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URN: urn:nbn:de:0030-drops-78608
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### Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points

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### 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(n^{5/2} 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(nk^{3/2} log k) time, and (ii) an approximation algorithm that finds, in O(n+(n/(k epsilon^5))*log^{5/2}(n/k)log((1/epsilon) log(n/k))) time, a rectangle of the given perimeter or diagonal that contains at least (1-epsilon)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.

### BibTeX - Entry

```@InProceedings{kaplan_et_al:LIPIcs:2017:7860,
author =	{Haim Kaplan and Sasanka Roy and Micha Sharir},
title =	{{Finding Axis-Parallel Rectangles of Fixed Perimeter or Area Containing the Largest Number of Points}},
booktitle =	{25th Annual European Symposium on Algorithms (ESA 2017)},
pages =	{52:1--52:13},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-049-1},
ISSN =	{1868-8969},
year =	{2017},
volume =	{87},
editor =	{Kirk Pruhs and Christian Sohler},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},