10.4230/LIPICS.STACS.2012.266
Koutis, Ioannis
Ioannis
Koutis
Levin, Alex
Alex
Levin
Peng, Richard
Richard
Peng
Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
2012
Article
Spectral sparsification
linear system solving
Dürr, Christoph
Christoph
Dürr
Wilke, Thomas
Thomas
Wilke
2012
2012-02-24
2012-02-24
2012-02-24
en
urn:nbn:de:0030-drops-34348
10.4230/LIPIcs.STACS.2012
978-3-939897-35-4
1868-8969
10.4230/LIPIcs.STACS.2012
LIPIcs, Volume 14, STACS 2012
29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
2013
14
25
266
277
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Dürr, Christoph
Christoph
Dürr
Wilke, Thomas
Thomas
Wilke
1868-8969
Leibniz International Proceedings in Informatics (LIPIcs)
2012
14
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
12 pages
653418 bytes
application/pdf
Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license
info:eu-repo/semantics/openAccess
We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.
LIPIcs, Vol. 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012), pages 266-277