Metric structures in L1: dimension, snowflakes, and average distortion

Abstract

We study the metric properties of finite subsets of L 1 . The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L 1 . We present some new observations concerning the relation of L 1 to dimension, topology, and Euclidean distortion. We show that every n -point subset of L 1 embeds into L 2 with average distortion O ( log n ) , yielding the first evidence that the conjectured worst-case bound of O ( log n ) is valid. We also address the issue of dimension reduction in L p for p ∈ ( 1 , 2 ) . We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the ℓ 1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251–260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in ℓ 1 , in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514–523], cannot be used to prove a lower bound for the non-linear case. This is accomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space.