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Voronoi diagrams of moving points in the plane and of lines in space: tight bounds for simple configurations

Amit Weisman, L. Paul Chew, Klara Kedem
2004 Information Processing Letters  
The Voronoi diagram of moving points in the plane Let P be a set of n points in the plane. Each point p i has a constant velocity v i assigned to it. At time t = 0 all points start moving.  ...  Future work Sharp complexity bounds for the Voronoi diagram of moving points in the plane and for the Voronoi diagram of lines in space are clearly hard to achieve.  ... 
doi:10.1016/j.ipl.2004.08.004 fatcat:56bxdlbl3rdmxaolfmiugvlkru

On Dynamic Generalized Voronoi Diagrams in the Euclidean Metric [chapter]

M. L. Gavrilova, J. Rokne
2001 Lecture Notes in Computer Science  
The problem of dynamic maintenance of the Voronoi diagram for a set of spheres moving independently in d-dimensional space is addressed in this paper.  ...  The maintenance of the generalized Voronoi diagram of spheres, moving alone the given trajectories, requires the calculation of topological events, that occur when d + 2 spheres become tangent to a common  ...  For the dynamic Voronoi diagram of a set of moving points in the plane only a few algorithms are known.  ... 
doi:10.1007/3-540-45545-0_78 fatcat:fastbknewfempg55mgewsvvthy

Angular Voronoi Diagram with Applications

Tetsuo Asano
2006 2006 3rd International Symposium on Voronoi Diagrams in Science and Engineering  
Given a set of line segments in the plane, we define an angular Voronoi diagram as follows: a point belongs to a Voronoi region of a line segment if the visual angle of the line segment from the point  ...  The Voronoi diagram is interesting in itself and different from an ordinary Voronoi diagram for a point set.  ...  Work of T.A. was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (B). We also thank K. Ushikubo for his survey and programming.  ... 
doi:10.1109/isvd.2006.9 dblp:conf/isvd/AsanoTKT06 fatcat:xehbl42zwrclbhtx47jhofdnqe

Voronoi diagrams for polygon-offset distance functions [chapter]

Gill Barequet, Matthew T. Dickerson, Michael T. Goodrich
1997 Lecture Notes in Computer Science  
In this paper we develop the concept of a polygon-o set distance function and show how to compute the respective nearest-and furthest-site Voronoi diagrams of point sites in the plane.  ...  We provide optimal deterministic O(n(log n + log m) + m)-time algorithms, where n is the number of points and m is the complexity of the underlying polygon, for computing compact representations of both  ...  The authors wish to thank J. Snoeyink and J. Bose for helpful discussions on Voronoi diagrams. We also thank V. Mirelli for introducing us the concept of polygon o sets and their applications.  ... 
doi:10.1007/3-540-63307-3_60 fatcat:y5loxmfovvgxfbrlxe5yhbs2jy

Queries on Voronoi diagrams of moving points

O. Devillers, M. Golin, K. Kedem, S. Schirra
1996 Computational geometry  
The first one asks "who is the nearest postman at time tq to a dog located at point Sq. In the second type a query dog is located at point sq at time tq, its speed is vq > Ivil (for all i = 1,...  ...  Suppose we are given n moving postmen described by their motion equations pi(t) = s~ + vit, i = 1,..., n, where s~ E R 2 is the position of the ith postman at time t = 0, and vi E R 2 is his velocity.  ...  The authors would like to thank Mike Goodrich for directing our attention to the applicability of [7] to solving moving-Voronoi queries.  ... 
doi:10.1016/0925-7721(95)00053-4 fatcat:hadspahg3fcpzont63wzdzg4qa

Voronoi diagrams for direction-sensitive distances

Oswin Aichholzer, Danny Z. Chen, D. T. Lee, Asish Mukhopadhyay, Evanthia Papadopoulou, Franz Aurenhammer
1997 Proceedings of the thirteenth annual symposium on Computational geometry - SCG '97  
l:ldL> or dislrihllt~>d till prL)til Of LXIIII 111 C1L,i:ll fi[i\l:lll!:lgC. !ht. L,~)p\,-rigJlt notic.c.  ...  Then SV(S) is the Euclidean closest-point Voranoi diagram of C(S). Various properties are known for the Voronoi diagram of circles in the plane; see [3, 6, 9] .  ...  In fact, for k <1, SV(S) has a combinatorial structure identical to the Euclidean Voronoi diagram of some set S' of point sites.  ... 
doi:10.1145/262839.263045 dblp:conf/compgeom/AichholzerACLMP97 fatcat:oyamz7nk2nhglgrsanvbzg2x6q

Near-quadratic bounds for the L1 Voronoi diagram of moving points

L.Paul Chew
1997 Computational geometry  
Given a set of n moving points in the plane, how many topological changes occur in the Voronoi diagram of the points?  ...  We show this belief to be true for the case of Voronoi diagrams based on the L1 (or L~) metric; the number of changes is shown to be O(nZc~(n)) where c~(n) grows so slowly it is effectively a small constant  ...  The problem of determining the number of changes in a Voronoi diagram of moving points in the plane is closely related the problem of determining the size of a Voronoi diagram of lines in 3-space.  ... 
doi:10.1016/0925-7721(95)00044-5 fatcat:o3stba2c2zewbgqtj2misg7cyy

