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A Lower Bound for Sampling Disjoint Sets

Authors Mika Göös, Thomas Watson

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  • Theory of computation → Communication complexity
Keywords
  • Communication complexity
  • set disjointness
  • sampling

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Abstract

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta<1, this requires Omega(n) communication even to get within statistical distance 1-beta^n of the target distribution. Previously, Ambainis, Schulman, Ta-Shma, Vazirani, and Wigderson (FOCS 1998) proved that Omega(sqrt{n}) communication is required to get within some constant statistical distance epsilon>0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}.

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Mika Göös and Thomas Watson. A Lower Bound for Sampling Disjoint Sets. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.51

Author Details

Mika Göös
  • Institute for Advanced Study, Princeton, NJ, USA
Thomas Watson
  • University of Memphis, TN, USA

Funding

  • Göös, Mika: Supported by NSF grant CCF-1412958.
  • Watson, Thomas: Supported by NSF grant CCF-1657377.

References

  1. Scott Aaronson. The Equivalence of Sampling and Searching. Theory of Computing Systems, 55(2):281-298, 2014. URL: https://doi.org/10.1007/s00224-013-9527-3.
  2. Scott Aaronson and Andris Ambainis. Quantum Search of Spatial Regions. Theory of Computing, 1(1):47-79, 2005. URL: https://doi.org/10.4086/toc.2005.v001a004.
  3. Scott Aaronson and Avi Wigderson. Algebrization: A New Barrier in Complexity Theory. ACM Transactions on Computation Theory, 1(1):2:1-2:54, 2009. URL: https://doi.org/10.1145/1490270.1490272.
  4. Amir Abboud, Aviad Rubinstein, and Ryan Williams. Distributed PCP Theorems for Hardness of Approximation in P. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 25-36. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.12.
  5. Josh Alman, Joshua Wang, and Huacheng Yu. Cell-Probe Lower Bounds from Online Communication Complexity. In Proceedings of the 50th Symposium on Theory of Computing (STOC), pages 1003-1012. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188862.
  6. Noga Alon, Yossi Matias, and Mario Szegedy. The Space Complexity of Approximating the Frequency Moments. Journal of Computer and System Sciences, 58(1):137-147, 1999. URL: https://doi.org/10.1006/jcss.1997.1545.
  7. Andris Ambainis, Leonard Schulman, Amnon Ta-Shma, Umesh Vazirani, and Avi Wigderson. The Quantum Communication Complexity of Sampling. SIAM Journal on Computing, 32(6):1570-1585, 2003. URL: https://doi.org/10.1137/S009753979935476.
  8. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial Pass Lower Bounds for Graph Streaming Algorithms. In Proceedings of the 51st Symposium on Theory of Computing (STOC), pages 265-276. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316361.
  9. László Babai, Peter Frankl, and Janos Simon. Complexity Classes in Communication Complexity Theory. In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS), pages 337-347. IEEE, 1986. URL: https://doi.org/10.1109/SFCS.1986.15.
  10. Ziv Bar-Yossef, T.S. Jayram, Ravi Kumar, and D. Sivakumar. An Information Statistics Approach to Data Stream and Communication Complexity. Journal of Computer and System Sciences, 68(4):702-732, 2004. URL: https://doi.org/10.1016/j.jcss.2003.11.006.
  11. Paul Beame and Dang-Trinh Huynh-Ngoc. Multiparty Communication Complexity and Threshold Circuit Size of AC⁰. In Proceedings of the 50th Symposium on Foundations of Computer Science (FOCS), pages 53-62. IEEE, 2009. URL: https://doi.org/10.1109/FOCS.2009.12.
