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Unformatted text preview: CS 601/IITB Global Min Cut Abhiram Ranade How easy is it to disconnect a network? This problem is formalized as the min-cut problem. It also arises as a subproblem in many optimization problems. Input: Undirected graph G = ( v; E ). Output: Smallest set of edges whose removal separates the graph into at least two pieces. Deterministic Algorithms: Fix a vertex as s and for all possible choices of other vertices as t , nd the minimum s t cut. Clearly this must be the global min-cut; since the global min-cut must separate s from some t . Each st-min-cut takes time O ( n 3 ) for a total time of O ( n 4 ). Randomized Algorithms: We will see a very simple randomized algorithm that returns the global min-cut with high probability in O ( n 4 ln n ) time. We will then re ne it to get algorithms which run respectively in O ( n 3 ln n ) and O ( n 2 ln 3 n ) time. The last is a substantial improvement over the deterministic algorithm, and nearly optimal, since for a dense graph we would need ( n 2 ) time just to examine all edges. This algorithm does not always return the correct answer, only most of the time. Such algorithms are called Monte Carlo algorithms. Algorithms that always return the correct answer but take a variable amount of time depending upon the random choices are in contrast called Las Vegas algorithms. 1 The basic algorithm For the algorithm we treat G as a multigraph, i.e. allow parallel edges between the same pair of vertices. The core of the algorithm is as follows: For i = 1 to n 2 1. Pick a random edge ( u; v ) from the current graph. 2. Merge vertices u; v into a single vertex w , and remove the edge ( u; v ). The vertex w will have edges to all erstwhile neighbours of u; v . Further, if u; v both had n u ; n v edges to some vertex x , then w will have n u + n v edges to x . This operation will be called an edge contraction....
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This note was uploaded on 01/04/2010 for the course CSE CS601 taught by Professor Prof.ranade during the Summer '09 term at IIT Bombay.
- Summer '09