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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 59 (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
A Unified Framework for Efficiently Processing Ranking Related Queries
"... The computation of klower envelope is a classical problem and has been very well studied for main memory nonindexed data. In this paper, we study the problem from the database perspective and present the first algorithm which utilizes the presence of the index and achieves access optimality, i.e., ..."
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The computation of klower envelope is a classical problem and has been very well studied for main memory nonindexed data. In this paper, we study the problem from the database perspective and present the first algorithm which utilizes the presence of the index and achieves access optimality, i.e., it accesses a node of the index only if the correctness of the results cannot be guaranteed without accessing this node. We also demonstrate the applications of klower envelope in ranking systems. Let an object be called valuable if it is one of the topk objects according to at least one linear scoring function. In this paper, we answer the following important questions that may be asked by different users: 1) I am not sure what scoring function I should use, therefore, return me the set of valuable objects so that I can select an object I like the most; 2) How can I modify the attributes (e.g., price) of my product such that it becomes a valuable object; 3) What are the preference functions for which a given object is among the topk objects. These three questions are formalized and called ksnippet, kdepth contour and reverse topk query, respectively. We propose a unified framework to solve these queries by utilizing klower envelope as a common foundation. Our main algorithm is access optimal for ksnippet and klower envelope computation. We also demonstrate its access optimality for the kdepth contour problem when k is smaller than the minimum number of objects in any leaf node of the index structure. Our algorithms outperform stateoftheart algorithms by more than an order of magnitude in terms of both CPU and I/O cost. 1.
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
A Unified Framework for Efficiently Processing Ranking Related Queries
"... The computation of klower envelope is a classical problem and has been very well studied for main memory nonindexed data. In this paper, we study the problem from the database perspective and present the first algorithm which utilizes the presence of the index and achieves access optimality, i.e., ..."
Abstract
 Add to MetaCart
(Show Context)
The computation of klower envelope is a classical problem and has been very well studied for main memory nonindexed data. In this paper, we study the problem from the database perspective and present the first algorithm which utilizes the presence of the index and achieves access optimality, i.e., it accesses a node of the index only if the correctness of the results cannot be guaranteed without accessing this node. We also demonstrate the applications of klower envelope in ranking systems. Let an object be called valuable if it is one of the topk objects according to at least one linear scoring function. In this paper, we answer the following important questions that may be asked by different users: 1) I am not sure what scoring function I should use, therefore, return me the set of valuable objects so that I can select an object I like the most; 2) How can I modify the attributes (e.g., price) of my product such that it becomes a valuable object; 3) What are the preference functions for which a given object is among the topk objects. These three questions are formalized and called ksnippet, kdepth contour and reverse topk query, respectively. We propose a unified framework to solve these queries by utilizing klower envelope as a common foundation. Our main algorithm is access optimal for ksnippet and klower envelope computation. We also demonstrate its access optimality for the kdepth contour problem when k is smaller than the minimum number of objects in any leaf node of the index structure. Our algorithms outperform stateoftheart algorithms by more than an order of magnitude in terms of both CPU and I/O cost. 1.