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：本文提出解决平面上点集间点对距离几何优化问题的新技术。给定包含$n$个点的平面点集$P$及整数$1 \leq k \leq \binom{n}{2}$，距离选择问题旨在找出$P$中所有点对间第$k$小的点间距离。此前最优确定性算法由Katz和Sharir提出，时间复杂度为$O(n^{4/3} \log^2 n)$[SIAM J. Comput. 1997及SoCG 1993]。本文将该算法改进至$O(n^{4/3} \log n)$。采用相似技术，我们同时改进了平面上两个点集间带捷径的双侧与单侧离散Fr\'{e}chet距离问题算法。对于双侧问题（单侧问题），我们将此前工作[Avraham, Filtser, Kaplan, Katz, and Sharir, ACM Trans. Algorithms 2015及SoCG 2014]的复杂度降低约$\log^2(m+n)$因子（$(m+n)^{\epsilon}$因子），其中$m$与$n$分别为两个输入点集规模。其他可借助本文技术改进解决方案的问题包括单位圆盘图的反向最短路径问题。本文所提技术具有较强通用性，预计将在未来研究中得到广泛应用。