- Journal Home
- Volume 42 - 2024
- Volume 41 - 2023
- Volume 40 - 2022
- Volume 39 - 2021
- Volume 38 - 2020
- Volume 37 - 2019
- Volume 36 - 2018
- Volume 35 - 2017
- Volume 34 - 2016
- Volume 33 - 2015
- Volume 32 - 2014
- Volume 31 - 2013
- Volume 30 - 2012
- Volume 29 - 2011
- Volume 28 - 2010
- Volume 27 - 2009
- Volume 26 - 2008
- Volume 25 - 2007
- Volume 24 - 2006
- Volume 23 - 2005
- Volume 22 - 2004
- Volume 21 - 2003
- Volume 20 - 2002
- Volume 19 - 2001
- Volume 18 - 2000
- Volume 17 - 1999
- Volume 16 - 1998
- Volume 15 - 1997
- Volume 14 - 1996
- Volume 13 - 1995
- Volume 12 - 1994
- Volume 11 - 1993
- Volume 10 - 1992
- Volume 9 - 1991
- Volume 8 - 1990
- Volume 7 - 1989
- Volume 6 - 1988
- Volume 5 - 1987
- Volume 4 - 1986
- Volume 3 - 1985
- Volume 2 - 1984
- Volume 1 - 1983
Cited by
- BibTex
- RIS
- TXT
Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, and numerical solution of symmetric positive definite linear systems. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm, which is called the universal greedy approach (UGA) for the graph sparsification problem. For a general spectral sparsification problem, e.g., the positive subset selection problem from a set of $m$ vectors in $\mathbb{R}^n$, we propose a nonnegative UGA algorithm which needs $O(mn^2+ n^3/\epsilon^2)$ time to find a $\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with positive coefficients with sparsity at most $\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$ is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm is established. For the graph sparsification problem, another UGA algorithm is proposed which can output a $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral sparsifier with $\lceil\frac{n}{\epsilon^2}\rceil$ edges in $O(m+n^2/\epsilon^2)$ time from a graph with $m$ edges and $n$ vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. $O(m^{1+o(1)})$ for some $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.
}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2201-m2021-0130}, url = {http://global-sci.org/intro/article_detail/jcm/21638.html} }Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, and numerical solution of symmetric positive definite linear systems. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm, which is called the universal greedy approach (UGA) for the graph sparsification problem. For a general spectral sparsification problem, e.g., the positive subset selection problem from a set of $m$ vectors in $\mathbb{R}^n$, we propose a nonnegative UGA algorithm which needs $O(mn^2+ n^3/\epsilon^2)$ time to find a $\frac{1+\epsilon/\beta}{1-\epsilon/\beta}$-spectral sparsifier with positive coefficients with sparsity at most $\lceil\frac{n}{\epsilon^2}\rceil$, where $\beta$ is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm is established. For the graph sparsification problem, another UGA algorithm is proposed which can output a $\frac{1+O(\epsilon)}{1-O(\epsilon)}$-spectral sparsifier with $\lceil\frac{n}{\epsilon^2}\rceil$ edges in $O(m+n^2/\epsilon^2)$ time from a graph with $m$ edges and $n$ vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. $O(m^{1+o(1)})$ for some $o(1)=O(\frac{(\log\log(m))^{2/3}}{\log^{1/3}(m)})$. Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.