Question: The Floyd-Warshall algorithm for ALL-PAIRS SHORTEST PATHS is shown in Figure 3.30. The only difference between it and the matrix-squaring algorithm we gafve in Section

The Floyd-Warshall algorithm for ALL-PAIRS SHORTEST PATHS is shown in Figure 3.30. The only difference between it and the matrix-squaring algorithm we gafve in Section 3.4.3 is that the loop over k is now outside the loops over i and j, rather than being the innermost. Surprisingly, this change means that we no longer need to repeat this process The Floyd-Warshall algorithm for ALL-PAIRS SHORTEST PATHS is shown in Figure 3.30. times with an additional outer loop, so the running time is now The only difference between it and the matrix-squaring algorithm we gafve in instead of Section 3.4.3 is that the loop over k is now outside the .

Prove that this algorithm works. Hint: state a loop invariant that holds after running the outer loop k times, and use induction on k.

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