L16-MST

# Then the set a e is part of a minimum spanning tree

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Unformatted text preview: f T . First, draw a picture of the situation: Now consider adding edge e to T . This creates a cycle from u to v to u. (why?) Kruskal-pseudocode(G) 1 A⌅⌥ 2 repeat V 1 times: 3 add to A the lightest edge e ⇧ E that does not create a cycle correctness Theorem 2 Suppose the set of edges A is part of a minimum spanning tree of G = (V, E ). Let (S, V S ) be any cut that respects A and let e be the edge with the minimum weight that crosses (S, V S ). Then the set A {e} is part of a minimum spanning tree. proof: by induction. in step 1, A is part of some MST. suppose that after k steps, A is part of some MST (line 2). in line 3, we add an edge e=(u,v) to A. Proof. By assumption, A ⇥ T for some minimum spanning tree T of G. Case 1 If A {e} ⇥ T , then the theorem is true already. Case 2 Suppose A {e} ⌃⇥ T . Let e = (u, v ). We shall construct a new tree T that contains A {e} by changing only a few edges of T . First, draw a picture of the situation: Now consider adding edge e to T . This creates a cycle from u to v to u. (why?) 3 cases for edge e. Case 1: e=(u,v) and both u,v are in A. 3 cases for edge e. Case 2: e=(u,v) and only u is in A. 3 cases for edge e. Case 3: e=(u,v) and neither u nor v are in A. 3 cases for edge e S S S analysis? 1 Minimum Spanning Tree Algorithm Kruskal-pseudocode(G) 1 A⌅⌥ 2 repeat V 1 times: 3 add to A the lightest edge e ⇧ E that does not create a cycle Theorem 2 Suppose the set of edges A is part of a minimum spanning tree of G = (V, E ). Let (S, V S ) be any cut that respects A and let e be the edge with the minimum weight that crosses (S, V S ). Then the set A {e} is part of a minimum spanning tree. Proof. By assumption, A ⇥ T for some minimum spanning tree T of G. Case 1 If A {e} ⇥ T , then the theorem is true already. Case 2 Suppose A {e} ⌃⇥ T . Let e = (u, v ). We shall construct a new tree T that contains A {e} by changing only a few edges of T . First, draw a picture of the situation: General-MST-Strategy(G = (V, E )) 1 A⇥⇤ 2 repeat V 1...
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## This note was uploaded on 02/25/2014 for the course CS 4102 taught by Professor Horton during the Spring '10 term at UVA.

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