Homework_3_sol

# Homework_3_sol - Homework 3 Problem 1 Show an example where...

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Homework 3 Problem 1: Show an example where Dijkstra’s algorithm fails. Problem 2: Run Dijkstra’s algorithm manually for the following graph. (A as a source) Problem 3: Suppose that a graph G has a minimum spanning tree already computed. How quickly can the minimum spanning tree be updated if a new vertex and incident edges are added to G

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The basic idea here is to remove the maximum weighted edge from a cycle to obtain a better tree. Algorithm: Let e be the lightest edge across the cut ( V- v, v ), where v is the new vertex added, then add e to the tree. Now, for each new edge ( v;w ) added in the graph do the following. Add the edge ( v;w ) to the tree, and then remove the maximum weight edge from the resulting cycle. The most important thing here is to find the maximum weighted edge in a cycle. A new added edge e will result in only one cycle in the tree. Correctness: Considering the fact that the new tree we obtained is a spanning tree and the most expensive edge in the cycle will never in the MST, we can see that the new tree
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Homework_3_sol - Homework 3 Problem 1 Show an example where...

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