MST_Handout - UML CS Analysis of Algorithms 91.503 Fall...

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UML CS Analysis of Algorithms 91.503 Fall, 2004 Minimum Spanning Trees In class we discussed a greedy algorithm to construct a Minimum Spanning Tree (MST) of an undirected graph G=(V,E). Greedy_MST(G) E’ sort edges of E by non-decreasing weight/cost Initialize T to be empty for each edge e’ in E’ do if T union e’ is acyclic then T T union e’ if Vertices(T) = V and T is a single tree then break out of loop return T How do we know that the resulting tree T is an MST of G? Let cost(e’) = cost of an edge and cost(T) = Σ e’ in T (cost(e’)). Theorem : Let T o be an MST of G. Then cost(T) <= cost(T o ). Proof : [based on Chartrand, Introductory Graph Theory ] If T = T o , then the theorem is trivially true. We therefore concentrate on the case in which T = T o . To prove the theorem in this case, we perform an iterative procedure designed to make T o increasingly similar to T; at the end of this procedure we will have transformed T o into T and this will prove the theorem. Each iteration of the procedure adds an edge to T
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MST_Handout - UML CS Analysis of Algorithms 91.503 Fall...

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