Planar Voronoi Diagrams for Sums of Convex Functions, Smoothed Distance and Dilation

@article{Dickerson2010PlanarVD,
  title={Planar Voronoi Diagrams for Sums of Convex Functions, Smoothed Distance and Dilation},
  author={Matthew Dickerson and David Eppstein and Kevin A. Wortman},
  journal={2010 International Symposium on Voronoi Diagrams in Science and Engineering},
  year={2010},
  pages={13-22}
}
We study Voronoi diagrams for distance functions that add together two convex functions, each taking as its argument the difference between Cartesian coordinates of two planar points. When the functions do not grow too quickly, then the Voronoi diagram has linear complexity and can be constructed in near-linear randomized expected time. Additionally, the level sets of the distances from the sites form a family of pseudocircles in the plane, all cells in the Voronoi diagram are connected, and… 

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References

SHOWING 1-10 OF 32 REFERENCES
Voronoi diagrams based on convex distance functions
TLDR
An asymptotically optimal algorithm for computing Voronoi diagrams based on convex distance functions that allows such diagrams to be defined for very general metrics and for distance measures that do not qualify as metrics.
Bregman Voronoi Diagrams
TLDR
A framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences, which allow one to define information-theoretic Vor onoi diagrams in statistical parametric spaces based on the relative entropy of distributions.
On the construction of abstract voronoi diagrams
TLDR
It is shown that the abstract Voronoi diagram of n sites in the plane can be constructed in timeO(n logn) by a randomized algorithm based on Clarkson and Shor's randomized incremental construction technique.
Randomized Incremental Construction of Abstract Voronoi Diagrams
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
TLDR
An upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets is obtained and can be applied to planning a collision-free translational motion of a convex polygonB amidst several polygonal obstacles.
Centroidal Voronoi Tessellations: Applications and Algorithms
TLDR
Some applications of centroidal Voronoi tessellations to problems in image compression, quadrature, finite difference methods, distribution of resources, cellular biology, statistics, and the territorial behavior of animals are given.
Minimum dilation stars
TLDR
This paper considers the problem of positioning the root of a star such that the dilation of the resulting star is minimal, and presents a deterministic O(n log n)-time algorithm for evaluating theDilation of a given star; a randomized O( n log n) expected- time algorithm for finding an optimal center in Rd; and for the case d = 2, a randomized N2α(n) log2n expected-time algorithm.
Encyclopedia of Distances
TLDR
This book begins with several metrics in classical geometry, then proceeds to applications of distance in fields like algebra and probability, eventually working through applied mathematics, computer science, physics and chemistry, social science, and even art and religion.
New bounds for lower envelopes in three dimensions, with applications to visibility in terrains
TLDR
An upper bound is obtained on the combinatorial complexity of the “lower envelope” of thespace of all rays in 3-space that lie above a given polyhedral terrain with edges with the additional property that the interiors of any triple of these surfaces intersect in at most two points.
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