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On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic

Authors Karthik C. S. , Pasin Manurangsi

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LIPIcs.ITCS.2019.17.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Closest Pair
  • Bichromatic Closest Pair
  • Contact Dimension
  • Fine-Grained Complexity

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Abstract

Given a set of n points in R^d, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the l_p-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when d=omega(log n) was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]).
In this paper, we show that for every p in R_{>= 1} cup {0}, under the Strong Exponential Time Hypothesis (SETH), for every epsilon>0, the following holds:
- No algorithm running in time O(n^{2-epsilon}) can solve the Closest Pair problem in d=(log n)^{Omega_{epsilon}(1)} dimensions in the l_p-metric. 
- There exists delta = delta(epsilon)>0 and c = c(epsilon)>= 1 such that no algorithm running in time O(n^{1.5-epsilon}) can approximate Closest Pair problem to a factor of (1+delta) in d >= c log n dimensions in the l_p-metric. 
In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to n^{epsilon} factor in the running time) for d=(log n)^{Omega_epsilon(1)} dimensions. 
Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of n points in n^{o(1)}-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product.
At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on n vertices with n^{2-epsilon} edges whose vertices can be realized as points in a (log n)^{Omega_epsilon(1)}-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].

Cite As Get BibTex

Karthik C. S. and Pasin Manurangsi. On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ITCS.2019.17

Author Details

Karthik C. S.
  • Weizmann Institute of Science, Rehovot, Israel
Pasin Manurangsi
  • University of California, Berkeley, USA

Funding

  • C. S., Karthik: Supported by Irit Dinur’s ERC-CoG grant 772839 and BSF grant 2014371.
  • Manurangsi, Pasin: Supported by NSF under Grants No. CCF 1655215 and CCF 1815434.

