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Unformatted text preview: v ) crosses a graph cut (S,VS) if
u
b 8 d 8 3 12 c 2
5 e 9 9 i
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a v⇥V S f 6 h example of a crossing
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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 minweight edge across (S, V ⇤ S)
then: A {e} is part of an m.s.t. example of theorem
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a 8 3 c 5 e 9 12 2
9 11 3 1 i f 6 h d
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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
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2 7 5 9 u 11 e h 3
3 8 6 v
c 1 9
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12 10 a correctness 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 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.
 Spring '10
 HORTON
 Algorithms

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