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In the last decade, the notion of metric embeddings with small distortion received wide attention in the literature, with applications in combinatorial optimization, discrete mathematics, functional analysis 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 metric space, given an embedding with small distortion. The better the distortion, the better is the effectiveness of the original algorithm applied to a new metric space. The goal that was recently studied by Kenyon, Rabani, and Sinclair [16] is to consider all possible embeddings and to try to find the best possible one, i.e., consider a single objective function over the space of all possible embeddings that minimizes the distortion. In this paper we continue this important direction. In particular, using a theorem of Albert and Atkinson [3], 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 √ 2, solving an open question posed by [16]. Further, we show an inherent limitation of algorithms using the “forbidden pattern ” based dynamic-programming approach that they cannot find optimal mapping if the optimal distortion is more than 7 + 4 √ 3( � 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

Year: 2009

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