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Suppose we also wish to compute the vertices on shortest paths in the algorithms of this section. Show how to compute the predecessor matrix $\Pi$ from the complete...

Question

Suppose we also wish to compute the vertices on shortest paths in the algorithms of this section. Show how to compute the predecessor matrix $\Pi$ from the completed matrix $L$ of shortest-path weights in $O\left(n^{3}\right)$ time.

Suppose we also wish to compute the vertices on shortest paths in the algorithms of this section. Show how to compute the predecessor matrix $\Pi$ from the completed matrix $L$ of shortest-path weights in $O\left(n^{3}\right)$ time.



Answers

Use Exercise 37 to prove Theorem $4 .[\text { Hint: Count the }$ number of paths with $n$ steps of the type described in Exercise $37 .$ Every such path must end at one of the points $(n-k, k)$ for $k=0,1,2, \ldots, n . ]$


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