Computing a largest empty anchored cylinder, and related problems

Abstract

Let $S$ be a set of $n$ points in $R^d$, 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 $\min_{p\in S} w(p) \cdot d(p,R)$ (resp. $\min_{p\in S} w(p) \cdot d(p,l)$) 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 problem