Small distortion and volume preserving embeddings for planar and Euclidean metrics

Abstract

A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v). Bourgain showed that any graphical metric could be embedded with distortion O(logn). Linial, London and Rabinovich and Aumman and Rabani used such embeddings to prove an O(log k) approximate max-flow min-cut theorem for k commodity flow problems. A generalization of embeddings that preserve distances between pairs-of points are embeddings that preserve volumes of larger sets. In particular, A (k, c)-volume respecting embedding of n-points in any metric space is a contraction where every subset of k points has within an ck-’ factor of its maximal possible k l-dimensional volume. Feige invented these embeddings in devising a polylogarithmic approximation algorithm for the bandwidth problem using these embeddings. Feige’s methods have subsequently been used by Vempala for approximating versions of the VLSI layout problem. Feise showed that a (k, O(10,g~‘~ n,/m)) volume r&ecting embedding‘ eksted.” Be -recently found improved (k, 0( mdk log k + log n)) volume respecting embeddings. For metrics arising from planar graphs (planar metrics), we give (k,O(m)) volume respecting contractions. As a corollary, we give embeddings for Permission to makkr digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. XC’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00 planar metrics with distortion O(e). This gives rise to an O(e)-approximate max-flow min-cut theorem for multicommodity flow problems in planar graphs. We also give an improved bound for volume respecting embeddings for Euclidean metrics. In particular, we give an (k,O(dog klog D)) volume respecting embedding where D is the ratio of the largest distance to the smallest distance in the metric. Our results give improvements for Feige’s and Vempala’s approximation algorithms for planar and Euclidean metrics. For volume respecting embeddings, our embeddings do not degrade very fast when preserving the volumes of large subsets. This may be useful in the future for approximation algorithms or if volume .respecting embeddings prove to be of independent interest.