mst - CMSC 451: Minimum Spanning Trees & Clustering...

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Unformatted text preview: CMSC 451: Minimum Spanning Trees & Clustering Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Sections 4.5–4.6 of Algorithm Design by Kleinberg & Tardos. Network Design You want to connect up several computers with a network, and you want to run as little wire as possible. It is feasible to directly connect only some pairs of computers. 5 1 2 1 3 1 2 3 Minimum Spanning Tree Problem Minimum Spanning Tree Problem Given • undirected graph G with vertices for each of n objects • weights d ( u , v ) on the edges giving the distance u and v , Find the subgraph T that connects all vertices and minimizes ∑ { u , v }∈ T d ( u , v ). T will be a tree. Why? Minimum Spanning Tree Problem Minimum Spanning Tree Problem Given • undirected graph G with vertices for each of n objects • weights d ( u , v ) on the edges giving the distance u and v , Find the subgraph T that connects all vertices and minimizes ∑ { u , v }∈ T d ( u , v ). T will be a tree. Why? If there was a cycle, we could remove any edge on the cycle to get a new subgraph T with smaller ∑ { u , v }∈ T d ( u , v ). MST History • Studied as far back as 1926 by Bor˚ uvka. • We’ll see algorithms that take O ( m log n ) time, where m is number of edges. • Best known algorithm takes time O ( m α ( m , n )), where α ( m , n ) is the “inverse Ackerman” function (grows very slowly). • Still open: Can you find a O ( m ) algorithm? Assumption We assume no two edges have the same edge cost....
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This note was uploaded on 01/13/2012 for the course CMSC 423 taught by Professor Staff during the Fall '07 term at Maryland.

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mst - CMSC 451: Minimum Spanning Trees & Clustering...

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