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) $\Omega(n^{4/3})$ 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.Comment: Accepted to STACS 202

Computing closest points for segments

Abstract We consider the proximity problem of computing for each of n line segments the closest point from a given set of n points in the plane. It generalizes Hopcroft's problem [11] and the nearest foreign neighbors problem [15]. We show that it can be solved in O(

Computing Closest Points for Segments

Abstract We consider the proximity problem of computing for each of n line segments the closest point from a given set of n points in the plane. We show that it can be solved in (i) O(n4=32O(log\Lambda n)) time, and (ii) O(n log2 n) time for the case of disjoint segments