Algorithms for computing closest points for segments

Abstract

Given a set P of n points and a set S of n segments in the plane, we consider the problem of computing for each segment of S its closest point in P. The previously best algorithm solves the problem in $n^{4/3}{2}^{O({\log }^{⁎}n)}$ time [Bespamyatnikh, 2003] and a lower bound (under a somewhat restricted model) $\Omega \left(n^{4/3}\right)$ has also been proved. In this paper, we present an $O\left(n^{4/3}\right)$ time algorithm, which matches the above lower bound. In addition, we also present data structures for solving the online version of the problem, i.e., given a query segment (or a line as a special case), find its closest point in P. Our new results improve the previous work.