A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Boissonnat, Jean Daniel; Devillers, Olivier; Teillaud, Monique

doi:10.1007/bf01228508

Cited by 37 publications

(26 citation statements)

References 15 publications

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“…A few attempts have been proposed to avoid the conflict graph and thus to obtain on-line algorithms in some specific applications [2], [3], [10]. The present paper extends these results and provides a general framework for on-line algorithms with good expected behavior when the objects are assumed to be introduced in random order.…”

Section: Introductionmentioning

confidence: 76%

Applications of random sampling to on-line algorithms in computational geometry

Boissonnat

Devillers

Schott³

et al. 1992

Discrete Comput Geom

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Abstract. This paper presents a general framework for the design and randomized analysis of geometric algorithms. These algorithms are on-line and the framework provides general bounds for their expected space and time complexities when averaging over all permutations of the input data. The method is general and can be applied to various geometric problems. The power of the technique is illustrated by new efficient on-line algorithms for constructing convex hulls and Voronoi diagrams in any dimension, Voronoi diagrams of line segments in the plane, arrangements of curves in the plane, and others.

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Section: Introductionmentioning

confidence: 76%

Applications of random sampling to on-line algorithms in computational geometry

Boissonnat

Devillers

Schott³

et al. 1992

Discrete Comput Geom

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“…In particular, it has been observed (see e.g., [7,1] and the books [11,12]) that collections of points whose circumspheres contain a fixed number of points in their interior play an important role in algorithms for generating higher-order Voronoi diagrams. For example, the notion of a Delaunay tree [1] can be viewed as a hierarchy of Delaunay configurations of varying degrees. The paper [1] in fact contains the observation in the first part of Proposition 2.1, concerning the existence of points v andv.…”

Section: Discussionmentioning

confidence: 99%

“…For example, the notion of a Delaunay tree [1] can be viewed as a hierarchy of Delaunay configurations of varying degrees. The paper [1] in fact contains the observation in the first part of Proposition 2.1, concerning the existence of points v andv.…”

Section: Discussionmentioning

confidence: 99%

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

Neamtu

2007

Trans. Amer. Math. Soc.

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Abstract. In the 1920s, B. N. Delaunay proved that the dual graph of the Voronoi diagram of a discrete set of points in a Euclidean space gives rise to a collection of simplices, whose circumspheres contain no points from this set in their interior. Such Delaunay simplices tessellate the convex hull of these points. An equivalent formulation of this property is that the characteristic functions of the Delaunay simplices form a partition of unity. In the paper this result is generalized to the so-called Delaunay configurations. These are defined by considering all simplices for which the interiors of their circumspheres contain a fixed number of points from the given set, in contrast to the Delaunay simplices, whose circumspheres are empty. It is proved that every family of Delaunay configurations generates a partition of unity, formed by the so-called simplex splines. These are compactly supported piecewise polynomial functions which are multivariate analogs of the well-known univariate B-splines. It is also shown that the linear span of the simplex splines contains all algebraic polynomials of degree not exceeding the degree of the splines.

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“…The underlying applications are characterized by nonstructured environments, little or not known, with constrained accesses or displacements, and the implementation of complex robotics systems which may be composed of several interacting robots and various sensory devices. In this context, the main activity of the project is computational geometry for robot motion planning (Avnaim & Boissonnat 1989;Boissonnat, Devillers, Teillaud, 1993;Devillers 1992;Devillers, Meiser, & Teillaud 1992).…”

Section: Introductionmentioning

confidence: 99%

Computer vision research at INRIA

Faugeras

1993

Int J Comput Vision

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A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

Cited by 37 publications

References 15 publications

Applications of random sampling to on-line algorithms in computational geometry

Applications of random sampling to on-line algorithms in computational geometry

Delaunay configurations and multivariate splines: A generalization of a result of B. N. Delaunay

Computer vision research at INRIA

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