Improved Algorithms for Distance Selection and Related Problems

Abstract

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 \(1 \le k \le \left( {\begin{array}{c}n\\ 2\end{array}}\right) \), 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 in SIAM J Comput 26(5):1384–1408, 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 Fréchet 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 et al. in ACM Trans Algorithms 11(4):29, 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.