Euclidean minimum spanning trees and bichromatic closest pairs
Discrete & Computational Geometry
Faster algorithms for some geometric graph problems in higher dimensions
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Topology control and routing in ad hoc networks: a survey
ACM SIGACT News
Minimum spanning trees in d dimensions
Nordic Journal of Computing
Planar Spanners and Approximate Shortest Path Queries among Obstacles in the Plane
ESA '96 Proceedings of the Fourth Annual European Symposium on Algorithms
Testing bipartiteness of geometric intersection graphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
Well-Separated Pair Decomposition for the Unit-Disk Graph Metric and Its Applications
SIAM Journal on Computing
Approximate distance queries in disk graphs
WAOA'06 Proceedings of the 4th international conference on Approximation and Online Algorithms
Compact and Low Delay Routing Labeling Scheme for Unit Disk Graphs
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
Localized spanner construction for ad hoc networks with variable transmission range
ACM Transactions on Sensor Networks (TOSN)
New constructions of SSPDs and their applications
Proceedings of the twenty-sixth annual symposium on Computational geometry
Planar hop spanners for unit disk graphs
ALGOSENSORS'10 Proceedings of the 6th international conference on Algorithms for sensor systems, wireless adhoc networks, and autonomous mobile entities
New constructions of SSPDs and their applications
Computational Geometry: Theory and Applications
Compact and low delay routing labeling scheme for Unit Disk Graphs
Computational Geometry: Theory and Applications
| Hi-index | 0.00 |
A disk graph is an intersection graph of a set of disks with arbitrary radii in the plane. In this paper, we consider the problem of efficient construction of sparse spanners of disk (ball) graphs with support for fast distance queries. These problems are motivated by issues arising from topology control and routing in wireless networks. We present the first algorithm for constructing spanners of ball graphs. For a ball graph in Rk, we construct a (1+ε)-spanner with O(nε-k+1) edges in O(n2l+δε-klogl S) expected time, using an efficient partitioning of the space into hypercubes and solving intersection problems. Here l = 1-1/([k/2]+2), δ is any positive constant, and S is the ratio between the largest and smallest radius. For the special case where all the balls have the same radius, we show that the spanner construction has complexity almost equivalent to the construction of a Euclidean minimum spanning tree. Previously known constructions of spanners of unit ball graphs have time complexity much closer to n2. Additionally, these spanners have a small vertex separator (hereditary), which is then exploited for fast answering of distance queries. The results on geometric graph separators might be of independent interest.