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Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 57 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
A SemiDynamic Construction of Higher Order Voronoï Diagrams and its Randomized Analysis
, 1990
"... The kDelaunay tree extends the Delaunay tree introduced in [1,2]. It is a hierarchical data structure that allows the semidynamic construction of the higher order Voronoi diagrams of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the kDelaunay ..."
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Cited by 7 (3 self)
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The kDelaunay tree extends the Delaunay tree introduced in [1,2]. It is a hierarchical data structure that allows the semidynamic construction of the higher order Voronoi diagrams of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the kDelaunay tree, and thus, of all the order < k Voronoi diagrams, can be done in O(n log n1 k3n) expected time and O(k2t expected storage in the plane, which is asymptotically optimal for fixed k. Our algorithm extends to d dimensional space with expected time complexity o and space complexity O The algorithm is simple and experimental results are given.
Computational geometry and its Application to Computer Graphics
, 1989
"... This paper gives a basic introduction into some of the techniques and data structures developed in computational geometry with emphasis on their use in computer graphics applications. Due to the limited space only global ideas are outlined and no details are presented. References are added, in parti ..."
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Cited by 3 (0 self)
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This paper gives a basic introduction into some of the techniques and data structures developed in computational geometry with emphasis on their use in computer graphics applications. Due to the limited space only global ideas are outlined and no details are presented. References are added, in particular in a subsection "Further Reading" at the end of each section, for those that are interested in learning more about the topics treated
Nordic Journal of Computing 1(1994), 268–272. A NOTE ON HIGHER ORDER VORONOI DIAGRAMS
"... Abstract. In this note we prove some facts about the number of segments of a bisector of two sites that are used in a higher order Voronoi diagram. CR Classification: F.2.1 Key words: computational geometry, Voronoi diagrams Let S be a set of n sites in the Euclidean plane. Informally, the Voronoi d ..."
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Abstract. In this note we prove some facts about the number of segments of a bisector of two sites that are used in a higher order Voronoi diagram. CR Classification: F.2.1 Key words: computational geometry, Voronoi diagrams Let S be a set of n sites in the Euclidean plane. Informally, the Voronoi diagram is a subdivision of the plane into regions such that each point of a region has the same closest site. Aurenhammer [3] discusses the importance of Voronoi diagrams to computer scientists and presents a survey of the Voronoi diagram and its variants, including mathematical properties and algorithms. The variant that we are interested in here is the kth order Voronoi diagram, denoted by Vk(S). The kth order Voronoi diagram is a partition of the plane into regions such that points in each region have the same k closest sites. There are many deterministic algorithms to compute Vk(S) [7, 5, 4, 2]. Lee, Chazelle and Edelsbrunner [7, 4] describe many properties of Vk(S). In this paper we are not concerned with algorithms to compute Vk(S), instead, we will prove some properties about the number of segments of a bisector of two sites that are used in Vk(S). Most algorithms for Voronoi diagrams and variants of Voronoi diagrams assume nondegeneracy: no four sites are cocircular. For our purposes, this also implies no three sites lie on the same line. We will first make this assumption and then state what happens when we have degeneracies. In order to prove some facts about the kth order Voronoi diagram, we will transform the diagram to a three dimensional arrangement of planes using the standard lifting projection [4, 6]. That is, a site pi with coordinates (x1, x2) is mapped to the plane Hi tangent to the paraboloid x3 = x2 1 + x22 at the point (x1, x2, x2 1 + x22). The arrangement of these planes is denoted by A(H). There is a onetoone correspondence between the edges in Vk(S) and the edges e in A(H) with k − 1 planes above e. Corresponding to the perpen