{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7075","name":"Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices","abstract":"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.","keywords":["Spectral sparsification","linear system solving"],"author":[{"@type":"Person","name":"Koutis, Ioannis","givenName":"Ioannis","familyName":"Koutis"},{"@type":"Person","name":"Levin, Alex","givenName":"Alex","familyName":"Levin"},{"@type":"Person","name":"Peng, Richard","givenName":"Richard","familyName":"Peng"}],"position":25,"pageStart":266,"pageEnd":277,"dateCreated":"2012-02-24","datePublished":"2012-02-24","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Koutis, Ioannis","givenName":"Ioannis","familyName":"Koutis"},{"@type":"Person","name":"Levin, Alex","givenName":"Alex","familyName":"Levin"},{"@type":"Person","name":"Peng, Richard","givenName":"Richard","familyName":"Peng"}],"copyrightYear":"2012","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2012.266","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6217","volumeNumber":14,"name":"29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)","dateCreated":"2012-02-24","datePublished":"2012-02-24","editor":[{"@type":"Person","name":"D\u00fcrr, Christoph","givenName":"Christoph","familyName":"D\u00fcrr"},{"@type":"Person","name":"Wilke, Thomas","givenName":"Thomas","familyName":"Wilke"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7075","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6217"}}}