Computing a largest empty anchored cylinder, and related problems

Abstract

Let S be a set of n points in IRd, and let each point p of S’ have a positive weight w(p). We consider the problem of computing a ray R emanating from the origin (resp. a line l through the origin) such that minpεsw(p). d(p, R) (resp. minpεsw(p) d (p, R)) is maximal. If all weights are one, this corresponds to computing a silo emanating from the origin (resp. a cylinder whose axis contains the origin) that does not contain any point of S and whose radius is maximal. For d=2, we show how to solve these problems in O(n log n) time, which is optimal in the algebraic computation tree model. For d=3, we give algorithms that are based on the parametric search technique and run in O(n log 5 n) time. The previous best known algorithms for these three-dimensional problems had almost quadratic running time. In the final part of the paper, we consider some related problems.