Discrete Mathematics with Graph Theory (3rd Edition) 321

Discrete Mathematics with Graph Theory (3rd Edition) 321 -...

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Chapter 11 319 of length two from Vi to Vi, and such a walk is only possible when the above configuration occurs. Thus L: b ii counts the number of such configurations twice. 8. (a) The final labels assigned by Dijkstra's algorithm, the improved version, are shown. The algorithm incorrectly determines that the shortest path from A to E has length 2, when the path ABCDE has length 1. It fails because Dijkstra's algorithm does not apply to digraphs where some arcs have negative weights. 9. (b) For k = 1 to n, where n is the number of vertices, the Bellman-Ford Algorithm proceeds by determining shortest paths along k arcs from VI to each other vertex. We show the successive stages of the algorithm in the table. Max. no. of arcs 1 2 3 4 5 B I,A I,A I,A I,A I,A Vertices C 00 3,B 3,B 3,B 3,B D 00 00, A 2,C 2,C 2,C E 2,A 2,A 2,A I,D I,D GGAU (a) The only abnormal fragment is G, so the chain ends
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Unformatted text preview: AU G with G. Since the abnormal fragment doesn't split, we CCUG look for Eulerian trails or circuits whose last vertex is G. The unique chain is UCAGCCUGGAUG. C AGC U AG UCAG (b) The only abnormal fragment is AUU, so the chain ends with this fragment. Since the abnormal fragment splits, we look for Eulerian trails or circuits whose U G final arc is labeled AUU. There are four chains: GUCGGGGUGAUU, GUGGGGUCGAUU, GGGGUGUCGAUU, GGGGUCGUGAUU. AU (c) The only abnormal fragment is AAG, so the chain ends with this. Since the abnormal fragment does not split, we look for Eulerian trails or circuits whose last U G vertex is AAG. There are eight chains: conditions are UGUCGCGAGCUAAG,UGUCGAGCGCUAAG, UGCGUCGAGCUAAG,UGAGCGUCGCUAAG, UCGUGCGAGCUAAG,UCGUGAGCGCUAAG, UCGCGUGAGCUAAG,UCGAGCGUGCUAAG. C AAG...
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