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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?) Kruskalpseudocode(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 Kruskalpseudocode(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: GeneralMSTStrategy(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.
 Spring '10
 HORTON
 Algorithms

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