## 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} logn) 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} logk) time, and (ii) an approximation algorithm that finds, in [formula-presented] time, a rectangle of the given perimeter 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 language | English |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Computational Geometry: Theory and Applications |

Volume | 81 |

DOIs | |

State | Published - Aug 2019 |

## Keywords

- Area
- Axis-parallel rectangle
- Diagonal
- Enclose points
- Perimeter