# 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 language English 28th International Symposium on Algorithms and Computation, ISAAC 2017 Takeshi Tokuyama, Yoshio Okamoto Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing 9783959770545 https://doi.org/10.4230/LIPIcs.ISAAC.2017.39 Published - 1 Dec 2017 28th International Symposium on Algorithms and Computation, ISAAC 2017 - Phuket, ThailandDuration: 9 Dec 2017 → 22 Dec 2017

### Publication series

Name Leibniz International Proceedings in Informatics, LIPIcs 92 1868-8969

### Conference

Conference 28th International Symposium on Algorithms and Computation, ISAAC 2017 Thailand Phuket 9/12/17 → 22/12/17

## Keywords

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

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