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

## Document Type

Conference Article

## Publication Title

Leibniz International Proceedings in Informatics, LIPIcs

## 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.

## DOI

10.4230/LIPIcs.ESA.2017.52

## Publication Date

9-1-2017

## Recommended Citation

Kaplan, Haim; Roy, Sasanka; and Sharir, Micha, "Finding axis-parallel rectangles of fixed perimeter or area containing the largest number of points" (2017). *Conference Articles*. 196.

https://digitalcommons.isical.ac.in/conf-articles/196