This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 6.896 Sublinear Time Algorithms September 23, 2004 Lecture 4 Lecturer: Ronitt Rubinfeld Scribe: Seth Gilbert Overview Approximating the weight of the minimum spanning tree Assume from last lecture : approx num cc( G, ) approximates the number of connected components in an n node graph, G , with degree d in time O ( d/ 4 ). This time : Determine the (approximate) weight of a minimum spanning tree (MST) for an n node graph, G , with degree d . Plan : Characterize minimum spanning tree in terms of number of connected components of certain subgraphs. Use approx num cc to estimate these values. Testing Bipartiteness Adjacency Matrix Model : The adjacency matrix model is defined. Problem Statement : The problem of testing for bipartitness is presented. Sketch : We sketch the algorithm to test for bipartitness and its proof. 1 Approximating the weight of the minimum spanning tree 1.1 Problem Statement Assume you are given: a graph, G = ( V, E ), of n nodes and m edges with degree d , G is connected, each edge ( i, j ) E is assigned an integer weight w ij { 1 , . . . , w } , and G is represented as an adjacency list augmented with weights for each edge. We define the minimum spanning tree (MST) weight as follows: MST ( G ) = min T spans G w ( T ) = X ( i,j ) T w ij Our goal is to output d MST ( G ) such that: (1 ) MST ( G ) d MST ( G ) (1 + ) MST ( G ) Comments Notice that the resulting algorithm provides no indication of how to actually construct the minimum spanning tree. Notice that n 1 MST ( G ), since each edge has weight at least 1, and there are n 1 edges in a minimum spanning tree....
View Full
Document
 Spring '04
 RonittRubinfeld
 Algorithms

Click to edit the document details