2,268 research outputs found
An Efficient Randomized Algorithm for Higher-Order Abstract Voronoi Diagrams
Given a set of n sites in the plane, the order-k Voronoi diagram is a planar subdivision such that all points in a region share the same k nearest sites. The order-k Voronoi diagram arises for the k-nearest-neighbor problem, and there has been a lot of work for point sites in the Euclidean metric. In this paper, we study order-k Voronoi diagrams defined by an abstract bisecting curve system that satisfies several practical axioms, and thus our study covers many concrete order-k Voronoi diagrams. We propose a randomized incremental construction algorithm that runs in O(k(n-k) log^2 n +n log^3 n) steps, where O(k(n-k)) is the number of faces in the worst case. Due to those axioms, this result applies to disjoint line segments in the L_p norm, convex polygons of constant size, points in the Karlsruhe metric, and so on. In fact, this kind of run time with a polylog factor to the number of faces was only achieved for point sites in the L_1 or Euclidean metric before
On the Complexity of Randomly Weighted Voronoi Diagrams
In this paper, we provide an bound on the expected
complexity of the randomly weighted Voronoi diagram of a set of sites in
the plane, where the sites can be either points, interior-disjoint convex sets,
or other more general objects. Here the randomness is on the weight of the
sites, not their location. This compares favorably with the worst case
complexity of these diagrams, which is quadratic. As a consequence we get an
alternative proof to that of Agarwal etal [AHKS13] of the near linear
complexity of the union of randomly expanded disjoint segments or convex sets
(with an improved bound on the latter). The technique we develop is elegant and
should be applicable to other problems
Diagramas de Voronoi de ordem k na geometria projetiva orientada
Orientador: Pedro Jussieu de RezendeDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta dissertação, apresentamos uma generalização do diagrama de Voronoi: consideramos diagramas de Voronoi de ordem k no plano projetivo orientado T². Este espaço admite retas orientadas assim como muitos outros conceitos geométricos fundamentais de maneira consistente. Neste contexto, demonstramos várias propriedades de diagramas de Voronoi, algumas delas intrínsecas a T². Por exemplo, o diagrama de Voronoi de ordem k de um conjunto de n sítios em T² tem um número exato de regiões e é antípoda do diagrama de Voronoi de ordem n - k do mesmo conjunto de sítios, para todo k : 1 < k < n. Finalmente, apresentamos uma generalização, de R² para T², de dois algoritmos para construção de diagramas de Voronoi de ordem k. O primeiro algoritmo constrói os diagramas de Voronoi de todas as ordens para busca dos k vizinhos mais próximos, em tempo e espaço ótimos; enquanto o segundo é um algoritmo incremental randomizado on-line para construir o diagrama de Voronoi de cada ordem, independentemente. Para este segundo algoritmo, apresentamos um novo método para localização de pontos, o qual reduz a complexidade de tempo por um fator logarítmico e que é muito mais simples que o original.Abstract: In this dissertation, we present a generalization of the Voronoi diagram: we consider order k Voronoi diagrams in the oriented projective plane T². This space handles oriented lines as well as many other fundamental geometric concepts in a consistent way. In this context, we show several properties of Voronoi diagrams, some of them intrinsic to T². For example, the order k Voronoi diagram of a set of n sites in T² has an exact number of regions. Furthermore, this diagram is antipodal to the order n - k Voronoi diagram of the same set of sites, for all k : 1 < k < n. Finally, we present a generalization, from R² to T², of two algorithms for constructing order k Voronoi diagrams. The first one constructs all Voronoi diagrams for k nearest neighbor search, in optimal time and space, and the other is an on-line randomized incremental algorithm for constructing each order k Voronoi diagram, independently. For this second algorithm, we present a new method for point location which improves the time complexity by a logarithmic factor and which is much simpler than the original one.MestradoMestre em Ciência da Computaçã
Searching edges in the overlap of two plane graphs
Consider a pair of plane straight-line graphs, whose edges are colored red
and blue, respectively, and let n be the total complexity of both graphs. We
present a O(n log n)-time O(n)-space technique to preprocess such pair of
graphs, that enables efficient searches among the red-blue intersections along
edges of one of the graphs. Our technique has a number of applications to
geometric problems. This includes: (1) a solution to the batched red-blue
search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an
algorithm to compute the maximum vertical distance between a pair of 3D
polyhedral terrains one of which is convex in O(n log n) time, where n is the
total complexity of both terrains; (3) an algorithm to construct the Hausdorff
Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n)
time and O(n+m) space, where n is the total number of points in all clusters
and m is the number of crossings between all clusters; (4) an algorithm to
construct the farthest-color Voronoi diagram of the corners of n axis-aligned
rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle
problem for n parallel line segments in the plane in optimal O(n log n) time.
All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure
Dense point sets have sparse Delaunay triangulations
The spread of a finite set of points is the ratio between the longest and
shortest pairwise distances. We prove that the Delaunay triangulation of any
set of n points in R^3 with spread D has complexity O(D^3). This bound is tight
in the worst case for all D = O(sqrt{n}). In particular, the Delaunay
triangulation of any dense point set has linear complexity. We also generalize
this upper bound to regular triangulations of k-ply systems of balls, unions of
several dense point sets, and uniform samples of smooth surfaces. On the other
hand, for any n and D=O(n), we construct a regular triangulation of complexity
Omega(nD) whose n vertices have spread D.Comment: 31 pages, 11 figures. Full version of SODA 2002 paper. Also available
at http://www.cs.uiuc.edu/~jeffe/pubs/screw.htm
Data Structures for Halfplane Proximity Queries and Incremental Voronoi Diagrams
We consider preprocessing a set of points in convex position in the
plane into a data structure supporting queries of the following form: given a
point and a directed line in the plane, report the point of that
is farthest from (or, alternatively, nearest to) the point among all points
to the left of line . We present two data structures for this problem.
The first data structure uses space and preprocessing
time, and answers queries in time, for any . The second data structure uses space and
polynomial preprocessing time, and answers queries in time. These
are the first solutions to the problem with query time and
space.
The second data structure uses a new representation of nearest- and
farthest-point Voronoi diagrams of points in convex position. This
representation supports the insertion of new points in clockwise order using
only amortized pointer changes, in addition to -time
point-location queries, even though every such update may make
combinatorial changes to the Voronoi diagram. This data structure is the first
demonstration that deterministically and incrementally constructed Voronoi
diagrams can be maintained in amortized pointer changes per operation
while keeping -time point-location queries.Comment: 17 pages, 6 figures. Various small improvements. To appear in
Algorithmic
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