11_mincut - Chapter 11 Min Cut By Sariel Har-Peled October 1 2007x To acknowledge the corn This purely American expression means to admit the

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Chapter 11 Min Cut By Sariel Har-Peled , October 1, 2007 ± To acknowledge the corn - This purely American expression means to admit the losing of an argu- ment, especially in regard to a detail; to retract; to admit defeat. It is over a hundred years old. Andrew Stewart, a member of Congress, is said to have mentioned it in a speech in 1828. He said that haystacks and cornfields were sent by Indiana, Ohio and Kentucky to Philadelphia and New York. Charles A. Wickli e, a member from Kentucky questioned the statement by commenting that haystacks and corn- fields could not walk. Stewart then pointed out that he did not mean literal haystacks and cornfields, but the horses, mules, and hogs for which the hay and corn were raised. Wickli e then rose to his feet, and said, "Mr. Speaker, I acknowledge the corn". – Funk, Earle, A Hog on Ice and Other Curious Expressions. 11.1 Min Cut 11.1.1 Problem Definition Let G = ( V , E ) be undirected graph with n vertices and m edges. We are interested in cuts in G . Definition 11.1.1 A cut in G is a partition of the vertices of V into two sets S and V \ S , where the edges of the cut are V \ S S ( S , V \ S ) = ± uv ² ² ² ² u S , v V \ S , and uv E ³ , where S , and V \ S , . We will refer to the number of edges in the cut ( S , V \ S ) as the size of the cut . For an example of a cut, see figure on the right. We are interested in the problem of computing the minimum cut (i.e., mincut ), that is, the cut in the graph with minimum cardinality. Specifically, we would like to find the set S V such that ( S , V \ S ) is as small as possible, and S is neither empty nor V \ S is empty. ± This work is licensed under the Creative Commons Attribution-Noncommercial 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc/3.0/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA. 1
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11.1.2 Some Definitions We remind the reader of the following concepts. The conditional probability of X given Y is Pr ± X = x | Y = y ² = Pr ± ( X = x ) ( Y = y ) ² / Pr ± Y = y ² . An equivalent, useful restatement of this is that Pr ± ( X = x ) ( Y = y ) ² = Pr ³ X = x ´ ´ ´ ´ Y = y µ · Pr ± Y = y ² . (11.1) Two events X and Y are independent , if Pr ± X = x Y = y ² = Pr [ X = x ] · Pr ± Y = y ² . In particular, if X and Y are independent, then Pr ³ X = x ´ ´ ´ ´ Y = y µ = Pr [ X = x ]. The following is easy to prove by induction using Eq. (11.1). Lemma 11.1.2 Let E 1 , . . . , E n be n events which are not necessarily independent. Then, Pr ± n i = 1 E i ² = Pr [ E 1 ] * Pr [ E 2 |E 1 ] * Pr [ E 3 |E 1 ∩ E 2 ] * . . . * Pr ³ E n ´ ´ ´ ´ E 1 . . . ∩ E n - 1 µ .
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This note was uploaded on 06/14/2009 for the course CS 473 taught by Professor Viswanathan during the Fall '08 term at University of Illinois at Urbana–Champaign.

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11_mincut - Chapter 11 Min Cut By Sariel Har-Peled October 1 2007x To acknowledge the corn This purely American expression means to admit the

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