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Discrete Mathematics with Graph Theory (3rd Edition) 309

# Discrete Mathematics with Graph Theory (3rd Edition) 309 -...

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Section 11.3 307 30. The answer is no, unless there are negative weight cycles. Suppose there is a walk .. ~ ... w c with a repeated vertex v. We may assume that the part of the walk from v back to v (which we have denoted C) is a cycle. If this walk has length less than u ... v ... u (removing C), then C must have had negative weight. 31. At Step 2, on each of n 2 occasions, the algorithm finds the minimum of n - 1 numbers. This requires n - 2 comparisons. In addition, one more comparison is required to determine the value of diU). Finally, Step 3 requires a further n comparisons. In all, our algorithm requires n 2 [(n - 2) + 1] + n = O(n 3 ) comparisons. 32. The proposed procedure fails since paths which use more edges will be rejected in favor of higher weight paths using fewer edges. In the digraph shown at the left, the proposal is first to add 1 to the weight of each arc. This gives the digraph on the right. Also shown on the right are the final labels obtained by the original version of Dijkstra's algorithm. The shortest distance (on the right) from
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