Extremal point queries with lines and line segments and related problems

摘要

We address a number of extremal point query problems when P is a set of n points in R-d, d >= 3 a constant, including the computation of the farthest point from a query line and the computation of the farthest point from each of the lines spanned by the points in P. In R-3, we give a data structure of size O(n(1+epsilon)), that can be constructed in 0(n(1+epsilon)) time and can report the farthest point of P from a query line segment in O(n(2/3+epsilon)) time, where epsilon > 0 is an arbitrarily small constant. Applications of our results also include: (1) Sub-cubic time algorithms for fitting a polygonal chain through an indexed set of points in R d, d > 3 a constant, and (2) A sub-quadratic time and space algorithm that, given P and an anchor point q, computes the minimum (maximum) area triangle defined by q with P {q}. (c) 2005 Elsevier B.V. All rights reserved.