Dominance product and high-dimensional closest pair under L∞

Omer Gold, Micha Sharir

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Given a set S of n points in Rd, the Closest Pair problem is to find a pair of distinct points in S at minimum distance. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L∞ Closest Pair problem, where d = nr for some r > 0, and the underlying metric is L∞. We improve and simplify previous results for L∞ Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in O(DP(n, d) log n) time, and by a randomized algorithm that runs in O(DP(n, d)) expected time, where DP(n, d) is the time bound for computing the dominance product for n points in Rd. That is a matrix D, such that D[i, j] = | {k | pi[k] ≤pj [k]} | ; this is the number of coordinates at which pj dominates pi. For integer coordinates from some interval [-M,M], we obtain an algorithm that runs in O min{Mnω(1,r,1), DP(n, d)} | time1, where ω(1, r, 1) is the exponent of multiplying an n × nr matrix by an nr × n matrix. We also give slightly better bounds for DP(n, d), by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).

Original languageEnglish
Title of host publication28th International Symposium on Algorithms and Computation, ISAAC 2017
EditorsTakeshi Tokuyama, Yoshio Okamoto
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770545
DOIs
StatePublished - 1 Dec 2017
Event28th International Symposium on Algorithms and Computation, ISAAC 2017 - Phuket, Thailand
Duration: 9 Dec 201722 Dec 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume92
ISSN (Print)1868-8969

Conference

Conference28th International Symposium on Algorithms and Computation, ISAAC 2017
Country/TerritoryThailand
CityPhuket
Period9/12/1722/12/17

Keywords

  • Closest pair
  • Dominance product
  • L∞
  • Matrix multiplication

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