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) Ω ( n 4 / 3 ) has also been proved. In this paper, we present an O ( n 4 / 3 ) 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.