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Abstract. This paper addresses the complexity of computing the smallest-radius infinite cylinder that encloses an input set of n points in 3-space. We show that the problem can be solved in time O(n 4 log O(1) n) in an algebraic complexity model. We also achieve a time of O(n 4 L · µ(L)) in a bit complexity model where L is the maximum bit size of input numbers and µ(L) is the complexity of multiplying two L bit integers.These and several other results highlight a general linearization technique which transforms nonlinear problems into some higher-dimensional but linear problems. The technique is reminiscent of the use of Plücker coordinates, and is used here in conjunction with Megiddo's parametric searching.We further report on experimental work comparing the practicality of an exact with that of a numerical strategy.

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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 min p ∈ s w(p) · d(p,R) ( resp. min p ∈ 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 log 4 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.