Deterministic Distance Selection in $O(n^{4/3})$ Time

Abstract

Let $P$ be a set of $n$ points in the plane. Given an integer $K$ with $1\le K\le \left(\genfrac{}{}{0ex}{}{n}{2}\right)$, the distance selection problem asks for the $K$-th smallest distance among all pairwise distances of the points of $P$. The problem has been studied extensively over the past four decades. Recently, Chan and Zheng (2023) gave a randomized algorithm with $O(n^{4/3})$ expected running time, while the best known deterministic algorithm runs in $O(n^{4/3}\log n)$ time. In this paper, we present a deterministic $O(n^{4/3})$-time algorithm for the problem, matching the best known randomized complexity. We also consider the more general bichromatic version of the problem, where two point sets $A$ and $B$ are given, with $m=|A|$ and $n=|B|$, and the goal is to find the $K$-th smallest distance among the $mn$ distances between points of $A$ and points of $B$. For this problem, the best previously known algorithm runs in $O((m\log n+n\log m+m^{2/3}{n}^{2/3})\log (m+n))$ time. We present a new deterministic algorithm with running time $O(m{\log }^{2}n+n{\log }^{2}m+m^{2/3}{n}^{2/3})$.

arXiv commentTo appear in SODA 2027