance, use boolean OR. Set reach[i][j] = true if there is a path from $i$ to $j$. Update: reach[i][j] = reach[i][j] || (reach[i][k] && reach[k][j]). This computes reachability in $O(n^3)$ time. It answers questions like "is there any path from $i$ to $j$?" without caring about distance.

Space complexity is $O(n^2)$ for the data structures used.