This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 01/13/2012 for the course CMSC 423 taught by Professor Staff during the Fall '07 term at Maryland.
 Fall '07
 staff

Click to edit the document details