Eunhui Park
David M. Mount
ABSTRACT
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.
- S. Arya and D. M. Mount. Approximate range searching. Comput. Geom. Theory Appl., 17:135--163, 2001. Google Scholar
Digital Library
- P. B. Callahan and S. R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. Assoc. Comput. Mach., 42:67--90, 1995. Google ScholarDigital Library
- K. L. Clarkson. Fast algorithms for the all nearest neighbors problem. In Proc. 24th Annu. IEEE Sympos. Found. Comput. Sci., pages 226--232, 1983. Google ScholarDigital Library
- J. Fischer and S. Har-Peled. Dynamic well-separated pair decomposition made easy. Proc. 17th Canad. Conf. Comput. Geom., pages 235--238, 2005.Google Scholar
- G. N. Frederickson. A data structure for dynamically maintaining rooted trees. Journal of Algorithms, 24:37--65, 1997. Google ScholarDigital Library
- J. Gao, L. Guibas, and A. Nguyen. Deformable spanners and applications. Comput. Geom. Theory Appl., 35(1):2--19, 2006. Google ScholarDigital Library
- L.-A. Gottlieb and L. Roditty. An optimal dynamic spanner for doubling metric spaces. In Proc. 16th Annu. European Sympos. Algorithms, volume 5193/2008, pages 478--489. Springer, 2008. Google ScholarDigital Library
- L. Greengard and V. Rokhlin. A fast algorithm for particle simulations. J. Comput. Phys., 73:325--348, 1987. Google ScholarDigital Library
- S. Har-Peled. Geometric Approximation Algorithms. American Mathematical Society, Providence, Rhode Island, 2011. Google ScholarDigital Library
- S. Har-Peled and M. Mendel. Fast construction of nets in low dimensional metrics and their applications. SIAM J. Comput., 35:1148--1184, 2006. Google ScholarDigital Library
- D. M. Mount and E. Park. A dynamic data structure for approximate range searching. In Proc. 26th Annu. Sympos. Comput. Geom., pages 247--256, 2010. Google ScholarDigital Library
- G. Narasimhan and M. Smid. Geometric Spanner Networks. Cambridge University Press, 2007. Google ScholarCross Ref
- G. Narasimhan and M. Zachariasen. Geometric minimum spanning trees via well-separated pair decompositions. J. Exper. Algorithmics, 6, 2001. Google ScholarDigital Library
- L. Roditty. Fully dynamic geometric spanners. Algorithmica, 62:1073--1087, 2012. Google ScholarDigital Library
- H. Samet. Foundations of Multidimensional and Metric Data Structures. Morgan-Kaufmann, 2006. Google ScholarDigital Library
- J. Sankaranarayanan and H. Samet. Query processing using distance oracles for spatial networks. IEEE Trans. on Knowledge and Data Engr., 22:1158--1175, 2010. Google ScholarDigital Library
- D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees. J. Comput. Sys. Sci., 26:362--391, 1983. Google ScholarDigital Library
- M. Smid. Closest point problems in computational geometry. In J.-R. Sack and J. Urrutia, editors, Handbook on Computational Geometry, pages 877--935. Elsevier Science, Amsterdam, 2000.Google ScholarCross Ref
- M. Smid. The well-separated pair decomposition and its applications. In T. Gonzalez, editor, Handbook of Approximation Algorithms and Metaheuristics, pages 53-1--53-12. Chapman & Hall/CRC, Boca Raton, 2007.Google ScholarCross Ref
Index Terms
- Output-sensitive well-separated pair decompositions for dynamic point sets
Recommendations
Pruning spanners and constructing well-separated pair decompositions in the presence of memory hierarchies
Given a geometric graph G=(S,E) in R^d with constant dilation t, and a positive constant @e, we show how to construct a (1+@e)-spanner of G with O(|S|) edges using O(sort(|E|)) memory transfers in the cache-oblivious model of computation. The main ...
Geometric Spanners for Weighted Point Sets
Let ( S , d ) be a finite metric space, where each element p S has a non-negative weight w ( p ). We study spanners for the set S with respect to the following weighted distance function: $$\mathbf{d}_{\omega}(p,q)=\left\{\begin{array}{ll}0&\...
Well-Separated Pair Decomposition for the Unit-Disk Graph Metric and Its Applications
We extend the classic notion of well-separated pair decomposition [P. B. Callahan and S. R. Kosaraju, J. ACM, 42 (1975), pp. 67--90] to the unit-disk graph metric: the shortest path distance metric induced by the intersection graph of unit disks. We ...
Comments