发表机构
University of Utah（犹他大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对平面点集的距离选择问题提出O(n^{4/3})时间的确定性算法，匹配最优随机复杂度，还给出双色版本问题的更优确定性算法。
AI 中文摘要
设P为平面上的n个点构成的集合，给定整数K满足1≤K≤C(n,2)，距离选择问题要求找出P中所有点对距离中的第K小距离。该问题在过去四十年间被广泛研究，Chan与Zheng（2023）近期提出了一种期望运行时间为O(n^{4/3})的随机算法，而已知最优确定性算法的运行时间为O(n^{4/3}log n)。本文针对该问题提出了一种运行时间为O(n^{4/3})的确定性算法，其复杂度与已知最优随机算法匹配。本文还考虑了更通用的双色版本问题：给定两个点集A和B，其中m=|A|，n=|B|，目标是找出A与B中点之间的mn个距离中的第K小距离。针对该问题，此前已知最优算法的运行时间为O((m log n + n log m + m^{2/3}n^{2/3})log(m+n))，本文提出了一种新的确定性算法，其运行时间为O(m log²n + n log²m + m^{2/3}n^{2/3})。
英文摘要
Let $P$ be a set of $n$ points in the plane. Given an integer $K$ with $1\le K\le {n\choose 2}$, 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 $O(m\log^2 n+n\log^2 m+m^{2/3}n^{2/3})$.
CommentsTo appear in SODA 2027
