Approximating Additive Distortion of Embeddings into Line Metrics
نویسنده
چکیده
We consider the problem of fitting metric data consisting of n points to a path (line) metric. Our objective is to minimize the total additive distortion of this mapping. The total additive distortion is the sum of errors in all pairwise distances in the input data. This problem has been shown to be NP-hard by [13]. We give an O(log n) approximation for this problem by using Garg et al.’s [10] algorithm for the multicut problem as a subroutine. Our algorithm also gives an O(log n) approximation for the Lp norm of the additive distortion.
منابع مشابه
On Dominated L 1 Metrics
We introduce and study a class l dom 1 () of l 1-embeddable metrics corresponding to a given metric. This class is deened as the set of all convex combinations of-dominated line metrics. Such metrics were implicitly used before in several constuctions of low-distortion embeddings into l p-spaces, such as Bourgain's embedding of an arbitrary metric on n points with O(log n) distortion. Our main ...
متن کاملAlgorithmic embeddings
We present several computationally efficient algorithms, and complexity results on low distortion mappings between metric spaces. An embedding between two metric spaces is a mapping between the two metric spcaes and the distortion of the embedding is the factor by which the distances change. We have pioneered theoretical work on relative (or approximation) version of this problem. In this setti...
متن کاملFinite Metric Spaces & Their Embeddings: Introduction and Basic Tools
Definition of (semi) metric. CS motivation. Finite metric spaces arise naturally in combinatorial objects, and algo-rithmic questions. For example, as the shortest path metrics on graphs. We will also see less obvious connections. Properties of finite metrics. The following properties have been investigated: Dimension , extendability of Lipschitz and Hölder functions, decomposability, Inequalit...
متن کاملParameterized Low-distortion Embeddings - Graph metrics into lines and trees
We revisit the issue of low-distortion embedding of metric spaces into the line, and more generally, into the shortest path metric of trees, from the parameterized complexity perspective. Low-distortion embeddings of a metric space into the line, or into some other “simple” metric space, (that is, a mapping that preserves distances, up to some small multiplicative factor called the distortion) ...
متن کاملLower Bounds on Embeddings of Planar Graphs into the l1 Metric
This paper presents an overview of existing bounds on l1-embeddings of planar metrics. A new family of graphs containing the K2,3 minor is introduced. Computational results on this family of graphs establish a new lower bound on the constant in the following conjecture: There exists an absolute constant Cmax > 0 such that every finite planar metric embeds into the l1 metric with distortion < Cm...
متن کامل