Finding axis-parallel rectangles of fixed perimeter or area containing the largest number of points

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

Abstract

Let P be a set of n points in the plane in general position, and consider the problem of finding an axis-parallel rectangle with a given perimeter, or area, or diagonal, that encloses the maximum number of points of P. We present an exact algorithm that finds such a rectangle in O(n5/2 log n) time, and, for the case of a fixed perimeter or diagonal, we also obtain (i) an improved exact algorithm that runs in O(nk3/2 log k) time, and (ii) an approximation algorithm that finds, in O ( n + n/kϵ5 log5/2 n/k log(1/ϵ log n/ k )) time, a rectangle of the given perimeter or diagonal that contains at least (1 - ϵ)k points of P, where k is the optimum value. We then show how to turn this algorithm into one that finds, for a given k, an axis-parallel rectangle of smallest perimeter (or area, or diagonal) that contains k points of P. We obtain the first subcubic algorithms for these problems, significantly improving the current state of the art.

Original languageEnglish
Title of host publication25th European Symposium on Algorithms, ESA 2017
EditorsChristian Sohler, Christian Sohler, Kirk Pruhs
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770491
DOIs
StatePublished - 1 Sep 2017
Event25th European Symposium on Algorithms, ESA 2017 - Vienna, Austria
Duration: 4 Sep 20176 Sep 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume87
ISSN (Print)1868-8969

Conference

Conference25th European Symposium on Algorithms, ESA 2017
Country/TerritoryAustria
CityVienna
Period4/09/176/09/17

Keywords

  • Area
  • Computational geometry
  • Geometric optimization
  • Perimeter
  • Rectangles

Fingerprint

Dive into the research topics of 'Finding axis-parallel rectangles of fixed perimeter or area containing the largest number of points'. Together they form a unique fingerprint.

Cite this