# Smallest k-Enclosing Rectangle Revisited

## TL;DR

This paper introduces faster algorithms for finding the smallest axis-aligned rectangle enclosing k points, including near quadratic, near linear, and approximation algorithms, along with complexity bounds and related problem variants.

## Contribution

It presents the first near quadratic time algorithm, a k-sensitive algorithm, and a near linear approximation for the k-enclosing rectangle problem, advancing computational geometry methods.

## Key findings

- First near quadratic time algorithm for the problem.
- A k-sensitive algorithm with near O(nk) time complexity.
- A near linear time (1+ε)-approximation algorithm for minimum area.

## Abstract

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-$O(n^{5/2})$-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.