scholarly journals Voronoi Diagrams of Moving Points

1998 ◽  
Vol 08 (03) ◽  
pp. 365-379 ◽  
Author(s):  
Gerhard Albers ◽  
Leonidas J. Guibas ◽  
Joseph S. B. Mitchell ◽  
Thomas Roos
Keyword(s):  
Euclidean Space ◽  
Voronoi Diagram ◽  
Upper Bound ◽  
Voronoi Diagrams ◽  
Maximum Length ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Moving Points ◽  
Over Time

Consider a set of n points in d-dimensional Euclidean space, d ≥ 2, each of which is continuously moving along a given individual trajectory. As the points move, their Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Voronoi diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, at a cost of O( log n) per event, while showing that the number of topological events has an upper bound of O(ndλs(n)), where λs(n) is the (nearly linear) maximum length of a (n,s)-Davenport-Schinzel sequence, and s is a constant depending on the motions of the point sites. In addition, we show that if only k points are moving (while leaving the other n - k points fixed), there is an upper bound of O(knd-1λs(n)+(n-k)dλ s(k)) on the number of topological events.

VORONOI DIAGRAMS FOR A TRANSPORTATION NETWORK ON THE EUCLIDEAN PLANE

2006 ◽  
Vol 16 (02n03) ◽  
pp. 117-144 ◽  
Author(s):  
SANG WON BAE ◽  
KYUNG-YONG CHWA
Keyword(s):  
Voronoi Diagram ◽  
Euclidean Plane ◽  
Transportation Network ◽  
Voronoi Diagrams ◽  
The Other ◽  
Constant Number ◽  
Time Path ◽  
The Given

This paper investigates geometric and algorithmic properties of the Voronoi diagram for a transportation network on the Euclidean plane. In the presence of a transportation network, the distance is measured as the length of the shortest (time) path. In doing so, we introduce a needle, a generalized Voronoi site. We present an O(nm2+ m3+ nm log n) algorithm to compute the Voronoi diagram for a transportation network on the Euclidean plane, where n is the number of given sites and m is the complexity of the given transportation network. Moreover, in the case that the roads in a transportation network have only a constant number of directions and speeds, we propose two algorithms; one needs O(nm + m2+ n log n) time with O(m(n + m)) space and the other O(nm log n + m2log m) time with O(n + m) space.

A ROBUST TOPOLOGY-ORIENTED INCREMENTAL ALGORITHM FOR VORONOI DIAGRAMS

1994 ◽  
Vol 04 (02) ◽  
pp. 179-228 ◽  
Author(s):  
KOKICHI SUGIHARA ◽  
MASAO IRI
Keyword(s):  
Numerical Computation ◽  
Voronoi Diagram ◽  
Voronoi Diagrams ◽  
The Other ◽  
Geometric Algorithms ◽  
Incremental Algorithm ◽  
Single Precision ◽  
Careful Choice ◽  
Numerical Errors ◽  
Incremental Method

The paper presents a robust algorithm for constructing Voronoi diagrams in the plane. The algorithm is based on an incremental method, but is quite new in that it is robust against numerical errors. Conventionally, geometric algorithms have been designed on the assumption that numerical errors do not take place, and hence they are not necessarily valid for real computers where numerical errors are inevitable. The algorithm to be proposed in this paper, on the other hand, is designed with the recognition that numerical errors are inevitable in real computation; i.e., in the proposed algorithm higher priority is placed on topological structures than on numerical values. As a result, the algorithm is "completely" robust in the sense that it always gives some output however poor the precision of numerical computation may be. In general, the output cannot be more than an approximation to the true Voronoi diagram which we should have got by infinite-precision computation. However, the algorithm is asymptotically correct in the sense that the output converges to the true diagram as the precision becomes higher. Moreover, careful choice of the way of numerical computation makes the algorithm stable enough; indeed the present version of the algorithm can construct in single-precision arithmetic a correct Voronoi diagram for one million generators randomly placed in the unit square in the plane.

scholarly journals The number of partitions of a set of N points in k dimensions induced by hyperplanes

1967 ◽  
Vol 15 (4) ◽  
pp. 285-289 ◽  
Author(s):  
E. F. Harding
Keyword(s):  
Euclidean Space ◽  
Half Space ◽  
General Position ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Unordered Pair

