Professor Borden proposes a new divide and conquer algorithm for computing minimum spanning trees, which goes as follows Given a graph G (V, E), partition the set V of vertices into two sets V 1 and V 2 such that V 1 and V 2 differ by at most 1 Let E 1 be the set of edges that are incident only on vertices in V 1 , and let E 2 be the set of edges that are incident only on vertices in V 2 Recursively solve a minimum spanning tree problem on each of the two subgraphs G 1 (V 1, E 1) and G 2 (V 2, E 2) Finally, select the minimum weight edge in E that crosses the cut (V 1, V 2 ), and use this edge to unite the resulting two minimum spanning trees into a single spanning tree Either argue that the algorithm correctly computes a minimum spanning tree of G, or provide an example for which the algorithm fails

Question: Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph G = (V, E), partition the

Professor Borden proposes a new divide-and-conquer algorithm for computing minimum spanning trees, which goes as follows. Given a graph G = (V, E), partition the set V of vertices into two sets V₁ and V₂ such that |V₁| and |V₂| differ by at most 1. Let E₁ be the set of edges that are incident only on vertices in V₁, and let E₂be the set of edges that are incident only on vertices in V₂. Recursively solve a minimum-spanning-tree problem on each of the two subgraphs G₁ = (V_1,E₁₎ and G₂ = (V_2,E₂₎. Finally, select the minimum-weight edge in E that crosses the cut (V_1, V₂), and use this edge to unite the resulting two minimum spanning trees into a single spanning tree.

Either argue that the algorithm correctly computes a minimum spanning tree of G, or provide an example for which the algorithm fails.

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