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The “nearest neighbor” relation, or more generally the “k nearest neighbors” relation, defined for a set of points in a metric space, has found many uses in computational geometry and clustering analysis, yet surprisingly little is known about some of its basic properties. In this paper, we consider some natural questions that are motivated by geometric embedding problems. We derive bounds on the...
Geometric algorithms are usually designed with continuous parameters in mind. When the underlying geometric space is intrinsically discrete, as is the case for computer graphics problems, such algorithms are apt to give invalid solutions if properties of a finite-resolution space are not taken into account. In this paper we discuss an approach for transforming geometric concepts and algorithms from...
The task of eliminating invisible parts when generating an image is one of the central problems in computer graphics. One approach achieves the desired obscuring effect by assigning priority numbers to the faces of an object. Image generation can be speeded up significantly in these priority schemes if priority information of the faces can be partly computed in advance, before a viewing position is...
Let Vk (n) be the minimum average number of pairwise comparisons needed to find the k-th largest of n numbers (k≥2), assuming that all n! orderings are equally likely. D. W. Matula proved that, for some absolute constant c, Vk(n)- n ≤ ck log log n as n → ∞. In the present paper, we show that there exists an absolute constant c′ ≫ 0 such that Vk(n) - n ≥ c′k log log n as n → ∞, proving a conjecture...
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