1. An arbitrary (k– 1)-dimensional hyperplane disconnects K-dimensional Euclidean space Ek into two disjoint half-spaces. If a set of N points in general position in Ek is given [nok +1 in a (k–1)-plane, no k in a (k–2)-plane, and so on], then the set is partitione into two subsets by the hyperplane, a point belonging to one or the other subset according to which half-space it belongs to; for this purpose the half-spaces are considered as an unordered pair.

scholarly journals APPLICATION OF THE THEORY OF OPTIMAL SET PARTITIONING BEFORE BUILDING MULTIPLICATIVELY WEIGHTED VORONOI DIAGRAM WITH FUZZY PARAMETERS

2020 ◽  
Vol 6 (2(71)) ◽  
pp. 30-35
Author(s):  
O.M. Kiseliova ◽  
O.M. Prytomanova ◽  
V.H. Padalko
Keyword(s):  
Euclidean Space ◽  
Finite Number ◽  
Nonsmooth Optimization ◽  
Voronoi Diagram ◽  
Optimization Problems ◽  
Optimal Location ◽  
Set Partitioning ◽  
Dimensional Euclidean Space ◽  
Optimal Set ◽  
Fuzzy Parameters

An algorithm for constructing a multiplicatively weighted Voronoi diagram involving fuzzy parameters with the optimal location of a finite number of generator points in a limited set of n-dimensional Euclidean space 𝐸𝑛 has been suggested in the paper. The algorithm has been developed based on the synthesis of methods of solving the problems of optimal set partitioning theory involving neurofuzzy technologies modifications of N.Z. Shor 𝑟 -algorithm for solving nonsmooth optimization problems.

scholarly journals Sums of distances between points of a sphere

1982 ◽  
Vol 5 (4) ◽  
pp. 707-714 ◽  
Author(s):  
Glyn Harman
Keyword(s):  
Euclidean Space ◽  
Unit Sphere ◽  
Upper Bound ◽  
Dimensional Euclidean Space ◽  
Spherical Caps

GivenNpoints on a unit sphere ink+1dimensional Euclidean space, we obtain an upper bound for the sum of all the distances they determine which improves upon earlier work by K. B. Stolarsky whenkis even. We use his method, but derive a variant of W. M. Schmidt's results for the discrepancy of spherical caps which is more suited to the present application.

scholarly journals NP-hardness of quadratic Euclidean 1-Mean and 1-Median 2-Clustering problem with the constraints on the cluster sizes

2019 ◽  
Vol 489 (4) ◽  
pp. 339-343
Author(s):  
A. V. Kel’manov ◽  
A. V. Pyatkin ◽  
V. I. Khandeev
Keyword(s):  
Euclidean Space ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Clustering Problem ◽  
Geometric Center ◽  
Finite Set ◽  
The Given ◽  
Np Hardness

In the paper, we consider a problem of clustering a finite set of N points in d-dimensional Euclidean space into two clusters minimizing the sum over all clusters of the intracluster sums of the distances between clusters elements and their centers. The center of one cluster is defined as centroid (geometric center). The center of the other one is a sought point in the input set. We analyze the variant of the problem with the given clusters sizes. We have proved the strong NP-hardness of this problem.

scholarly journals Construction of a generalized Voronoi diagram with optimal placement of generator points based on the theory of optimal set partitioning

2020 ◽  
Vol 53 (1) ◽  
pp. 109-112
Author(s):  
E.M. Kiseleva ◽  
L.L. Hart ◽  
O.M. Prytomanova ◽  
S.V. Zhuravel
Keyword(s):  
Euclidean Space ◽  
Voronoi Diagram ◽  
Quality Criterion ◽  
Set Partitioning ◽  
Optimal Placement ◽  
Dimensional Euclidean Space ◽  
Bounded Set ◽  
Optimal Set ◽  
Continuous Problem ◽  
Generalized Voronoi Diagram

