Consider the following divide-and-conquer algorithm for computing minimum spanni

Question

Consider the following divide-and-conquer algorithm for computing minimum spanning trees: Given a graph G (V, E), partition the set V of vertices into two sets Vi and ½ such that l Vil and MI differ by at most 1. Let El be the set of edges that are incident only on vertices of V, aud let E2 be the set of edges that are incident only on vertices of V2. Recursively solve a minimum-spanning-tree problem on each of the two subgraphs Gi - (Vi, Ei) and G2 (½ ??). Finally, select the minimum-weight edge in E that crosses the cut (?,V2) and use this edge to unite the resulting two minimum spanning trees into a single spanning tree. Prove that this algorithm correctly computes a minimum spanning tree of G, or provide an example for which the algorithm fails.

Explanation / Answer

Solution:

The stated algorithm might fail in the case given below:

and our resulting graph will not be a spanning tree.

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