Covering many points with a small-area box
Mark de Berg, Sergio Cabello, Otfried Cheong, David Eppstein,, Christian Knauer

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 be a set of points in the plane. We show how to find, for a given integer , the smallest-area axis-parallel rectangle that covers points of in time. We also consider the problem of, given a value , covering as many points of as possible with an axis-parallel rectangle of area at most . For this problem we give a probabilistic -approximation that works in near-linear time: In time we find an axis-parallel rectangle of area at most that, with high probability, covers at least points, where is the maximum possible number of points that could be covered.
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