l4 - 6.896 Sublinear Time Algorithms September 23, 2004...

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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....
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l4 - 6.896 Sublinear Time Algorithms September 23, 2004...

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