Multi-embedding and path approximation of metric spaces

Yair Bartal*, Manor Mendel

*Corresponding author for this work

Research output: Contribution to conferencePaperpeer-review

12 Scopus citations

Abstract

Metric embeddings have become a frequent tool in the design of algorithms. The applicability is often dependent on how high the embedding's distortion is. For example embedding into ultrametrics (or arbitrary trees) requires linear distortion. Using probabilistic metric embeddings, the bound reduces to O(log n log log n). Yet, the lower bound is still logarithmic. We make a step further in the direction of by-passing this difficulty. We define "multi-embeddings" of metric spaces where a point is mapped onto a set of points, while keeping the target metric being of polynomial size and preserving the distortion of paths. The distortion obtained with such multi-embeddings into ultrametrics is at most O(log Δ log log Δ) where Δ is the (normalized) diameter, and probabilistically O(log n log log log n). In particular, for expander graphs, we are able to obtain constant distortions embeddings into trees vs. the Ω(log n) lower bound for all previous notions of embeddings. We demonstrate the algorithmic application of the new embeddings by obtaining improvements for two well-known problems: group Steiner tree and metrical task systems.

Original languageAmerican English
Pages424-433
Number of pages10
StatePublished - 2003
EventConfiguralble Computing: Technology and Applications - Boston, MA, United States
Duration: 2 Nov 19983 Nov 1998

Conference

ConferenceConfiguralble Computing: Technology and Applications
Country/TerritoryUnited States
CityBoston, MA
Period2/11/983/11/98

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