L16-MST

1 t 2 repeat v 1 times 3 add to t the lightest edge e

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Unformatted text preview: v ) crosses a graph cut (S,V-S) if u b 8 d 8 3 12 c 2 5 e 9 9 i 11 3 1 S g 7 10 a v⇥V S f 6 h example of a crossing b 8 d g 7 10 a 8 3 c 5 e 9 12 2 9 11 3 1 i f 6 h definition: respect cut theorem cut theorem suppose the set of edges A is part of an m.s.t. let (S, V ⇤ S) be any cut that respects A . let edge e be the min-weight edge across (S, V ⇤ S) then: A {e} is part of an m.s.t. example of theorem b 8 d g 7 10 a 8 3 c 5 e 9 12 2 9 11 3 1 i f 6 h d 8 5 g 9 7 h 8 2 11 e 6 i 3 b 3 10 9 a 12 f 1 c 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: Now consider adding edge e to T . This creates a cycle from u to v to u. (why?) Let e⇧ ⌥= e be the edge on this cycle that crosses (S, V S). (why must such an edge e⇧ exist?) Let T ⇧ = T {e⇧ } + {e}. Since T ⇧ has V 1 edges (why?) and since T ⇧ is proof of cut thm d 8 g 2 7 5 9 u 11 e h 3 3 8 6 v c 1 9 b i f 12 10 a correctness 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 o...
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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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