Question

i. Consider a weighted version of the directed graph in (a.), given below. Carry out the steps of Dijkstra's shortest path algorithm starting at vertex S, and consequently give the shortest path from S to vertex T and its length. ii. For a graph G = (V,E), what is the worst-case time complexity of the version of Dijkstra's shortest path algorithm examined in lectures? (Your answer should be in terms of |V| and/or |E|.) c. Consider an arbitrary weighted directed graph G = (V, E) with weights w : E â†’ â„ (that is, the weights are real numbers). Assume that when you run Dijkstra's shortest path algorithm on G from some vertex s âˆˆ V to another vertex t âˆˆ V, you get shortest path Xt. Now, consider the same graph G = (V,E) with weights that are double the original weights (i.e. with weights w' : E â†’ â„ such that w(e) = 2w(e) for any e âˆˆ E). Will running Dijkstra's shortest path algorithm on graph G with weights w' from s to t also produce the same shortest path Xt? If so, explain why (in no more than 5 lines). If not, give a counter-example.

          i. Consider a weighted version of the directed graph in (a.), given below. Carry out the steps of Dijkstra's shortest path algorithm starting at vertex S, and consequently give the shortest path from S to vertex T and its length.

ii. For a graph G = (V,E), what is the worst-case time complexity of the version of Dijkstra's shortest path algorithm examined in lectures? (Your answer should be in terms of |V| and/or |E|.)

c. Consider an arbitrary weighted directed graph G = (V, E) with weights w : E â†’ â„ (that is, the weights are real numbers). Assume that when you run Dijkstra's shortest path algorithm on G from some vertex s âˆˆ V to another vertex t âˆˆ V, you get shortest path Xt. Now, consider the same graph G = (V,E) with weights that are double the original weights (i.e. with weights w' : E â†’ â„ such that w(e) = 2w(e) for any e âˆˆ E). Will running Dijkstra's shortest path algorithm on graph G with weights w' from s to t also produce the same shortest path Xt? If so, explain why (in no more than 5 lines). If not, give a counter-example.

b 8 marks i 6 marks consider a weighted version of the directed graph in a given below carry out the steps of dijkstras shortest path algorithm starting at vertex sand consequently give the 79392

Added by Alexander H.

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