A fast algorithm for particle simulations
Journal of Computational Physics
A data structure for dynamic trees
Journal of Computer and System Sciences
A data structure for dynamically maintaining rooted trees
Journal of Algorithms
Computational Geometry: Theory and Applications
Geometric Minimum Spanning Trees via Well-Separated Pair Decompositions
Journal of Experimental Algorithmics (JEA)
Foundations of Multidimensional and Metric Data Structures (The Morgan Kaufmann Series in Computer Graphics and Geometric Modeling)
Fast Construction of Nets in Low-Dimensional Metrics and Their Applications
SIAM Journal on Computing
Geometric Spanner Networks
Fast algorithms for the all nearest neighbors problem
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
An Optimal Dynamic Spanner for Doubling Metric Spaces
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Deformable spanners and applications
Computational Geometry: Theory and Applications
A dynamic data structure for approximate range searching
Proceedings of the twenty-sixth annual symposium on Computational geometry
Query Processing Using Distance Oracles for Spatial Networks
IEEE Transactions on Knowledge and Data Engineering
Geometric Approximation Algorithms
Geometric Approximation Algorithms
Fully Dynamic Geometric Spanners
Algorithmica
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The well-separated pair decomposition (WSPD) is a fundamental structure in computational geometry. Given a set P of n points in d-dimensional space and a positive separation parameter s, an s-WSPD is a concise representation of all the O(n2) pairs of P requiring only O(sdn) storage. The WSPD has numerous applications in spatial data processing, such as computing spanner graphs, minimum spanning trees, shortest-path oracles, and statistics on interpoint distances. We consider the problem of maintaining a WSPD when points are inserted to or deleted from P. Worst-case arguments suggest that the addition or deletion of a single point could result in the generation (or removal) up to Ω(sd) pairs, which can be unacceptably high in many applications. Fortunately, the actual number of well separated pairs can be significantly smaller in practice, particularly when the points are well clustered. This suggests the importance of being able to respond to insertions and deletions in a manner that is output sensitive, that is, whose running time depends on the actual number of pairs that have been added or removed. We present the first output-sensitive algorithms for maintaining a WSPD of a point set under insertion and deletion. We show that our algorithms are nearly optimal, in the sense that these operations can be performed in time that is roughly equal to the number of changes to the WSPD.