{"id":184602,"date":"2010-01-06T00:00:00","date_gmt":"2010-01-11T23:01:02","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/twice-ramanujan-sparsifiers\/"},"modified":"2016-08-22T11:27:22","modified_gmt":"2016-08-22T18:27:22","slug":"twice-ramanujan-sparsifiers","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/twice-ramanujan-sparsifiers\/","title":{"rendered":"Twice-Ramanujan Sparsifiers"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs.<\/p>\n<p>In particular, we prove that for every <i>d > 1<\/i> and every undirected, weighted graph <i>G = (V,E,w)<\/i> on <i>n<\/i> vertices, there exists a weighted graph <i>H=(V,F,&tilde;{w})<\/i> with at most <i>dn <\/i> edges such that for every<\/p>\n<p><i>x &isin; R<sup>V<\/sup><\/i>,<br \/>\n\\[ x<sup>T<\/sup> L<sub>G<\/sub> x \\leq x<sup>T<\/sup> L<sub>H<\/sub> x \\leq ( d+1+2\\sqrt{d} \/ d+1-2\\sqrt{d} ) x<sup>T<\/sup> L<sub>G<\/sub> x \\]<\/p>\n<p>where <i>L<sub>G<\/sub><\/i> and <i>L<sub>H<\/sub><\/i> are the Laplacian matrices of <i>G<\/i> and <i>H<\/i>, respectively. Thus, <i>H<\/i> approximates <i>G<\/i> spectrally at least as well as a Ramanujan expander with <i>dn\/2<\/i> edges approximates the complete graph.<\/p>\n<p>We give an elementary deterministic polynomial time algorithm for constructing <i>H<\/i>.<\/p>\n<p>Joint work with Josh Batson and Dan Spielman.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We prove that every graph has a spectral sparsifier with a number of edges linear in its number of vertices. As linear-sized spectral sparsifiers of complete graphs are expanders, our sparsifiers of arbitrary graphs can be viewed as generalizations of expander graphs. In particular, we prove that for every d > 1 and every undirected, [&hellip;]<\/p>\n","protected":false},"featured_media":195504,"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-184602","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/6hmzV9wuLMw","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184602","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\/184602\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195504"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184602"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184602"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184602"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184602"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184602"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184602"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184602"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184602"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184602"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}