An O(n log n) Algorithm for Rectilinear Minimal Spanning Trees
Journal of the ACM (JACM)
Two-Dimensional Voronoi Diagrams in the Lp-Metric
Journal of the ACM (JACM)
Computing the Largest Empty Rectangle
STACS '84 Proceedings of the Symposium of Theoretical Aspects of Computer Science
Retraction: A new approach to motion-planning
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Generalized voronoi diagrams and geometric searching.
Generalized voronoi diagrams and geometric searching.
Fast heuristics for minimum length rectangular partitions of polygons
SCG '86 Proceedings of the second annual symposium on Computational geometry
There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
A new efficient motion-planning algorithm for a rod in polygonal space
SCG '86 Proceedings of the second annual symposium on Computational geometry
A sweepline algorithm for Voronoi diagrams
SCG '86 Proceedings of the second annual symposium on Computational geometry
On the geodesic Voronoi diagram of point sites in a simple polygon
SCG '87 Proceedings of the third annual symposium on Computational geometry
Constrained Delaunay triangulations
SCG '87 Proceedings of the third annual symposium on Computational geometry
Placing the largest similar copy of a convex polygon among polygonal obstacles
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Computing the minimum Hausdorff distance for point sets under translation
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Solving query-retrieval problems by compacting Voronoi diagrams
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
The upper envelope of Voronoi surfaces and its applications
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Multiplicatively weighted crystal growth Voronoi diagrams (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Transitions in geometric minimum spanning trees (extended abstract)
SCG '91 Proceedings of the seventh annual symposium on Computational geometry
Voronoi diagrams—a survey of a fundamental geometric data structure
ACM Computing Surveys (CSUR)
Bottleneck Steiner Trees in the Plane
IEEE Transactions on Computers
On the topological shape of planar Voronoi diagrams
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Convex distance functions in 3-space are different
SCG '93 Proceedings of the ninth annual symposium on Computational geometry
Piecewise linear paths among convex obstacles
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Voronoi diagrams and containment of families of convex sets on the plane
Proceedings of the eleventh annual symposium on Computational geometry
Voronoi diagrams for direction-sensitive distances
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Probabilistic analysis for combinatorial functions of moving points
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
Voronoi diagram in statistical parametric space by Kullback-Leibler divergence
SCG '97 Proceedings of the thirteenth annual symposium on Computational geometry
IEEE Transactions on Parallel and Distributed Systems
A zero-skew clock routing scheme for VLSI circuits
ICCAD '92 Proceedings of the 1992 IEEE/ACM international conference on Computer-aided design
On bisectors for different distance functions
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Voronoi diagrams of lines in 3-space under polyhedral convex distance functions
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
On the all-pairs Euclidean short path problem
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
A practical algorithm for computing the Delaunay triangulation for convex distance functions
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Planning the shortest path for a disc in O(n2log n) time
SCG '85 Proceedings of the first annual symposium on Computational geometry
A tight bound for the complexity of voroni diagrams under polyhedral convex distance functions in 3D
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Optimizing Constrained Offset and Scaled Polygonal Annuli
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
Java applets for the dynamic visualization of Voronoi diagrams
Computer Science in Perspective
Abstract Voronoi diagram in 3-space
Journal of Computer and System Sciences
Computing the visibility graph of points within a polygon
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Robustness of k-gon Voronoi diagram construction
Information Processing Letters
Querying approximate shortest paths in anisotropic regions
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Approximate shortest paths in anisotropic regions
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
A simple and efficient kinetic spanner
Proceedings of the twenty-fourth annual symposium on Computational geometry
Divide-and-conquer for Voronoi diagrams revisited
Proceedings of the twenty-fifth annual symposium on Computational geometry
Abstract Voronoi diagrams revisited
Computational Geometry: Theory and Applications
A Scheme for Computing Minimum Covers within Simple Regions
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
A theoretical structure for computational geometry: regions of point-free overlapping circles
ISCGAV'09 Proceedings of the 9th WSEAS international conference on Signal processing, computational geometry and artificial vision
A simple and efficient kinetic spanner
Computational Geometry: Theory and Applications
Robustness of k-gon Voronoi diagram construction
Information Processing Letters
Kinetic stable Delaunay graphs
Proceedings of the twenty-sixth annual symposium on Computational geometry
Divide-and-conquer for Voronoi diagrams revisited
Computational Geometry: Theory and Applications
Optimal cover of points by disks in a simple polygon
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Querying Approximate Shortest Paths in Anisotropic Regions
SIAM Journal on Computing
Submatrix maximum queries in Monge matrices and Monge partial matrices, and their applications
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
ICCSA'06 Proceedings of the 2006 international conference on Computational Science and Its Applications - Volume Part V
Skew jensen-bregman voronoi diagrams
Transactions on Computational Science XIV
Optimal Cover of Points by Disks in a Simple Polygon
SIAM Journal on Computing
Kinetic pie delaunay graph and its applications
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Convex Distance Functions In 3-Space Are Different
Fundamenta Informaticae
Kinetic data structures for all nearest neighbors and closest pair in the plane
Proceedings of the twenty-ninth annual symposium on Computational geometry
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We present an “expanding waves” view of Voronoi diagrams that allows such diagrams to be defined for very general metrics and for distance measures that do not qualify as metrics. If a pebble is dropped into a still pond, circular waves move out from the point of impact. If n pebbles are dropped simultaneously, the places where wave fronts meet define the Voronoi diagram on the n points of impact.The Voronoi diagram for any normed metric, including the Lp metrics, can be obtained by changing the shape of the wave front from a circle to the shape of the “circle” in that metric. (For example, the “circle” in the L1 metric is diamond shaped.) For any convex wave shape there is a corresponding convex distance function. If the shape is not symmetric about its center (a triangle, for example) then the resulting distance function is not a metric, although it can still be used to define a Voronoi diagram.Like Voronoi diagrams based on the Euclidean metric, the Voronoi diagrams based on other normed metrics can be used to solve various closest-point problems (all-nearest-neighbors, minimum spanning trees, etc.). Some of these problems also make sense under convex distance functions which are not metrics. In particular, the “largest empty circle” problem becomes the “largest empty convex shape” problem, and “motion planning for a disc” becomes “motion planning for a convex shape”. These problems can both be solved quickly given the Voronoi diagram. We present an asymptotically optimal algorithm for computing Voronoi diagrams based on convex distance functions.