Smallest k-Enclosing Rectangle Revisited

Authors Timothy M. Chan, Sariel Har-Peled

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Author Details

Timothy M. Chan
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA
Sariel Har-Peled
  • Dept. of Computer Science, University of Illinois at Urbana-Champaign, USA

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Timothy M. Chan and Sariel Har-Peled. Smallest k-Enclosing Rectangle Revisited. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


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(n^{5/2})-time algorithm by Kaplan et al. [Haim Kaplan et al., 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 either perimeter or area) that can make the time bound sensitive to k, giving near O(n k) time. We also present a near linear time (1+epsilon)-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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Geometric optimization
  • outliers
  • approximation algorithms
  • conditional lower bounds


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