lecture_04 (1) - Minimum (weight) spanning trees Input : A...

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Unformatted text preview: Minimum (weight) spanning trees Input : A connected undirected graph G = (V, E); each edge e in E has a weight or cost, denoted w( e ). To find : MST = a subset of edges T ⊆ E such that T is a spanning tree whose sum of edge weights is minimized . 1 a c b d e f 2 3 6 5 4 4 6 1 a c b d e f 2 6 5 4 2 7 6 Total weight = 22 Reading assignment: • Dasgupta et al. : 5.1 • Cormen et al. : Chapter 23 Assignment 1: 6 pm today Applications MST is fundamental problem with diverse applications. s Network design: telephone, electrical, hydraulic, TV cable, computer, road s Approximation algorithms for NP-hard problems s traveling salesperson problem, Steiner tree 2 s Indirect applications. s max bottleneck paths s LDPC codes for error correction s image registration with Renyi entropy s learning salient features for real-time face verification s reducing data storage in sequencing amino acids in a protein s Cluster analysis. A greedy algorithm for MST Prim's algorithm . Start with an arbitrary node s and greedily grow a tree T from s outward. Repeatedly add the “lightest” (min-weight) edge e that is between a vertex in T and a vertex outside T. w 5 1 Start with w: w 1 a c b d f 2 3 6 4 4 6 2 7 a c b d f 2 A greedy algorithm for MST Prim's algorithm . Start with an arbitrary node s and greedily grow a tree T from s outward. Repeatedly add the “lightest” (min-weight) edge e that is between a vertex in T and a vertex outside T. w 5 1 Start with w: Correctness. s Would different start nodes give trees of different cost ? No . Always the same cost. s Does the tree reported have the smallest possible cost ? Yes , it is a MST. Why? a c b d f 2 3 6 4 4 6 2 7 Start with d: A greedy algorithm for MST Prim's algorithm . Start with an arbitrary node s and greedily grow a tree T from s outward. Repeatedly add the “lightest” (min-weight) edge e that is between a vertex in T and a vertex outside T....
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This note was uploaded on 03/01/2010 for the course CS 1234 taught by Professor Chan during the Spring '10 term at University of the Bío-Bío.

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lecture_04 (1) - Minimum (weight) spanning trees Input : A...

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