  12. Paul Beame, Toniann Pitassi, Nathan Segerlind, and Avi Wigderson. A Strong Direct Product Theorem for Corruption and the Multiparty Communication Complexity of Disjointness. Computational Complexity, 15(4):391-432, 2006. URL: https://doi.org/10.1007/s00037-007-0220-2.
  13. Christopher Beck, Russell Impagliazzo, and Shachar Lovett. Large Deviation Bounds for Decision Trees and Sampling Lower Bounds for AC⁰-Circuits. In Proceedings of the 53rd Symposium on Foundations of Computer Science (FOCS), pages 101-110. IEEE, 2012. URL: https://doi.org/10.1109/FOCS.2012.82.
  14. Avraham Ben-Aroya, Oded Regev, and Ronald de Wolf. A Hypercontractive Inequality for Matrix-Valued Functions with Applications to Quantum Computing and LDCs. In Proceedings of the 49th Symposium on Foundations of Computer Science (FOCS), pages 477-486. IEEE, 2008. URL: https://doi.org/10.1109/FOCS.2008.45.
  15. Itai Benjamini, Gil Cohen, and Igor Shinkar. Bi-Lipschitz Bijection Between the Boolean Cube and the Hamming Ball. In Proceedings of the 55th Symposium on Foundations of Computer Science (FOCS), pages 81-89. IEEE, 2014. URL: https://doi.org/10.1109/FOCS.2014.17.
  16. Lucas Boczkowski, Iordanis Kerenidis, and Frédéric Magniez. Streaming Communication Protocols. ACM Transactions on Computation Theory, 10(4):19:1-19:21, 2018. URL: https://doi.org/10.1145/3276748.
  17. Ralph Bottesch, Dmitry Gavinsky, and Hartmut Klauck. Correlation in Hard Distributions in Communication Complexity. In Proceedings of the 19th International Workshop on Randomization and Computation (RANDOM), pages 544-572. Schloss Dagstuhl, 2015. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.544.
  18. Mark Braverman, Faith Ellen, Rotem Oshman, Toniann Pitassi, and Vinod Vaikuntanathan. A Tight Bound for Set Disjointness in the Message-Passing Model. In Proceedings of the 54th Symposium on Foundations of Computer Science (FOCS), pages 668-677. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.77.
  19. Mark Braverman, Ankit Garg, Young Kun-Ko, Jieming Mao, and Dave Touchette. Near-Optimal Bounds on the Bounded-Round Quantum Communication Complexity of Disjointness. SIAM Journal on Computing, 47(6):2277-2314, 2018. URL: https://doi.org/10.1137/16M1061400.
  20. Mark Braverman, Ankit Garg, Denis Pankratov, and Omri Weinstein. From Information to Exact Communication. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 151-160. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488628.
  21. Mark Braverman and Ankur Moitra. An Information Complexity Approach to Extended Formulations. In Proceedings of the 45th Symposium on Theory of Computing (STOC), pages 161-170. ACM, 2013. URL: https://doi.org/10.1145/2488608.2488629.
  22. Mark Braverman and Rotem Oshman. On Information Complexity in the Broadcast Model. In Proceedings of the 34th Symposium on Principles of Distributed Computing (PODC), pages 355-364. ACM, 2015. URL: https://doi.org/10.1145/2767386.2767425.
  23. Mark Braverman and Rotem Oshman. A Rounds vs. Communication Tradeoff for Multi-Party Set Disjointness. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 144-155. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.22.
  24. Joshua Brody, Amit Chakrabarti, Ranganath Kondapally, David Woodruff, and Grigory Yaroslavtsev. Beyond Set Disjointness: The Communication Complexity of Finding the Intersection. In Proceedings of the 33rd Symposium on Principles of Distributed Computing (PODC), pages 106-113. ACM, 2014. URL: https://doi.org/10.1145/2611462.2611501.
