Covering many points with a small-area box

TL;DR

This paper presents efficient algorithms for finding small-area rectangles covering a specified number of points or covering as many points as possible within a given area, with provable approximation guarantees.

Contribution

It introduces new algorithms for minimal-area k-point coverage and near-linear time approximation for maximum coverage under area constraints.

Findings

Optimal algorithms for smallest-area rectangles covering k points.

Probabilistic approximation algorithm for maximum points covered within an area.

Near-linear time complexity for the approximation algorithm.

Abstract

Let $P$ be a set of $n$ points in the plane. We show how to find, for a given integer $k>0$, the smallest-area axis-parallel rectangle that covers $k$ points of $P$ in $O(nk^2 \log n+ n\log^2 n)$ time. We also consider the problem of, given a value $\alpha>0$, covering as many points of $P$ as possible with an axis-parallel rectangle of area at most $\alpha$. For this problem we give a probabilistic $(1-\varepsilon)$-approximation that works in near-linear time: In $O((n/\varepsilon^4)\log^3 n \log (1/\varepsilon))$ time we find an axis-parallel rectangle of area at most $\alpha$ that, with high probability, covers at least $(1-\varepsilon)\mathrm{\kappa^*}$ points, where $\mathrm{\kappa^*}$ is the maximum possible number of points that could be covered.