References

  1. Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP Theorems for Hardness of Approximation in P. CoRR, abs/1706.06407, 2017. URL: http://arxiv.org/abs/1706.06407.
  2. Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP Theorems for Hardness of Approximation in P. In FOCS, pages 25-36, 2017. Google Scholar
  3. Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs. Discrete & Computational Geometry, 6:407-422, 1991. Preliminary version in SoCG'90. URL: http://dx.doi.org/10.1007/BF02574698.
  4. Nir Ailon and Bernard Chazelle. The Fast Johnson-Lindenstrauss Transform and Approximate Nearest Neighbors. SIAM J. Comput., 39(1):302-322, 2009. Preliminary version in STOC'06. URL: http://dx.doi.org/10.1137/060673096.
  5. Josh Alman, Timothy M. Chan, and R. Ryan Williams. Polynomial Representations of Threshold Functions and Algorithmic Applications. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 467-476, 2016. URL: http://dx.doi.org/10.1109/FOCS.2016.57.
  6. Josh Alman and Ryan Williams. Probabilistic Polynomials and Hamming Nearest Neighbors. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 136-150, 2015. URL: http://dx.doi.org/10.1109/FOCS.2015.18.
  7. Ethem Alpaydin. Introduction to Machine Learning. The MIT Press, 2nd edition, 2010. Google Scholar
  8. Alexei E. Ashikhmin, Alexander Barg, and Serge G. Vladut. Linear Codes with Exponentially Many Light Vectors. J. Comb. Theory, Ser. A, 96(2):396-399, 2001. URL: http://dx.doi.org/10.1006/jcta.2001.3206.
  9. Michael Ben-Or. Lower Bounds for Algebraic Computation Trees (Preliminary Report). In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 80-86, 1983. URL: http://dx.doi.org/10.1145/800061.808735.
  10. Jon Louis Bentley. Multidimensional Divide-and-Conquer. Commun. ACM, 23(4):214-229, 1980. URL: http://dx.doi.org/10.1145/358841.358850.
  11. Jon Louis Bentley and Michael Ian Shamos. Divide-and-Conquer in Multidimensional Space. In Proceedings of the 8th Annual ACM Symposium on Theory of Computing, May 3-5, 1976, Hershey, Pennsylvania, USA, pages 220-230, 1976. URL: http://dx.doi.org/10.1145/800113.803652.
  12. Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., and Pasin Manurangsi. Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH. In ICALP, pages 17:1-17:15, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2018.17.
  13. Yonatan Bilu and Nathan Linial. Monotone maps, sphericity and bounded second eigenvalue. J. Comb. Theory, Ser. B, 95(2):283-299, 2005. URL: http://dx.doi.org/10.1016/j.jctb.2005.04.005.
  14. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More. In FOCS, pages 743-754, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.74.
  15. Lijie Chen. On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product. In 33rd Computational Complexity Conference, CCC 2018, June 22-24, 2018, San Diego, CA, USA, pages 14:1-14:45, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2018.14.
  16. Lijie Chen. Toward Super-Polynomial Size Lower Bounds for Depth-Two Threshold Circuits. CoRR, abs/1805.10698, 2018. URL: http://arxiv.org/abs/1805.10698.
  17. Lijie Chen and Ryan Williams. An Equivalence Class for Orthogonal Vectors. To appear in SODA, 2019. Google Scholar
  18. Qi Cheng and Daqing Wan. A Deterministic Reduction for the Gap Minimum Distance Problem. IEEE Trans. Information Theory, 58(11):6935-6941, 2012. URL: http://dx.doi.org/10.1109/TIT.2012.2209198.
  19. Edith Cohen and David D. Lewis. Approximating Matrix Multiplication for Pattern Recognition Tasks. J. Algorithms, 30(2):211-252, 1999. URL: http://dx.doi.org/10.1006/jagm.1998.0989.
  20. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Third Edition. The MIT Press, 3rd edition, 2009. Google Scholar
  21. Roee David, Karthik C. S., and Bundit Laekhanukit. On the Complexity of Closest Pair via Polar-Pair of Point-Sets. In 34th International Symposium on Computational Geometry, SoCG 2018, June 11-14, 2018, Budapest, Hungary, pages 28:1-28:15, 2018. URL: http://dx.doi.org/10.4230/LIPIcs.SoCG.2018.28.
  22. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: http://dx.doi.org/10.1007/978-1-4471-5559-1.
  23. Ilya Dumer, Daniele Micciancio, and Madhu Sudan. Hardness of approximating the minimum distance of a linear code. IEEE Trans. Information Theory, 49(1):22-37, 2003. URL: http://dx.doi.org/10.1109/TIT.2002.806118.
  24. Peter Frankl and Hiroshi Maehara. On the Contact Dimensions of Graphs. Discrete & Computational Geometry, 3:89-96, 1988. URL: http://dx.doi.org/10.1007/BF02187899.
  25. François Le Gall. Powers of tensors and fast matrix multiplication. In International Symposium on Symbolic and Algebraic Computation, ISSAC'14, Kobe, Japan, July 23-25, 2014, pages 296-303, 2014. URL: http://dx.doi.org/10.1145/2608628.2608664.
  26. E. N. Gilbert. A comparison of signalling alphabets. Bell System Technical Journal, 31:504-522, 1952. Google Scholar
  27. Omer Gold and Micha Sharir. Dominance Products and Faster Algorithms for High-Dimensional Closest Pair under dollarL_backslashinftydollar. CoRR, abs/1605.08107, 2016. URL: http://arxiv.org/abs/1605.08107.
  28. Tomislav Hengl. Finding the right pixel size. Computers &Geosciences, 32(9):1283-1298, 2006. URL: http://dx.doi.org/10.1016/j.cageo.2005.11.008.
  29. Klaus H. Hinrichs, Jürg Nievergelt, and Peter Schorn. Plane-Sweep Solves the Closest Pair Problem Elegantly. Inf. Process. Lett., 26(5):255-261, 1988. URL: http://dx.doi.org/10.1016/0020-0190(88)90150-0.
  30. Piotr Indyk. Dimensionality reduction techniques for proximity problems. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, January 9-11, 2000, San Francisco, CA, USA., pages 371-378, 2000. URL: http://dl.acm.org/citation.cfm?id=338219.338582.
