Note: Computing closest and farthest points for a query segment

Authors:
Michael Segal;Eli Zeitlin
Affiliations:
Communication Systems Engineering Department, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel;School of Computer Science, Tel-Aviv University, Tel-Aviv 69978, Israel
Venue:
Theoretical Computer Science
Year:
2008

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Abstract

In this paper we present an improved algorithm for finding k closest (farthest) points for a given arbitrary query segment. We show how to preprocess a planar set P of n given points in O(n^2logn) expected time (or, alternatively, in O(n^2log^2n) deterministic time) and a subquadratic space, in order to report k closest points to an arbitrary given query line segment in O(k+log^2nloglogn) time. Here, for the first time, the data structure that provides polylogarithmic query time and uses subquadratic space is presented. We also show an algorithm for reporting the k farthest points from an arbitrary given query line segment.