Computing a Largest Empty Anchored Cylinder, and Related Problems

Let S be a set of n points in ℝd, 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 minp∈S w(p) · d(p, R) (resp. minp∈S w(p) · 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 log5n) 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.

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