Terminal Embeddings
Michael Elkin, Arnold Filtser, Ofer Neiman

TL;DR
This paper introduces terminal embeddings that preserve distances involving specific terminal points in a metric space, with distortion depending only on the number of terminals, and applies these embeddings to improve approximation algorithms for sparsest-cut problems.
Contribution
The paper develops new terminal embedding techniques with distortion bounds depending only on the number of terminals, and applies these to enhance sparsest-cut approximation algorithms.
Findings
Embeddings preserve terminal distances with distortion depending only on |K|.
Existence of embeddings that approximate all pairs while improving terminal pair distortion.
Application to improve sparsest-cut approximation ratio to O(\u221a{\log |K|})
Abstract
In this paper we study {\em terminal embeddings}, in which one is given a finite metric (or a graph ) and a subset of its points are designated as {\em terminals}. The objective is to embed the metric into a normed space, while approximately preserving all distances among pairs that contain a terminal. We devise such embeddings in various settings, and conclude that even though we have to preserve pairs, the distortion depends only on , rather than on . We also strengthen this notion, and consider embeddings that approximately preserve the distances between {\em all} pairs, but provide improved distortion for pairs containing a terminal. Surprisingly, we show that such embeddings exist in many settings, and have optimal distortion bounds both with respect to and with respect to . Moreover,…
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