Smallest k-Enclosing Rectangle Revisited

Research output: Contribution to journalArticlepeer-review

Abstract

Given a set of n points in the plane, and a parameter k, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing k points. We present the first near quadratic time algorithm for this problem, improving over the previous near-O(n5 / 2) -time algorithm by Kaplan et al. (25th European Symposium on Algorithms. Leibniz Int Proc Inform, vol. 87, # 52. Leibniz-Zent Inform, Wadern, 2017). We provide an almost matching conditional lower bound, under the assumption that (min , +) -convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for both perimeter and area) that can make the time bound sensitive to k, giving near O(nk) time. We also present a near linear time (1 + ε) -approximation algorithm to the minimum area of the optimal rectangle containing k points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.

Original languageEnglish (US)
JournalDiscrete and Computational Geometry
DOIs
StateAccepted/In press - 2020

Keywords

  • Approximation algorithms
  • Conditional lower bounds
  • Geometric optimization
  • Outliers

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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