Smallest k-Enclosing Rectangle Revisited

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-$O(n^{5\mathrm{/}2})$-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 $O(nk)$ time. We also present a near linear time $(1+\epsilon )$-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.