Testing Embeddability Between Metric Spaces
نویسندگان
چکیده
Let L ≥ 1, > 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 Lbilipschitz 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 -far from being L-bilipschitz embeddable into (N, ρ) if the above inequality fails on at least an fraction of pairs (x, y) ∈ M ×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 far from being L-bilipschitz embeddable into (N, ρ). When (M,d) is -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 O( √ ln |N | |M | (|M | 2 + |N |)) on the number of queries for determining with onesided error whether (M,d) is L-bilipschitz embeddable into (N, ρ) or -far from being L-bilipschitz embeddable into (N, ρ). For the special case of finite |M | = |N |, the above upper bound evaluates to O(|N | √ ln |N | ). We also prove a lower bound of Ω(|N |) even for the special case when |M | = |N | Research supported in part by NSC grant 95-2213-E-002-044. Copyright c ©2008, Australian Computer Society, Inc. This paper appeared at the Fourteenth Computing: The Australasian Theory Symposium (CATS2008), Wollongong, NSW, Australia. Conferences in Research and Practice in Information Technology (CRPIT), Vol. 77, James Harland and Prabhu Manyem, Ed. Reproduction for academic, not-for profit purposes permitted provided this text is included. 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 √ ln |N | , 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).
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