  31. Piotr Indyk, Moshe Lewenstein, Ohad Lipsky, and Ely Porat. Closest Pair Problems in Very High Dimensions. In Automata, Languages and Programming: 31st International Colloquium, ICALP 2004, Turku, Finland, July 12-16, 2004. Proceedings, pages 782-792, 2004. URL: http://dx.doi.org/10.1007/978-3-540-27836-8_66.
  32. Piotr Indyk and Rajeev Motwani. Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality. In Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, May 23-26, 1998, pages 604-613, 1998. URL: http://dx.doi.org/10.1145/276698.276876.
  33. William Johnson and Joram Lindenstrauss. Extensions of Lipschitz mappings into a Hilbert space. In Conference in modern analysis and probability (New Haven, Conn., 1982), volume 26 of Contemporary Mathematics, pages 189-206. American Mathematical Society, 1984. Google Scholar
  34. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the Parameterized Complexity of Approximating Dominating Set. In STOC, 2018. To appear. Google Scholar
  35. Samir Khuller and Yossi Matias. A Simple Randomized Sieve Algorithm for the Closest-Pair Problem. Inf. Comput., 118(1):34-37, 1995. URL: http://dx.doi.org/10.1006/inco.1995.1049.
  36. Jon Kleinberg and Éva Tardos. Algorithm Design. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 2005. Google Scholar
  37. Jon M. Kleinberg. Two Algorithms for Nearest-Neighbor Search in High Dimensions. In Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, May 4-6, 1997, pages 599-608, 1997. URL: http://dx.doi.org/10.1145/258533.258653.
  38. Drago Krznaric, Christos Levcopoulos, and Bengt J. Nilsson. Minimum Spanning Trees in d Dimensions. Nord. J. Comput., 6(4):446-461, 1999. Google Scholar
  39. Bingkai Lin. The Parameterized Complexity of k-Biclique. In SODA, pages 605-615, 2015. Google Scholar
  40. Udi Manber. Introduction to Algorithms: A Creative Approach. Addison-Wesley Longman Publishing Co., Inc., Boston, MA, USA, 1989. Google Scholar
  41. Daniele Micciancio. Locally Dense Codes. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 90-97, 2014. URL: http://dx.doi.org/10.1109/CCC.2014.17.
  42. Rajeev Motwani, Assaf Naor, and Rina Panigrahy. Lower Bounds on Locality Sensitive Hashing. SIAM J. Discrete Math., 21(4):930-935, 2007. URL: http://dx.doi.org/10.1137/050646858.
  43. Ryan O'Donnell, Yi Wu, and Yuan Zhou. Optimal Lower Bounds for Locality-Sensitive Hashing (Except When q is Tiny). TOCT, 6(1):5:1-5:13, 2014. URL: http://dx.doi.org/10.1145/2578221.
  44. Janos Pach. Decomposition of multiple packing and covering. Diskrete Geometrie, 2 Kolloq. Math. Inst. Univ. Salzburg:169-178, 1980. Google Scholar
  45. Franco P. Preparata and Michael I. Shamos. Computational Geometry: An Introduction. Springer-Verlag New York, Inc., New York, NY, USA, 1985. Google Scholar
  46. Michael O. Rabin. Probabilistic Algorithms. In Proceedings of a Symposium on New Directions and Recent Results in Algorithms and Complexity, Computer Science Department, Carnegie-Mellon University, April 7-9, 1976, pages 21-39, 1976. Google Scholar
  47. Ilya Razenshteyn. High-Dimensional Similarity Search and Sketching: Algorithms and Hardness. PhD Thesis, MIT, 2017. Google Scholar
  48. Irving S. Reed and Gustave Solomon. Polynomial Codes over Certain Finite Fields. Journal of the Society for Industrial and Applied Mathematics (SIAM), 8(2):300-304, 1960. Google Scholar
  49. Jan Reiterman, Vojtech Rödl, and Edita Sinajová. Embeddings of Graphs in Euclidean Spaces. Discrete & Computational Geometry, 4:349-364, 1989. URL: http://dx.doi.org/10.1007/BF02187736.
  50. Aviad Rubinstein. Hardness of approximate nearest neighbor search. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 1260-1268, 2018. URL: http://dx.doi.org/10.1145/3188745.3188916.
  51. Michael Ian Shamos and Dan Hoey. Closest-Point Problems. In 16th Annual Symposium on Foundations of Computer Science, Berkeley, California, USA, October 13-15, 1975, pages 151-162, 1975. URL: http://dx.doi.org/10.1109/SFCS.1975.8.
  52. Gregory Valiant. Finding Correlations in Subquadratic Time, with Applications to Learning Parities and the Closest Pair Problem. J. ACM, 62(2):13:1-13:45, 2015. URL: http://dx.doi.org/10.1145/2728167.
  53. R. R. Varshamov. Estimate of the number of signals in error correcting codes. Dokl. Akad. Nauk SSSR, 117:739-741, 1957. Google Scholar
  54. Serge Vlăduţ. Lattices with exponentially large kissing numbers. arXiv preprint, 2018. URL: http://arxiv.org/abs/1802.00886.
  55. Ryan Williams. On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-Linear vs Barely-Subquadratic Complexity. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018, New Orleans, LA, USA, January 7-10, 2018, pages 1207-1215, 2018. URL: http://dx.doi.org/10.1137/1.9781611975031.78.
  56. Raymond Chi-Wing Wong, Yufei Tao, Ada Wai-Chee Fu, and Xiaokui Xiao. On Efficient Spatial Matching. In Proceedings of the 33rd International Conference on Very Large Data Bases, University of Vienna, Austria, September 23-27, 2007, pages 579-590, 2007. URL: http://www.vldb.org/conf/2007/papers/research/p579-wong.pdf.
  57. Andrew Chi-Chih Yao. Lower Bounds for Algebraic Computation Trees with Integer Inputs. SIAM J. Comput., 20(4):655-668, 1991. Preliminary version in FOCS'89. URL: http://dx.doi.org/10.1137/0220041.
  58. Charles T. Zahn. Graph-Theoretical Methods for Detecting and Describing Gestalt Clusters. IEEE Trans. Computers, 20(1):68-86, 1971. URL: http://dx.doi.org/10.1109/T-C.1971.223083.
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