There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
An O(n log n) algorithm for the all-nearest-neighbors problem
Discrete & Computational Geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Data structures for mobile data
Journal of 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
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first annual symposium on Computational geometry
Multidimensional divide-and-conquer
Communications of the ACM
Static and kinetic geometric spanners with applications
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Journal of Computer and System Sciences - Special issue on PODS 2000
Kinetic and dynamic data structures for convex hulls and upper envelopes
Computational Geometry: Theory and Applications
Fast algorithms for the all nearest neighbors problem
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Kinetic and dynamic data structures for closest pair and all nearest neighbors
ACM Transactions on Algorithms (TALG)
A simple and efficient kinetic spanner
Computational Geometry: Theory and Applications
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
Journal of the ACM (JACM)
Kinetic stable Delaunay graphs
Proceedings of the twenty-sixth annual symposium on Computational geometry
Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
On topological changes in the delaunay triangulation of moving points
Proceedings of the twenty-eighth annual symposium on Computational geometry
Kinetic Euclidean minimum spanning tree in the plane
Journal of Discrete Algorithms
Kinetic pie delaunay graph and its applications
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Kinetic and stationary point-set embeddability for plane graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Hi-index | 0.00 |
This paper presents a kinetic data structure (KDS) for solutions to the all nearest neighbors problem and the closest pair problem in the plane. For a set P of n moving points where the trajectory of each point is an algebraic function of constant maximum degree s, our kinetic algorithm uses O(n) space and O(n log n) preprocessing time, and processes O(n2β22s+2(n)log n) events with total processing time O(n2β22s+2(n)log2 n), where βs(n) is an extremely slow-growing function. In terms of the KDS performance criteria, our KDS is efficient, responsive (in an amortized sense), and compact. Our deterministic kinetic algorithm for the all nearest neighbors problem improves by an O(log2 n) factor the previous randomized kinetic algorithm by Agarwal, Kaplan, and Sharir. The improvement is obtained by using a new sparse graph representation, the Pie Delaunay graph, to reduce the problem to one-dimensional range searching, as opposed to using two-dimensional range searching as in the previous work.