In this paper, we propose new techniques for solving geometric optimization problems involving interpoint distances of a point set in the plane. Given a set $P$ of $n$ points in the plane and an integer $1 \leq k \leq \binom{n}{2}$, the distance selection problem is to find the $k$-th smallest interpoint distance among all pairs of points of $P$. The previously best deterministic algorithm solves the problem in $O(n^{4/3} \log^2 n)$ time [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]. In this paper, we improve their algorithm to $O(n^{4/3} \log n)$ time. Using similar techniques, we also give improved algorithms on both the two-sided and the one-sided discrete Fr\'{e}chet distance with shortcuts problem for two point sets in the plane. For the two-sided problem (resp., one-sided problem), we improve the previous work [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014] by a factor of roughly $\log^2(m+n)$ (resp., $(m+n)^{\epsilon}$), where $m$ and $n$ are the sizes of the two input point sets, respectively. Other problems whose solutions can be improved by our techniques include the reverse shortest path problems for unit-disk graphs. Our techniques are quite general and we believe they will find many other applications in future.

翻译：摘要：本文提出了解决平面点集间点距几何优化问题的新技术。给定平面内 $n$ 个点的集合 $P$ 及整数 $1 \leq k \leq \binom{n}{2}$，距离选取问题要求找出 $P$ 中所有点对间第 $k$ 小的点距。此前最优确定性算法以 $O(n^{4/3} \log^2 n)$ 时间复杂度解决该问题 [Katz and Sharir, SIAM J. Comput. 1997 and SoCG 1993]。本文将其优化至 $O(n^{4/3} \log n)$ 时间复杂度。借助类似技术，我们还改进了平面内两点集间带捷径的双侧与单侧离散弗雷歇距离问题的算法。对于双侧问题（单侧问题），我们以约 $\log^2(m+n)$（或 $(m+n)^{\epsilon}$）的因子改进了先前工作 [Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015 and SoCG 2014]，其中 $m$ 和 $n$ 分别为两输入点集的大小。其他可通过本文技术改进的问题包括单位圆盘图的反向最短路径问题。本文提出的技术具有较强通用性，预计将在未来获得更多应用。