Abstract
The inability to answer proximity queries efficiently for spaces of dimension d > 2 has led to the study of approximation to proximity problems. Several techniques have been proposed to address different approximate proximity problems. In this paper, we present a new and unified approach to proximity searching, which provides efficient solutions for several problems: spherical range queries, idempotent spherical range queries, spherical emptiness queries, and nearest neighbor queries. In contrast to previous data structures, our approach is simple and easy to analyze, providing a clear picture of how to exploit the particular characteristics of each of these problems. As applications of our approach, we provide simple and practical data structures that match the best previous results up to logarithmic factors, as well as advanced data structures that improve over the best previous results for all aforementioned proximity problems.
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Arya, S., da Fonseca, G.D., Mount, D.M.: Tradeoffs in approximate range searching made simpler. In: Proc. 21st SIBGRAPI, pp. 237–244 (2008)
Arya, S., Malamatos, T.: Linear-size approximate Voronoi diagrams. In: Proc. 13th Ann. ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 147–155 (2002)
Arya, S., Malamatos, T., Mount, D.M.: Space-efficient approximate Voronoi diagrams. In: Proc. 34th Ann. ACM Symp. Theory of Comput. (STOC), pp. 721–730 (2002)
Arya, S., Malamatos, T., Mount, D.M.: Space-time tradeoffs for approximate spherical range counting. In: Proc. 16th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 535–544 (2005)
Arya, S., Malamatos, T., Mount, D.M.: On the importance of idempotence. In: Proc. 38th ACM Symp. on Theory of Comput. (STOC), pp. 564–573 (2006)
Arya, S., Malamatos, T., Mount, D.M.: Space-time tradeoffs for approximate nearest neighbor searching. J. ACM 57, 1–54 (2009)
Arya, S., Mount, D.M.: Approximate range searching. Comput. Geom. 17, 135–163 (2001)
Arya, S., Mount, D.M., Netanyahu, N.S., Silverman, R., Wu, A.Y.: An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM 45(6), 891–923 (1998)
Bespamyatnikh, S.N.: Dynamic algorithms for approximate neighbor searching. In: Proc. 8th Canad. Conf. Comput. Geom. (CCCG), pp. 252–257 (1996)
Chan, T.M.: Approximate nearest neighbor queries revisited. Discrete Comput. Geom. 20, 359–373 (1998)
Chan, T.M.: Closest-point problems simplified on the ram. In: Proc. 13th Annu. ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 472–473 (2002)
Chan, T.M.: Faster core-set constructions and data-stream algorithms in fixed dimensions. Comput. Geom. 35(1), 20–35 (2006)
Clarkson, K.L.: An algorithm for approximate closest-point queries. In: Proc. 10th Annu. ACM Symp. Comput. Geom. (SoCG), pp. 160–164 (1994)
da Fonseca, G.D., Mount, D.M.: Approximate range searching: The absolute model. Comput. Geom. 43(4), 434–444 (2010)
Duncan, C.A., Goodrich, M.T., Kobourov, S.: Balanced aspect ratio trees: Combining the advantages of k-d trees and octrees. J. Algorithms 38, 303–333 (2001)
Eppstein, D., Goodrich, M.T., Sun, J.Z.: The skip quadtree: a simple dynamic data structure for multidimensional data. In: Proc. 21st ACM Symp. Comput. Geom. (SoCG), pp. 296–305 (2005)
Funke, S., Malamatos, T., Ray, R.: Finding planar regions in a terrain: in practice and with a guarantee. Internat. J. Comput. Geom. Appl. 15(4), 379–401 (2005)
Har-Peled, S.: Notes on geometric approximation algorithms, http://valis.cs.uiuc.edu/~sariel/teach/notes/aprx/
Har-Peled, S.: A replacement for Voronoi diagrams of near linear size. In: Proc. 42nd Ann. Symp. Foundations of Computer Science (FOCS), pp. 94–103 (2001)
Har-Peled, S., Mazumdar, S.: Fast algorithms for comput. the smallest k-enclosing circle. Algorithmica 41(3), 147–157 (2005)
Matoušek, J.: Range searching with efficient hierarchical cutting. Discrete Comput. Geom. 10, 157–182 (1993)
Matoušek, J., Schwarzkopf, O.: On ray shooting in convex polytopes. Discrete Comput. Geom. 10, 215–232 (1993)
Sabharwal, Y., Sen, S., Sharma, N.: Nearest neighbors search using point location in balls with applications to approximate Voronoi decompositions. J. Comput. Sys. Sci. 72, 955–977 (2006)
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Arya, S., da Fonseca, G.D., Mount, D.M. (2010). A Unified Approach to Approximate Proximity Searching. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6346. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15775-2_32
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DOI: https://doi.org/10.1007/978-3-642-15775-2_32
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