{"id":184054,"date":"2005-07-21T00:00:00","date_gmt":"2009-10-31T13:20:02","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/lower-bounds-for-linear-degeneracy-testing\/"},"modified":"2016-09-09T10:01:06","modified_gmt":"2016-09-09T17:01:06","slug":"lower-bounds-for-linear-degeneracy-testing","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/lower-bounds-for-linear-degeneracy-testing\/","title":{"rendered":"Lower Bounds for Linear Degeneracy Testing"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Consider the following fundamental problem, called r-linear-degeneracy-testing (rLDT): Given an input of n real numbers, do any r of them sum up to 0?<\/p>\n<p>This problem is fundamental in computational geometry as it captures a broad notion of degeneracy, that is, input that satisfies a measure-zero property (not being  in \u201cgeneral position\u201d).  The case r=3 (a.k.a. 3-SUM) is well known and many important computational geometric problems reduce to it.  Its exact complexity is not known and considered a notorious open problem.<\/p>\n<p>The talk will discuss lower bounds for rLDT under the linear decision tree model of computation, a natural geometric model which allows the algorithm to compare the input against hyperplanes in n dimensional space.  Erickson proved that under a restricted version of this model, there is a tight bound of n<sup>r\/2<\/sup>.  I will describe Erickson\u2019s important result and show how to extend it to a more general model of computation.  This generalization, though modest, posed a challenge for many years and demanded new techniques.  The solution I will present gives rise to interesting new problems in linear algebra and combinatorics and strengthens the bold connection between error correcting codes and complexity.  I will describe the new techniques that we developed, their limits, and interesting open problems that may help in further understanding the complexity of this fundamental problem.<\/p>\n<p>The talk will be self-contained and no prior background in computational geometry is required, just basic notions of linear algebra.<\/p>\n<p>This is joint work with my advisor Bernard Chazelle.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the following fundamental problem, called r-linear-degeneracy-testing (rLDT): Given an input of n real numbers, do any r of them sum up to 0? This problem is fundamental in computational geometry as it captures a broad notion of degeneracy, that is, input that satisfies a measure-zero property (not being in \u201cgeneral position\u201d). The case r=3 [&hellip;]<\/p>\n","protected":false},"featured_media":195299,"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-184054","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/y_EPVu_CXHE","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184054","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\/184054\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195299"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184054"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184054"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184054"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184054"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184054"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184054"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184054"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184054"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184054"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}