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Discrete Mathematics with Graph Theory (3rd Edition) 308

Discrete Mathematics with Graph Theory (3rd Edition) 308 -...

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306 1 (C,S) D -2 (G,2) F Solutions to Exercises 26. (a) Applying the first version of Dijkstra's algorithm to the digraph produces the labels shown. Only the labels at A and B are correct. There is a negative weight cycle HGJCIH from which every vertex except A and B is accessible. So, repeating the cycle various numbers of times before heading to a vertex (except A or B) changes the labels of that vertex, so the labels on every vertex (except A and B) are meaningless. (b) This time Bellman-Ford does not help, because of the negative weight cycle. C 1 (C,S) D (B,~ 2 E (J, -7) -2 (G,O) F 27. [BB] As described in the text, the second last vertex on a shortest path is p(j). Since a shortest path to Vj makes use of a shortest path to p(j) following by edge p(j)Vj, the third last vertex on a shortest path to Vj is p(p(j)), this being the second last vertex on a shortest path to p(j). So the vertices of a shortest path to Vj, in reverse order, are Vj, p(j), p(p(j)), p(p(p(j))), ... , VI.
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