Lec5_RandomAlgs

Lec5_RandomAlgs - COT 6936: Topics in Algorithms! Giri...

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COT 6936: Topics in Algorithms Giri Narasimhan ECS 254A / EC 2443; Phone: x3748 giri@cs.fiu.edu http://www.cs.fiu.edu/~giri/teach/COT6936_S12.html https://moodle.cis.fiu.edu/v2.1/course/view.php?id= 174
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COT 6936: Topics in Algorithms Randomized Algorithms
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Cut-Sets & Min-Cuts Example 1: ({a,b,c,d}, {e,f,g}) Weight = 19 Example 2: ({a,b,g}, {c,d,e,f}) Weight = 30 Example 3: ({a}, {b,c,d,e,f,g}) Weight = 5 2/8/12 COT 6936 3 How is this different from Min-Cuts we considered in the context of Networks Flows? Undirected graphs No source or sink vertex “Robustness” parameter Global Min-Cut can be computed in poly time
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Edge Contraction [ Karger, 1992 ] 2/8/12 COT 6936 4 http://en.wikipedia.org/wiki/Edge_contraction
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Edge Contractions and Min-Cuts • Lemma : If you are not contracting an edge from the cut-set, edge contractions do not affect the size of min-cuts. • Observation : Most edges are not part of the min-cut. • Idea : Use randomization [ Karger, 1992 ] 2/8/12 COT 6936 5
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Min-Cuts in the Internet Graph 2/8/12 COT 6936 6 June 1999 Internet graph, Bill Cheswick http://research.lumeta.com/ches/map/gallery/index.html
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Randomized Algorithms: Min-Cut • Algorithm : – Pick a random edge and contract it until only 2 vertices are remaining. – Report edges connecting the 2 remaining vertices as the min cut • Analysis – Assume that the Min-cut is of size k – Prob { edge is not in Min-cut } 1 – 2/n ( why ?) – Prob { Min-cut is output } 2/n(n – 1) ( why ?) 2/8/12 COT 6936 7
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Analysis: Min-Cut Algorithm (Cont’d) • Observation: – If Min-Cut is of size k , then minimum degree of every vertex is k . (Why?) • Number of edges in graph
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Lec5_RandomAlgs - COT 6936: Topics in Algorithms! Giri...

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