Volume in general metric spaces

Ittai Abraham, Yair Bartal, Ofer Neiman, Leonard J. Schulman

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

2 Scopus citations

Abstract

A central question in the geometry of finite metric spaces is how well can an arbitrary metric space be "faithfully preserved" by a mapping into Euclidean space. In this paper we present an algorithmic embedding which obtains a new strong measure of faithful preservation: not only does it (approximately) preserve distances between pairs of points, but also the volume of any set of k points. Such embeddings are known as volume preserving embeddings. We provide the first volume preserving embedding that obtains constant average volume distortion for sets of any fixed size. Moreover, our embedding provides constant bounds on all bounded moments of the volume distortion while maintaining the best possible worst-case volume distortion. Feige, in his seminal work on volume preserving embeddings defined the volume of a set S = {v 1, ..., v k } of points in a general metric space: the product of the distances from v i to { v 1, ..., v i - 1 }, normalized by 1/(k-1)!, where the ordering of the points is that given by Prim's minimum spanning tree algorithm. Feige also related this notion to the maximal Euclidean volume that a Lipschitz embedding of S into Euclidean space can achieve. Syntactically this definition is similar to the computation of volume in Euclidean spaces, which however is invariant to the order in which the points are taken. We show that a similar robustness property holds for Feige's definition: the use of any other order in the product affects volume 1/(k - 1) by only a constant factor. Our robustness result is of independent interest as it presents a new competitive analysis for the greedy algorithm on a variant of the online Steiner tree problem where the cost of buying an edge is logarithmic in its length. This robustness property allows us to obtain our results on volume preserving embeddings.

Original languageEnglish
Title of host publicationAlgorithms, ESA 2010 - 18th Annual European Symposium, Proceedings
Pages87-99
Number of pages13
EditionPART 2
DOIs
StatePublished - 22 Nov 2010
Externally publishedYes
Event18th Annual European Symposium on Algorithms, ESA 2010 - Liverpool, United Kingdom
Duration: 6 Sep 20108 Sep 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 2
Volume6347 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference18th Annual European Symposium on Algorithms, ESA 2010
Country/TerritoryUnited Kingdom
CityLiverpool
Period6/09/108/09/10

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