Simple $O(n~log^{2}~n)$ Algorithms for the Planar 2-Center Problem

Abstract

The planar 2-center problem for a set S of points in the plane asks for two congruent circular disks of the minimum radius, whose union covers all points of S. In this paper, we present a simple $O(n \log ^2 n)$ time algorithm for the planar 2-center problem, improving upon the previously known bound by a factor of $(\log \log n)^2$. We first describe an $O(n \log ^2 n)$ time solution to a restricted 2-center problem in which the given points are in convex position (i.e., they are the vertices of a convex polygon), and then extend it to the case of arbitrarily given points. The novelty of our algorithms is their simplicity: our algorithms use only binary search and the known algorithms for computing the smallest enclosing disk of a point set, avoiding the use of relatively complicated parametric search, which is the base for most planar 2-center algorithms. Our work sheds more light on the (open) problem of developing an $O(n \log n)$ time algorithm for the planar 2-center problem.

The work by Tan was partially supported by JSPS KAKENHI Grant Number 15K00023, and the work by Jiang was partially supported by National Natural Science Foundation of China under grant 61173034.

Appendix: Note on the Kim and Shin algorithm for covering a set of points in convex position

In this appendix, we point out a mistake that occurs in the previously announced $O(n \log ^2 n)$ time algorithm, by Kim and Shin, for covering a set A of points in convex position [8].

We use the same notation as given in [8]. For any $i, j \in A$, let $r^1_{i,j}$ ($r^2_{i,j}$) be the radius of the smallest disk containing $\langle i, j-1 \rangle$ ($\langle j, i-1 \rangle$). Then, the optimum solution $r^{*} = \min _{i, j \in A} \max \{ r^1_{i, j}, r^2_{i, j} \}$.

Let $r_0^{*} = \min _{j \in A} \max \{ r^1_{0, j}, r^2_{0, j} \}$, and let k be the point such that $r_0^{*} = \max \{ r^1_{0, k}, r^2_{0, k} \}$. Then, it was stated in Lemma 7 of [8] that “For any i, j such that $i, j \in \langle 0, k-1 \rangle$ or $i, j \in \langle k, n-1 \rangle$, $\max \{ r^1_{i,j}, r^2_{i,j} \} > r^{*}_0$”.

By Lemma 7 of [8], all the pairs of $i, j \in \langle 0, k-1 \rangle$ or $i, j \in \langle k, n-1 \rangle$ needn’t be considered. Kim and Shin then described a divide-and-conquer procedure to compute all the pairs of i and j such that $i \in \langle 0, k-1 \rangle$ and $j \in \langle k, n-1 \rangle$, in the paragraph immediately following Lemma 7 of [8]. Note that the number of the pairs of i and j, which are found by their divide-and-conquer procedure, is O(n) (other than $O(\log n)$). To compute $r^{*}$, these O(n) pairs of i and j have further to be handled (e.g., using parametric search to compute the smallest enclosing disks for them). However, it is NOT considered in [8]. Therefore, the $O(n \log ^2 n)$ time solution to the planar 2-center problem for a set of points in convex position was not obtained in [8].