Extremal point queries with lines and line segments and related problems

Abstract

We address a number of extremal point query problems when P is a set of n points in ${\mathbb{R}}^d$, $d⩾3$ 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 ${\mathbb{R}}^3$, we give a data structure of size $\mathrm{O}(n^{1+\varepsilon })$, that can be constructed in $\mathrm{O}(n^{1+\varepsilon })$ time and can report the farthest point of P from a query line segment in $\mathrm{O}(n^{2/3+\varepsilon })$ time, where $\varepsilon >0$ 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 ${\mathbb{R}}^d$, $d⩾3$ 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 $P\setminus \{q\}$.