Spectral Sparsification of Graph

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Moradi, Somayyeh | 2013

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 45157 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Daneshgar, Amir
Abstract:
The running time of almost every algorithm in the graph theory depend on the number of edges. Thus, these algorithms are often too slow when the input graphs are dense. Therefore, it is useful to reduce the number of edges by sparsification. In fact, sparsification is the task of approximating a graph G = (V;E) by another graph ~G = (V; ~E) so that ~E E (j~Ej jEj) and ~G maintain a main prefixed property of G. Depending on these properties several notions of graph sparsification have been proposed. In this thesis we study a notion of sparsification that is called spectral sparsification which is based on the contributions of Daniel A. Spielman et.al..In this notion of sparsification eigenvalues of ~G is approximately the same as those of G. In this thesis, we explain that every weighted graph G = (V;E;w) has a spectral sparsifier with O(n logc1 n) edges that can be computed in O(mlogc2 n) time, where n = jV j, m = jEj and c1, c2 are some absolute constants
Keywords:
Sampling ; Laplacian Matrix ; Conductivity ; Spectral Sparsification ; Normalized Laplacian Matrix