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# Dominance Product and High-Dimensional Closest Pair under L_infty

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LIPIcs.ISAAC.2017.39.pdf
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## Cite As

Omer Gold and Micha Sharir. Dominance Product and High-Dimensional Closest Pair under L_infty. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 39:1-39:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ISAAC.2017.39

## Abstract

Given a set $S$ of $n$ points in \mathbb{R}^d, the Closest Pair problem is to find a pair of distinct points in S at minimum distance. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L_\infty Closest Pair problem, where d=n^r for some r > 0, and the underlying metric is L_\infty. We improve and simplify previous results for L_\infty Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in O(DP(n,d)\log n) time, and by a randomized algorithm that runs in O(DP(n,d)) expected time, where DP(n,d) is the time bound for computing the dominance product for n points in \mathbb{R}^d. That is a matrix D, such that D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|; this is the number of coordinates at which p_j dominates p_i. For integer coordinates from some interval [-M, M], we obtain an algorithm that runs in \tilde{O}\left(\min\{Mn^{\omega(1,r,1)},\, DP(n,d)\}\right) time, where \omega(1,r,1) is the exponent of multiplying an n \times n^r matrix by an n^r \times n matrix. We also give slightly better bounds for DP(n,d), by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).
##### Keywords
• Closest Pair
• Dominance Product
• L_infty Proximity
• Rectangular Matrix Multiplication

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## References

1. Nir Ailon and Bernard Chazelle. The fast Johnson-Lindenstrauss transform and approximate nearest neighbors. SIAM J. Comput., 39(1):302-322, 2009.
2. Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM, 51(1):117-122, 2008.
3. Michael Ben-Or. Lower bounds for algebraic computation trees. In Proc. of the 15th Annu. ACM Sympos. on Theory of Computing (STOC), pages 80-86, 1983.
4. Jon Louis Bentley. Multidimensional divide-and-conquer. Commun. ACM, 23(4):214-229, 1980.
5. Jon Louis Bentley and Michael Ian Shamos. Divide-and-conquer in multidimensional space. In Proc. of the 8th Annu. ACM Sympos. on Theory of Computing (STOC), pages 220-230, 1976.
6. Timothy M. Chan. Geometric applications of a randomized optimization technique. Discrete & Computational Geometry, 22(4):547-567, 1999.
7. Timothy M. Chan and Ryan Williams. Deterministic APSP, orthogonal vectors, and more: Quickly derandomizing Razborov-Smolensky. In Proc. of the 27th Annu. ACM-SIAM Sympos. on Discrete Algorithms (SODA), pages 1246-1255, 2016.
8. Ran Duan and Seth Pettie. Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In Proc. of the 20th Annu. ACM-SIAM Sympos. on Discrete Algorithms (SODA), pages 384-391, 2009.
9. Steve Fortune and John Hopcroft. A note on Rabin’s nearest-neighbor algorithm. Inform. Process. Lett., 8(1):20-23, 1979.
10. Greg N. Frederickson and Donald B. Johnson. The complexity of selection and ranking in x + y and matrices with sorted columns. Journal of Computer and System Sciences, 24(2):197 - 208, 1982.
11. Michael L. Fredman. How good is the information theory bound in sorting? Theoret. Comput. Sci, 1(4):355-361, 1976.
12. François Le Gall. Faster algorithms for rectangular matrix multiplication. In 53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012, New Brunswick, NJ, USA, October 20-23, 2012, pages 514-523. IEEE Computer Society, 2012. URL: http://dx.doi.org/10.1109/FOCS.2012.80.
13. Xiaohan Huang and Victor Y. Pan. Fast rectangular matrix multiplication and applications. J. Complexity, 14(2):257-299, 1998.
14. Piotr Indyk, Moshe Lewenstein, Ohad Lipsky, and Ely Porat. Closest pair problems in very high dimensions. In Josep Díaz, Juhani Karhumäki, Arto Lepistö, and Donald Sannella, editors, Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings, volume 3142 of Lecture Notes in Computer Science, pages 782-792. Springer, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27836-8_66.
15. François Le Gall. Powers of tensors and fast matrix multiplication. In Proc. 39th International Sympos. on Symbolic and Algebraic Computation (ISSAC), pages 296-303, 2014.
16. François Le Gall. Faster algorithms for rectangular matrix multiplication. CoRR, abs/1204.1111, 2012.
17. François Le Gall and Florent Urrutia. Improved Rectangular Matrix Multiplication using Powers of the Coppersmith-Winograd Tensor. ArXiv e-prints, August 2017. URL: http://arxiv.org/abs/1708.05622.
18. Jiří Matoušek. Computing dominances in Eⁿ. Inform. Process. Lett., 38(5):277-278, 1991.
19. Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Springer-Verlag New York, NY, 1985.
20. Michael Rabin. Probabilistic algorithms. In Algorithms and Complexity, Recent Results and New Directions, Academic Press, pages 21-39, 1976.
21. Michael Ian Shamos. Geometric complexity. In Proc. of 7th Annu. ACM Sympos. on Theory of Computing (STOC), pages 224-233, 1975.
22. Virginia Vassilevska, Ryan Williams, and Raphael Yuster. All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory of Computing, 5(1):173-189, 2009.
23. Virginia Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Howard J. Karloff and Toniann Pitassi, editors, Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 887-898. ACM, 2012. URL: http://dx.doi.org/10.1145/2213977.2214056.
24. Raphael Yuster. Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In Claire Mathieu, editor, Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2009, New York, NY, USA, January 4-6, 2009, pages 950-957. SIAM, 2009. URL: http://dl.acm.org/citation.cfm?id=1496770.1496873.
25. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM, 49(3):289-317, 2002.
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