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Which of the following can be the best algorithm(s) for all pair of the shortest path problem?

I. ‘V’ invocations of Dijkstra algorithm ⇒ Ο(VE logV).
II. ‘V’ invocations of Bellman-Ford algorithm ⇒ Ο(V2 E).
III. ‘1’ invocations of Floyd-Warshall algorithm ⇒ Ο(V3).
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2 Answers

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I think answer should be only III i.e. ‘1’ invocations of Floyd-Warshall algorithm ⇒ Ο(V3).

Reason for saying this –

As we know for complete graph number of edges , E = V(V-1)/2 nearly equal to V^2 , so for complete graph

I.‘V’ invocations of Dijkstra algorithm ⇒ Ο(VE logV) becomes O(V^3logV)

II. ‘V’ invocations of Bellman-Ford algorithm ⇒ Ο(V2 E) becomes O(V^4)

III. ‘1’ invocations of Floyd-Warshall algorithm ⇒ Ο(V3) remains O(V^3) only.

Hence considering best cases for all types of graph one must prefer option III 

 

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Only III should be the answer because in the worst case(i.e. complete graph) Floyd-Warshall performs better than any other algorithms which exist right now.

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