The problem of construction of a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of \textit{n}-dimensional Euclidean space is considered. A method is proposed for solving such a problem based on the formulation of the corresponding continuous problem of optimal partitioning of a set in \textit{n}-dimensional Euclidean space with a partition quality criterion that provides the corresponding form of the Voronoi diagram. Further, to solve such a problem, the developed mathematical and algorithmic apparatus is used, the part of which is Shor's \textit{r}-algorithm.

scholarly journals Construction of a multiplicatively weighted diagram of a crow with fuzzy parameters

2019 ◽  
Author(s):  
O. M. Kiselova ◽  
O. M. Prytomanova ◽  
S. V. Dzyuba ◽  
V. G. Padalko
Keyword(s):  
Euclidean Space ◽  
Finite Number ◽  
Voronoi Diagram ◽  
Optimization Problems ◽  
Optimal Location ◽  
Set Partitioning ◽  
Dimensional Euclidean Space ◽  
Bounded Set ◽  
Optimal Set ◽  
Fuzzy Parameters

An algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is proposed in the paper. The algorithm is based on the formulation of a continuous set partitioning problem from En into non-intersecting subsets with a partitioning quality criterion providing the corresponding form of Voronoi diagram. Algorithms for constructing the classical Voronoi diagram and its various generalizations, which are based on the usage of the methods of the optimal set partitioning theory, have several advantages over the other used methods: they are out of thedependence of En space dimensions, which containing a partitioned bounded set into subsets, independent of the geometry of the partitioned sets, the algorithm’s complexity is not growing under increasing of number of generator points, it can be used for constructing the Voronoi diagram with optimal location of the points and others. The ability of easily construction not only already known Voronoi diagrams but also the new ones is the result of this general-purpose approach. The proposed in the paper algorithm for constructing a multiplicatively weighted Voronoi diagram in the presence of fuzzy parameters with optimal location of a finite number of generator points in a bounded set of n-dimensional Euclidean space En is developed using a synthesis of methods for solving optimal set partitioning problems, neurofuzzy technologies and modifications of the Shor’s r-algorithm for solving non-smooth optimization problems.

How to see a cube moving into its mirror image

2012 ◽  
Vol 96 (537) ◽  
pp. 471-479
Author(s):  
Ester Dalvit ◽  
Domenico Luminati
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Rigid Motion ◽  
Mirror Image ◽  
The Other ◽  
Dimensional Euclidean Space ◽  
Other Hand ◽  
Rigid Motions

In n-dimensional Euclidean space no reflection with respect to a hyperplane can be realised by a rigid motion. But this is possible if we allow rigid motions in (n + 1)-dimensional space. These notes show a way to visualise a rigid motion of a cube in 4-dimensional space that flips the cube ‘as the page of a book’.The two terms rigid motion and isometry are sometimes used as synonyms. Yet they do refer to different concepts. The first one has a purely kinematic connotation: the swing of a door or the movement of a piece of furniture pushed over the floor are described by rigid motions. On the other hand to ensure that two figures are isometric it is enough that there exists a correspondence between their points that maintains the relative distances.

VORONOI DIAGRAMS IN A RIVER

1992 ◽  
Vol 02 (01) ◽  
pp. 29-48 ◽  
Author(s):  
KOKICHI SUGIHARA
Keyword(s):  
Voronoi Diagram ◽  
Topological Structure ◽  
Voronoi Diagrams ◽  
Uniform Flow ◽  
The Other ◽  
Plane Sweep ◽  
Water Flows ◽  
Sweep Algorithm ◽  
A Site ◽  
Generalized Voronoi Diagram

A new generalized Voronoi diagram is defined on the surface of a river with uniform flow; a point belongs to the territory of a site if and only if a boat starting from the site can reach the point faster than a boat starting from any other site. If the river runs slower than the boat, the Voronoi diagram has the same topological structure as the ordinary Voronoi diagram, and hence can be constructed from the ordinary Voronoi diagram by a certain transformation. If the river runs faster than the boat, on the other hand, the topological structure of the diagram becomes different from the ordinary one, but it can be constructed by the plane sweep technique. Moreover, Fortune’s plane sweep algorithm for constructing the ordinary Voronoi diagram can be interpreted as the algorithm for constructing the Voronoi diagram in a river in which the water flows at the same speed as the boat.

Sign in / Sign up

Export Citation Format

Share Document