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Lecture15

# Lecture15 - CME 305 Discrete Mathematics and Algorithms...

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CME 305: Discrete Mathematics and Algorithms Instructor: Professor Amin Saberi ([email protected]) February 25, 2010 Lecture 11: Randomized Algorithms II 1 Randomized Algorithm for Finding a Perfect Matching In this lecture, we extend the randomized algorithm from last lecture that determined the existence of a perfect matching in a graph G ( V, E ), to actually finding a perfect matching in G . The key computational step will be a matrix inversion. First we establish a key (and surprising) property of subsets of random numbers: Definition: A set system ( S, F ) consists of a finite set S of elements, S = { x 1 , x 2 , . . . , x n } and a family F of subsets of S , F = { S 1 , . . . , S k } , ∅ ⊂ S j S . For each element of S , assign weight w i to x i , where w i is chosen uniformly at random and independently from { 1 , . . . , 2 n } . Denote the weight of set S j to be x i S j w i . Lemma 1 (Isolating lemma) The probability that there is a unique minimum weight set is at least 1 / 2 . Proof: Fix the weight of all elements except x i . Given F , define the threshold for element x i to be the number α i such that if w i > α i then x i is in no set with minimum weight, and if w i α i then x i is in some set with minimum weight. Clearly, if w i < α i , then x i is in every set with minimum weight. The only ambiguity occurs when w i = α i .

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Lecture15 - CME 305 Discrete Mathematics and Algorithms...

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