569 research outputs found
A Randomized Incremental Algorithm for the Hausdorff Voronoi Diagram of Non-crossing Clusters
In the Hausdorff Voronoi diagram of a family of \emph{clusters of points} in
the plane, the distance between a point and a cluster is measured as
the maximum distance between and any point in , and the diagram is
defined in a nearest-neighbor sense for the input clusters. In this paper we
consider %El."non-crossing" \emph{non-crossing} clusters in the plane, for
which the combinatorial complexity of the Hausdorff Voronoi diagram is linear
in the total number of points, , on the convex hulls of all clusters. We
present a randomized incremental construction, based on point location, that
computes this diagram in expected time and expected
space. Our techniques efficiently handle non-standard characteristics of
generalized Voronoi diagrams, such as sites of non-constant complexity, sites
that are not enclosed in their Voronoi regions, and empty Voronoi regions. The
diagram finds direct applications in VLSI computer-aided design.Comment: arXiv admin note: substantial text overlap with arXiv:1306.583
From Proximity to Utility: A Voronoi Partition of Pareto Optima
We present an extension of Voronoi diagrams where when considering which site
a client is going to use, in addition to the site distances, other site
attributes are also considered (for example, prices or weights). A cell in this
diagram is then the locus of all clients that consider the same set of sites to
be relevant. In particular, the precise site a client might use from this
candidate set depends on parameters that might change between usages, and the
candidate set lists all of the relevant sites. The resulting diagram is
significantly more expressive than Voronoi diagrams, but naturally has the
drawback that its complexity, even in the plane, might be quite high.
Nevertheless, we show that if the attributes of the sites are drawn from the
same distribution (note that the locations are fixed), then the expected
complexity of the candidate diagram is near linear.
To this end, we derive several new technical results, which are of
independent interest. In particular, we provide a high-probability,
asymptotically optimal bound on the number of Pareto optima points in a point
set uniformly sampled from the -dimensional hypercube. To do so we revisit
the classical backward analysis technique, both simplifying and improving
relevant results in order to achieve the high-probability bounds
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
Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions
We describe the implementation of algorithms to construct and maintain
three-dimensional dynamic Delaunay triangulations with kinetic vertices using a
three-simplex data structure. The code is capable of constructing the geometric
dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points
is triangulated. Time evolution of the triangulation is not only governed by
kinetic vertices but also by a changing number of vertices. We use
three-dimensional simplex flip algorithms, a stochastic visibility walk
algorithm for point location and in addition, we propose a new simple method of
deleting vertices from an existing three-dimensional Delaunay triangulation
while maintaining the Delaunay property. The dual Dirichlet tessellation can be
used to solve differential equations on an irregular grid, to define partitions
in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly
nomenclature), referee suggestions included, typos corrected, bibliography
update
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
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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