{"id":168218,"date":"2008-08-01T00:00:00","date_gmt":"2008-08-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/improved-algorithms-for-optimal-embeddings\/"},"modified":"2018-10-16T21:12:43","modified_gmt":"2018-10-17T04:12:43","slug":"improved-algorithms-for-optimal-embeddings","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/improved-algorithms-for-optimal-embeddings\/","title":{"rendered":"Improved algorithms for optimal embeddings"},"content":{"rendered":"<div id=\"abstract\" class=\"ytab x-tabs-item-body\">\n<div class=\"tabbody\">\n<div>\n<p>In the last decade, the notion of metric embeddings with small distortion has received wide attention in the literature, with applications in combinatorial optimization, discrete mathematics, and bio-informatics. The notion of embedding is, given two metric spaces on the same number of points, to find a bijection that minimizes maximum Lipschitz and bi-Lipschitz constants. One reason for the popularity of the notion is that algorithms designed for one metric space can be applied to a different one, given an embedding with small distortion. The better distortion, the better the effectiveness of the original algorithm applied to a new metric space.<\/p>\n<p>The goal recently studied by Kenyon et al. [2004] is to consider all possible embeddings between two <i>finite<\/i> metric spaces and to find the best possible one; that is, consider a single objective function over the space of all possible embeddings that minimizes the distortion. In this article we continue this important direction. In particular, using a theorem of Albert and Atkinson [2005], we are able to provide an algorithm to find the optimal bijection between two line metrics, provided that the optimal distortion is smaller than 13.602. This improves the previous bound of 3 + 2&sqrt;2, solving an open question posed by Kenyon et al. [2004]. Further, we show an inherent limitation of algorithms using the \u201cforbidden pattern\u201d based dynamic programming approach, in that they cannot find optimal mapping if the optimal distortion is more than 7 + 4&sqrt;3 (\u2243 13.928). Thus, our results are almost optimal for this method. We also show that previous techniques for general embeddings apply to a (slightly) more general class of metrics.<\/p>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>In the last decade, the notion of metric embeddings with small distortion has received wide attention in the literature, with applications in combinatorial optimization, discrete mathematics, and bio-informatics. The notion of embedding is, given two metric spaces on the same number of points, to find a bijection that minimizes maximum Lipschitz and bi-Lipschitz constants. One [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[13561],"msr-publication-type":[193715],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168218","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-algorithms","msr-locale-en_us"],"msr_publishername":"ACM - Association for Computing Machinery","msr_edition":"ACM Transactions on 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