Question

Consider the following algorithm for finding the minimum span- ning tree of a connected graph G = (V, E, w(·)) where w : E → R+ is a weight function.

(1) TE

(2) while T is not a tree

(3) let C be an arbitrary cycle in T

(4) let e be the heaviest edge in C (ties are broken arbitrarily)

(5) T← T\{e}

(6) return T

Does the algorithm always return a minimum spanning tree T of G? If your answer is yes, prove it; if your answer is no, give an instance G for which the algorithm does not return a minimum spanning tree.

Answer 1

The algorithm does not always return a minimum spanning tree as proven by a counterexample. It may return a suboptimal solution in some cases.

Step-by-step explanation:

The algorithm does not always return a minimum spanning tree. There are instances where the algorithm may return a spanning tree with a weight greater than the weight of the minimum spanning tree.

For example, consider a graph G with three vertices A, B, C and three edges with weights 1, 2, and 3. The algorithm would first select the edge with weight 3 as the heaviest edge in the cycle, break it, and return a spanning tree with a weight of 3. However, the minimum spanning tree of G has a weight of 1 by selecting the edge with weight 1.

The algorithm may not select the heaviest edge in a cycle that is part of the minimum spanning tree, leading to a suboptimal solution.