{"id":185221,"date":"2010-09-02T00:00:00","date_gmt":"2010-09-08T07:49:48","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/spectral-graph-sparsification-part-2-an-omlog2-n-algorithm-for-solving-sdd-systems\/"},"modified":"2016-08-22T11:27:25","modified_gmt":"2016-08-22T18:27:25","slug":"spectral-graph-sparsification-part-2-an-omlog2-n-algorithm-for-solving-sdd-systems","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/spectral-graph-sparsification-part-2-an-omlog2-n-algorithm-for-solving-sdd-systems\/","title":{"rendered":"Spectral graph sparsification Part 2: [An O(mlog<sup>2<\/sup> n) algorithm for solving SDD systems]"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We present the fastest known algorithm for solving symmetric diagonally dominant (SDD) systems. If the number of the non-zeros in the system matrix is m, and the desired error is err, the algorithm runs in time O(mlog<sup>2<\/sup> m log(1\/err) ).<\/p>\n<p>The heart of the algorithm is a spectral sparsification algorithm, which on an input of a graph A with n nodes and m edges returns a graph B with n-1+m\/k edges, such that for all real vectors x,<\/p>\n<p>1 (x<sup>T<\/sup> B x)\/(x<sup>T<\/sup> A x) O(klog<sup>2<\/sup> n).<\/p>\n<p>The algorithm is randomized and runs in O(mlog<sup>2<\/sup> n) expected time. It is based on a simple and practical sampling procedure of independent interest.<\/p>\n<p>The sparsification algorithm and the solver simplify greatly the groundbreaking work of Spielman and Teng who were the first to design nearly linear time algorithms for SDD systems.<\/p>\n<p>Joint work with Gary Miller and Richard Peng<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We present the fastest known algorithm for solving symmetric diagonally dominant (SDD) systems. If the number of the non-zeros in the system matrix is m, and the desired error is err, the algorithm runs in time O(mlog2 m log(1\/err) ). The heart of the algorithm is a spectral sparsification algorithm, which on an input of [&hellip;]<\/p>\n","protected":false},"featured_media":195707,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-185221","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/_SSHyP4_-Sw","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185221","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/185221\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195707"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=185221"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=185221"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=185221"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=185221"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=185221"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=185221"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=185221"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=185221"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=185221"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=185221"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}