research-article

Approximating Minimization Diagrams and Generalized Proximity Search

Authors:

Sariel Har-Peled,

Nirman KumarAuthors Info & Claims

SIAM Journal on Computing, Volume 44, Issue 4

Pages 944 - 974

https://doi.org/10.1137/140959067

Published: 01 January 2015 Publication History

Abstract

We investigate the classes of functions whose minimization diagrams can be approximated efficiently in $\mathbb{R}^d$. We present a general framework and a data-structure that can be used to approximate the minimization diagram of such functions. The resulting data-structure has near linear size and can answer queries in logarithmic time. Applications include approximating the Voronoi diagram of multiplicatively weighted points, but the new technique also works for more general distance functions. For example, we get such data-structures for metrics induced by convex bodies, and the nearest furthest-neighbor distance to a set of point sets. Interestingly, our framework also works for distance functions that do not obey the triangle inequality. For many of these functions no near linear size approximation was known before.

References

[1]

P. K. Agarwal, B. Aronov, S. Har-Peled, J. M. Phillips, K. Yi, and W. Zhang, Nearest neighbor searching under uncertainty II, in Proceedings of the 32nd ACM Symposium on Principles of Database Systems, 2013, pp. 115--126.

[2]

P. K. Agarwal, A. Efrat, S. Sankararaman, and W. Zhang, Nearest-neighbor searching under uncertainty, in Proceedings of the 31st ACM Symposium on Principles of Database Systems, 2012, pp. 225--236.

[3]

P. K. Agarwal and J. Matoušek, Ray shooting and parametric search, SIAM J. Comput., 22 (1993), pp. 794--806.

[4]

C. Aggarwal, Managing and Mining Uncertain Data, Springer-Verlag, New York, 2009.

[5]

A. Andoni and P. Indyk, Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions, Commun. ACM, 51 (2008), pp. 117--122.

[6]

S. Arya and T. Malamatos, Linear-size approximate Voronoi diagrams, in Proceedings of the 13th ACM-SIAM Symposium on Discrete Algorithms, 2002, pp. 147--155.

[7]

S. Arya, T. Malamatos, and D. M. Mount, Space-time tradeoffs for approximate nearest neighbor searching, J. Assoc. Comput. Mach., 57 (2009), pp. 1--54.

[8]

S. Arya, D. M. Mount, N. S. Netanyahu, R. Silverman, and A. Y. Wu, An optimal algorithm for approximate nearest neighbor searching in fixed dimensions, J. Assoc. Comput. Mach., 45 (1998), pp. 891--923.

[9]

S. Basu, R. Pollack, and M. F. Roy, Algorithms in Real Algebraic Geometry, Algorithms Comput. Math. 10, Springer, Berlin, 2006.

[10]

M. de Berg, O. Cheong, M. van Kreveld, and M. H. Overmars, Computational Geometry: Algorithms and Applications, 3rd ed., Springer-Verlag, Berlin, 2008.

[11]

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 (1995), pp. 67--90.

[12]

T. M. Chan, Optimal partition trees, in Proceedings of the 26th Annual Symposium on Computational Geometry, 2010, pp. 1--10.

[13]

K. L. Clarkson, A randomized algorithm for closest-point queries, SIAM J. Comput., 17 (1988), pp. 830--847.

[14]

K. L. Clarkson, Nearest-neighbor searching and metric space dimensions, in Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, G. Shakhnarovich, T. Darrell, and P. Indyk, eds., MIT Press, Cambridge, MA, 2006, pp. 15--59.

[15]

N. N. Dalvi, C. Ré, and D. Suciu, Probabilistic databases: Diamonds in the dirt, Commun. ACM, 52 (2009), pp. 86--94.

[16]

J. Erickson, New lower bounds for Hopcroft's problem, Discrete Comput. Geom., 16 (1996), pp. 389--418.

[17]

S. Har-Peled, Constructing approximate shortest path maps in three dimensions, SIAM J. Comput., 28 (1999), pp. 1182--1197.

[18]

S. Har-Peled, A replacement for Voronoi diagrams of near linear size, in Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, 2001, pp. 94--103.

[19]

S. Har-Peled, Geometric Approximation Algorithms, Math. Surveys Monogr. 173, AMS, Providence, RI, 2011.

[20]

S. Har-Peled, P. Indyk, and R. Motwani, Approximate nearest neighbors: Towards removing the curse of dimensionality, Theory Comput., 8 (2012), pp. 321--350.

[21]

S. Har-Peled and N. Kumar, Approximating minimization diagrams and generalized proximity search, in Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 2013, pp. 717--726.

[22]

P. Indyk and R. Motwani, Approximate nearest neighbors: Towards removing the curse of dimensionality, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, 1998, pp. 604--613.

[23]

J. Matoušek, Efficient partition trees, Discrete Comput. Geom., 8 (1992), pp. 315--334.

[24]

S. Meiser, Point location in arrangements of hyperplanes, Inform. Comput., 106 (1993), pp. 286--303.

[25]

R. Motwani, A. Naor, and R. Panigrahi, Lower bounds on locality sensitive hashing, in Proceedings of the 22nd Annual Symposium on Computational Geometry, 2006, pp. 154--157.

[26]

M. Sharir and P. K. Agarwal, Davenport-Schinzel Sequences and Their Geometric Applications, Cambridge University Press, New York, 1995.

Cited By

Mount D(2019)New Directions in Approximate Nearest-Neighbor SearchingAlgorithms and Discrete Applied Mathematics10.1007/978-3-030-11509-8_1(1-15)Online publication date: 14-Feb-2019
https://dl.acm.org/doi/10.1007/978-3-030-11509-8_1
Aronov BKatz M(2018)Batched Point Location in SINR Diagrams via Algebraic ToolsACM Transactions on Algorithms10.1145/320967814:4(1-29)Online publication date: 9-Aug-2018
https://dl.acm.org/doi/10.1145/3209678

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cover image SIAM Journal on Computing

SIAM Journal on Computing Volume 44, Issue 4

DOI:10.1137/smjcat.44.4

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© 2015, Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2015

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Cited By

Mount D(2019)New Directions in Approximate Nearest-Neighbor SearchingAlgorithms and Discrete Applied Mathematics10.1007/978-3-030-11509-8_1(1-15)Online publication date: 14-Feb-2019
https://dl.acm.org/doi/10.1007/978-3-030-11509-8_1
Aronov BKatz M(2018)Batched Point Location in SINR Diagrams via Algebraic ToolsACM Transactions on Algorithms10.1145/320967814:4(1-29)Online publication date: 9-Aug-2018
https://dl.acm.org/doi/10.1145/3209678

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