Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Lectures on Discrete Geometry
Algorithmic Applications of Low-Distortion Geometric Embeddings
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Embeddings of finite metrics
Low distortion maps between point sets
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
A metric index for approximate string matching
Theoretical Computer Science
Geometry of Cuts and Metrics
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Let L ≥ 1, ε 0 be real numbers, (M, d) be a finite metric space and (N, ρ) be a metric space (Rudin 1976). The metric space (M, d) is said to be L-bilipschitz embeddable into (N, ρ) if there is an injective function f : M → N with 1/L·d(x, y)≤ρ(f(x),f(y))≥L·d(x, y) for all x, y ∈ N (Farb & Mosher 1999, David & Semmes 2000, Croom 2002). In this paper, we also say that (M, d) is ε-far from being L-bilipschitz embeddable into (N, ρ) if the above inequality fails on at least an ε fraction of pairs (x, y) ∈ M x M for every injective function f : M → N. Below, a query to a metric space consists of asking for the distance between a pair of points chosen for that query. We study the number of queries to metric spaces (M, d) and (N, ρ) needed to answer whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ε-far from being L-bilipschitz embeddable into (N, ρ). When (M, d) is ε-far from being L-bilipschitz embeddable into (N, ρ), we allow an o(1) probability of error (i.e., returning the wrong answer "L-bilipschitz embeddable"). However, we allow no error when (M, d) is L-bilipschitz embeddable into (N, ρ). That is, algorithms with only one-sided errors are considered in this paper. When |M| ≤ |N| are finite, we give an upper bound of [EQUATION] on the number of queries for determining with onesided error whether (M, d) is L-bilipschitz embeddable into (N, ρ) or ε-far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M| = |N|, the above upper bound evaluates to [EQUATION]. We also prove a lower bound of Ω(|N|3/2) even for the special case when |M| = |N| are finite and L = 1, which coincides with testing isometry between finite metric spaces (Croom 2002). For finite |M| = |N|, the upper and lower bounds thus match up to a multiplicative factor of at most [EQUATION] which depends only sublogarithmically in |N|. We also investigate the case when (N, ρ) is not necessarily finite. Our results are based on techniques developed in an earlier work on testing graph isomorphism (Fischer & Matsliah 2006).