Crystal Voronoi Diagram and Its Applications to Collision-Free Paths [chapter]

Kei Kobayashi, Kokichi Sugihara
2001 Lecture Notes in Computer Science  
This paper studies the multiplicatively weighted crystal-growth Voronoi diagram, which describes the partition of the plane into crystals with different growth speeds.  ...  This type of the Voronoi diagram is defined, and its basic properties are investigated. The analytic equation describing the boundary curve is given for a simple case.  ...  This work is supported by the Grant-in-Aid for Scientific Research of the Japanese Ministry of Education, Science, Sports and Culture.  ... 
doi:10.1007/3-540-45545-0_85 fatcat:ege2mban3vfbfakdjs4xdxrnc4

A parallel algorithm for constructing Voronoi diagrams based on point-set adaptive grouping

Jiechen Wang, Can Cui, Yikang Rui, Liang Cheng, Yingxia Pu, Wenzhou Wu, Zhenyu Yuan
2013 Concurrency and Computation  
The binary tree splitting method is used to adaptively group the point set in the plane and construct sub-Voronoi diagrams for each group.  ...  in each block, and the number of boundary points.  ...  As the number of points increases, the algorithm incurs a non-linear increase in time consumption. There is a more effective strategy of constructing a Voronoi diagram for a point set in the plane.  ... 
doi:10.1002/cpe.3005 fatcat:if5hgtfwcvhsnczjkv75t5clga

Compact Voronoi Diagrams for Moving Convex Polygons [chapter]

Leonidas J. Guibas, Jack Snoeyink, Li Zhang
2000 Lecture Notes in Computer Science  
We describe a kinetic data structure for maintaining a compact Voronoi-like diagram of convex polygons moving around in the plane.  ...  We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96] .  ...  In this paper we study the kinetic maintenance of the compact Voronoi diagram for disjoint moving convex polygons in the plane.  ... 
doi:10.1007/3-540-44985-x_30 fatcat:bb63fodayja6xipo7jclv7ylhm

Rapidly converging solution for p-centers in nonconvex regions

Monika MANGLA, Deepak GARG
2017 Turkish Journal of Electrical Engineering and Computer Sciences  
This paper aims to locate p resources in a nonconvex demand plane having n demand points.  ...  The objective of the location problem is to find the location for these p resources so that the distance from each of n demand points to its nearest resource is minimized, thus simulating a p -center problem  ...  The Voronoi diagram (VD) of P is the subdivision of the plane into n cells, one for each site.  ... 
doi:10.3906/elk-1601-186 fatcat:7jig3uwqzrcifeqfjjcrd2a4u4

Dynamic weighted Voronoi diagrams and weighted minimax matching of two corresponding point sets

Keiko Imai, Hiroshi Imai
1998 Optimization Methods and Software  
A weighted geometric fitting problem between two corresponding sets of points is to minimize the maximum weighted distance between two corresponding pairs of points by translating and rotating one set  ...  In this paper, we show a new reduction of the problem to the two-dimensional Davenport-Schinzel sequences, and provide a much simpler proof for the almost cubic bound.  ...  This work was supported in part by the Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan.  ... 
doi:10.1080/10556789808805714 fatcat:lbnksjjkdjhlrcwjh7opqenf5u

Dilation, smoothed distance, and minimization diagrams of convex functions [article]

Matthew Dickerson, David Eppstein, Kevin A. Wortman
2010 arXiv   pre-print
For sufficiently closely spaced points in the plane, the Voronoi diagram of smoothed distance has linear complexity and can be computed efficiently.  ...  in the diagram from each of the others forms an arrangement of pseudolines in the plane.  ...  ACKNOWLEDGEMENTS This work was supported in part by NSF grant 0830403 and by the Office of Naval Research under grant N00014-08-1-1015.  ... 
arXiv:0812.0607v2 fatcat:ifbgwo3ggvfe3ncramvb2zashy

Simplified Voronoi Diagrams for Motion Planning of Quadratically-Solvable Gough–Stewart Platforms [chapter]

Rubén Vaca, Joan Aranda, Federico Thomas
2012 Latest Advances in Robot Kinematics  
of 4 × 4 determinants involving the homogeneous coordinates of sets of four points.  ...  This Boolean combination induces a measure of distance to obstacles in Configuration Space from which a simplified Voronoi diagram can be derived.  ...  -Point-Plane determinants: They involve three attachments of the moving platform and one of the fixed base.  ... 
doi:10.1007/978-94-007-4620-6_20 dblp:conf/ark/VacaAT12 fatcat:chlv5du3w5fctanej7zxlxs3ne

Ready, Set, Go! The Voronoi diagram of moving points that start from a line

Vladlen Koltun
2004 Information Processing Letters  
It is an outstanding open problem of computational geometry to prove a nearquadratic upper bound on the number of combinatorial changes in the Voronoi diagram of points moving at a common constant speed  ...  along linear trajectories in the plane.  ...  Introduction Given a set P of n points in the plane, moving at a common constant speed along linear trajectories, we wish to estimate the number of changes that occur through time in the combinatorial  ... 
doi:10.1016/j.ipl.2003.11.012 fatcat:tp7hxrexdfbbvc4g74wcxtd2su
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