# Smallest k-Enclosing Rectangle Revisited

arXiv: https://arxiv.org/abs/1903.06785 · HTML: https://mlsanity.com/paper/1903.06785/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Mar 2019.

## Built on by (most cited first, 1 of 1)

- [Towards the mathematical foundation of the minimum enclosing ball and related problems](https://mlsanity.com/paper/2402.06629/) (2024)

## Abstract (arXiv)

Given a set of n points in the plane, and a parameter k, we consider the problem of computing the minimum (perimeter or area) axis-aligned rectangle enclosing k points. We present the first near quadratic time algorithm for this problem, improving over the previous near-O(n5/2)-time algorithm by Kaplan etal [KRS17]. We provide an almost matching conditional lower bound, under the assumption that (min,+)-convolution cannot be solved in truly subquadratic time. Furthermore, we present a new reduction (for either perimeter or area) that can make the time bound sensitive to k, giving near O(n k)  time. We also present a near linear time (1+varepsilon)-approximation algorithm to the minimum area of the optimal rectangle containing k points. In addition, we study related problems including the 3-sided, arbitrarily oriented, weighted, and subset sum versions of the problem.