Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Faster algorithms for some geometric graph problems in higher dimensions
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
Dynamic algorithms for geometric spanners of small diameter: randomized solutions
Computational Geometry: Theory and Applications
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Journal of Algorithms
A Divide-and-Conquer Algorithm for Min-Cost Perfect Matching in the Plane
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Computational Geometry: Theory and Applications - Special issue on the 14th Canadian conference on computational geometry CCCG02
Range Aggregate Processing in Spatial Databases
IEEE Transactions on Knowledge and Data Engineering
Geometric Spanner Networks
European Journal of Combinatorics
Region-fault tolerant geometric spanners
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Range-aggregate query problems involving geometric aggregation operations
Nordic Journal of Computing
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Delaunay triangulations of imprecise pointsin linear time after preprocessing
Proceedings of the twenty-fourth annual symposium on Computational geometry
Randomized and deterministic algorithms for geometric spanners of small diameter
SFCS '94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science
Approximating largest convex hulls for imprecise points
Journal of Discrete Algorithms
Spanners of complete k-partite geometric graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
New constructions of SSPDs and their applications
Computational Geometry: Theory and Applications
Range-aggregate queries for geometric extent problems
CATS '13 Proceedings of the Nineteenth Computing: The Australasian Theory Symposium - Volume 141
Hi-index | 0.00 |
A Semi-Separated Pair Decomposition (SSPD), with parameter s 1, of a set $S\subset {\mathbb R}^d$ is a set {(A i ,B i )} of pairs of subsets of S such that for each i , there are balls $D_{A_i}$ and $D_{B_i}$ containing A i and B i respectively such that $d(D_{A_i},D_{B_i}) \geq s \cdot$min ( radius$(D_{A_i}$) , radius$(D_{B_i})$ ), and for any two points p , q *** S there is a unique index i such that p *** A i and q *** B i or vice-versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set $S\subset {\mathbb R}^d$ of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with ${\mathcal O}(n\log n/(t-1)^d)$ edges which can be computed in ${\mathcal O}(n\log n/(t-1)^d)$ time. If all balls have the same radius, the number of edges reduces to ${\mathcal O}(n/(t-1)^d)$. Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in ${\mathcal O}(n^2\log^2 n)$ time using ${\mathcal O}(n\log n)$ space and answers a query in ${\mathcal O}(n^{1/2+\epsilon})$ time, for any *** 0. By reducing the preprocessing time to ${\mathcal O}(n^{1+\epsilon})$ and using ${\mathcal O}(n\log^2 n)$ space, the query can be answered in ${\mathcal O}(n^{3/4+\epsilon})$ time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k -partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.