This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.

Abstract. We introduce a new technique for solving problems of the following form: preprocess a set ofobjects so that those satisfying a given property with respect to a query object canbe listed very effectively. Well-known problems that fall into this category include range search, point enclosure, intersection, and near-neighbor problems. The approach which we take is very general and rests on a new concept called filtering search.We show on a number ofexamples how it can be used to improve the complexity ofknown algorithms and simplify their implementations as well. In particular, filtering search allows us to improve on the worst-case complexity ofthe best algorithms known so far for solving the problems mentioned above. Key words, computational geometry, database, data structures, filtering search, retrieval problems

Let be a set of points in in general position, i.e., no points on a common-flat,. A-set of is a set of points in that can be separated from by a hyperplane. A-facet of is an oriented-simplex spanned by points in which has exactly points from on the positive side of its affine hull. If is a planar point set and is even, a halving edge is an undirected edge between two points, such that the connecting line has the same number of points on either side. The number! "$ # of-sets is twice the number of halving edges. Inspired by Dey’s recent proof of a new bound on the number of-sets we show that

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the order-k Voronoi diagram in O(n log n + nk log k) expected time.

We present algorithms for five interdistance enumeration problems that take as input a set S of n points in IR d (for a fixed but arbitrary dimension d) and as output enumerate pairs of points in S satisfying various conditions. We present: an O(n log n + k) time and O(n) space algorithm that takes as additional input a distance # and outputs all k pairs of points in S separated by a distance of # or less; an O(n log n + k log k) time and O(n+k) space algorithm that enumerates in non-decreasing order the k closest pairs of points in S; an O(n log n + k) time algorithm for the same problem without any order restrictions; an O(nk log n) time and O(n) space algorithm that enumerates in nondecreasing order the nk pairs representing the k nearest neighbors of each point in S; and an O(n log n + kn) time algorithm for the same problem without any order restrictions. The algorithms combine a modification of the planar approach of Dickerson, Drysdale, and Sack [11] with the ...

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The Delaunay Tree is a hierarchical data structure that was introduced in [BT86]. It is defined from the Delaunay triangulation and, roughly speaking, represents a triangulation as a hierarchy of balls. It allows a semi-dynamic construction of the Delaunay triangulation of a finite set of n points in any dimension. In this paper, we prove that a randomized construction of the Delaunay Tree (and thus, of the Delaunay triangulation) can be done in O(n log n) expected time in the plane and in O i n d d 2 e j expected time in d-dimensional space. These results are optimal for fixed d. The algorithm is extremely simple and experimental results are given.

We use here the results on the influence graph[1] to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(nlog n) to O(n log ⋆ n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).

In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations and arithmetic degree.

In light of recent developments, this paper re-examines the fundamental geometric problem of how to construct the k-level in an arrangement of n lines in the plane. ffl The author's recent dynamic data structure for planar convex hulls improves a decade-old sweep-line algorithm by Edelsbrunner and Welzl, which now runs in O(n log m+m log 1+" n) deterministic time and O(n) space, where m is the output size and " is any positive constant. We discuss simplification of the data structure in this particular application, by viewing the problem kinetically. ffl Har-Peled recently announced a randomized algorithm with an expected running time of O((n + m)ff(n) log n). We observe that a version of an earlier randomized incremental algorithm by Agarwal, de Berg, Matousek, and Schwarzkopf yields almost the same result. ffl The current combinatorial bound by Dey shows that m = O(nk 1=3 ) in the worst case. We give an algorithm that guarantees O(n log n + nk 1=3 ) expected time. 1 Introd...