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Nearly Tight Bounds on Testing of Metric Properties

Yiqiao Bao, Sampath Kannan, Erik Waingarten

TL;DR

This paper proves a lower bound of matching dependence on $n$ of $\Omega (n^{2/3})$, ruling out any property testers with $o(n^{2/3})$ query complexity unless their dependence on $1/\epsilon$ is super-polynomial.

Abstract

Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree metrics and ultrametrics. For general metrics, our paper is the first to consider these questions. We prove an upper bound of $O(n^{2/3}/ε^{4/3})$ on the query complexity for this problem. Our algorithm is simple, but the analysis requires great care in bounding the variance on the number of violating triangles in a sample. When $ε$ is a slowly decreasing function of $n$ (rather than a constant, as is standard), we prove a lower bound of matching dependence on $n$ of $Ω(n^{2/3})$, ruling out any property testers with $o(n^{2/3})$ query complexity unless their dependence on $1/ε$ is super-polynomial. Next, we turn to tree metrics and ultrametrics. While there were known upper and lower bounds, we considerably improve these bounds showing essentially tight bounds of $\tilde{O}(1/ε)$ on the sample complexity. We also show a lower bound of $Ω( 1/ε^{4/3} )$ on the query complexity. Our upper bounds are derived by doing a more careful analysis of a natural, simple algorithm. For the lower bounds, we construct distributions on NO instances, where it is hard to find a witness showing that these are not ultrametrics.

Nearly Tight Bounds on Testing of Metric Properties

TL;DR

This paper proves a lower bound of matching dependence on of , ruling out any property testers with query complexity unless their dependence on is super-polynomial.

Abstract

Given a non-negative matrix viewed as a set of distances between points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree metrics and ultrametrics. For general metrics, our paper is the first to consider these questions. We prove an upper bound of on the query complexity for this problem. Our algorithm is simple, but the analysis requires great care in bounding the variance on the number of violating triangles in a sample. When is a slowly decreasing function of (rather than a constant, as is standard), we prove a lower bound of matching dependence on of , ruling out any property testers with query complexity unless their dependence on is super-polynomial. Next, we turn to tree metrics and ultrametrics. While there were known upper and lower bounds, we considerably improve these bounds showing essentially tight bounds of on the sample complexity. We also show a lower bound of on the query complexity. Our upper bounds are derived by doing a more careful analysis of a natural, simple algorithm. For the lower bounds, we construct distributions on NO instances, where it is hard to find a witness showing that these are not ultrametrics.

Paper Structure

This paper contains 30 sections, 33 theorems, 26 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Testing Metrics.
  4. Tree Metrics and Ultrametrics.
  5. Related Work
  6. Technical Overview
  7. Metric Testing Upper Bound.
  8. Metric Testing Lower Bound.
  9. Tree Metric and Ultrametric Testing Upper Bound.
  10. Tree Metric and Ultrametric Testing Lower Bound
  11. Preliminaries
  12. Metric Testing Upper Bound: Theorem \ref{['thm:testing-metric']}
  13. CheckViolation Sub-routine: Proof of Lemma \ref{['lem:check-vio']}
  14. Finding A Good Subset of Violating Triangles: Proof of Lemma \ref{['lem:goodSet']}
  15. Metric Testing Lower Bound---Theorem \ref{['thm:testing-metric-lb']}
  16. ...and 15 more sections

Key Result

Theorem 1.1

For any large enough $n \in \mathbbm N$ and any $\varepsilon \in (0,1)$, there exists a randomized algorithm that receives query access to an unknown matrix $M \in \mathbbm R^{n\times n}$ and makes $O(n^{2/3} / \varepsilon^{4/3})$ queries with the following guarantee: Furthermore, the algorithm is non-adaptive (i.e., queries made do not depend on answers to prior queries).

Figures (1)

  • Figure :

Theorems & Definitions (57)

  • Theorem 1.1: Testing Metrics---Upper Bound
  • Theorem 1.2: Testing Metrics---Lower Bound
  • Theorem 1.3: Testing Tree Metrics and Ultrametrics---Upper bound
  • Theorem 1.4: Testing Tree Metrics and Ultrametrics---Lower bound
  • Definition 2.1
  • Definition 2.2: Property Testing Algorithm for $n\times n$ Matrices
  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.1
  • Definition 3.2: Triangle Degree
  • ...and 47 more