Algorithms for Computing Closest Points for Segments

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((n4/3)) has also been proved. In this paper, we present an O(n(4/3)) time algorithm and thus solve the problem optimally (under the restricted model). 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.