  25. Harry Buhrman, Richard Cleve, and Avi Wigderson. Quantum vs. Classical Communication and Computation. In Proceedings of the 30th Symposium on Theory of Computing (STOC), pages 63-68. ACM, 1998. URL: https://doi.org/10.1145/276698.276713.
  26. Harry Buhrman, David Garcia-Soriano, Arie Matsliah, and Ronald de Wolf. The Non-Adaptive Query Complexity of Testing k-Parities. Chicago Journal of Theoretical Computer Science, 2013(6):1-11, 2013. URL: https://doi.org/10.4086/cjtcs.2013.006.
  27. Amit Chakrabarti, Subhash Khot, and Xiaodong Sun. Near-Optimal Lower Bounds on the Multi-Party Communication Complexity of Set Disjointness. In Proceedings of the 18th Conference on Computational Complexity, pages 107-117. IEEE, 2003. URL: https://doi.org/10.1109/CCC.2003.1214414.
  28. Arkadev Chattopadhyay and Anil Ada. Multiparty Communication Complexity of Disjointness. Technical Report TR08-002, Electronic Colloquium on Computational Complexity (ECCC), 2008. URL: https://eccc.weizmann.ac.il//eccc-reports/2008/TR08-002/.
  29. Lijie Chen. On The Hardness of Approximate and Exact (Bichromatic) Maximum Inner Product. In Proceedings of the 33rd Computational Complexity Conference (CCC), pages 14:1-14:45. Schloss Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.14.
  30. Yuval Dagan, Yuval Filmus, Hamed Hatami, and Yaqiao Li. Trading Information Complexity for Error. Theory of Computing, 14(1):1-73, 2018. URL: https://doi.org/10.4086/toc.2018.v014a006.
  31. Anirban Dasgupta, Ravi Kumar, and D. Sivakumar. Sparse and Lopsided Set Disjointness via Information Theory. In Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pages 517-528. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_44.
  32. Anindya De and Thomas Watson. Extractors and Lower Bounds for Locally Samplable Sources. ACM Transactions on Computation Theory, 4(1):3:1-3:21, 2012. URL: https://doi.org/10.1145/2141938.2141941.
  33. Yuval Filmus, Hamed Hatami, Yaqiao Li, and Suzin You. Information Complexity of the AND Function in the Two-Party and Multi-Party Settings. In Proceedings of the 23rd International Computing and Combinatorics Conference (COCOON), pages 200-211. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-62389-4_17.
  34. Dmitry Gavinsky. Communication Complexity of Inevitable Intersection. Technical Report abs/1611.08842, arXiv, 2016. URL: http://arxiv.org/abs/1611.08842.
  35. Dmitry Gavinsky and Alexander Sherstov. A Separation of NP and coNP in Multiparty Communication Complexity. Theory of Computing, 6(1):227-245, 2010. URL: https://doi.org/10.4086/toc.2010.v006a010.
  36. Oded Goldreich, Shafi Goldwasser, and Asaf Nussboim. On the Implementation of Huge Random Objects. SIAM Journal on Computing, 39(7):2761-2822, 2010. URL: https://doi.org/10.1137/080722771.
  37. Mika Göös, Shachar Lovett, Raghu Meka, Thomas Watson, and David Zuckerman. Rectangles Are Nonnegative Juntas. SIAM Journal on Computing, 45(5):1835-1869, 2016. URL: https://doi.org/10.1137/15M103145X.
  38. Mika Göös, Toniann Pitassi, and Thomas Watson. Zero-Information Protocols and Unambiguity in Arthur-Merlin Communication. Algorithmica, 76(3):684-719, 2016. URL: https://doi.org/10.1007/s00453-015-0104-9.
  39. Mika Göös and Thomas Watson. Communication Complexity of Set-Disjointness for All Probabilities. Theory of Computing, 12(9):1-23, 2016. URL: https://doi.org/10.4086/toc.2016.v012a009.
  40. Vince Grolmusz. The BNS Lower Bound for Multi-Party Protocols Is Nearly Optimal. Information and Computation, 112(1):51-54, 1994. URL: https://doi.org/10.1006/inco.1994.1051.
  41. André Gronemeier. Asymptotically Optimal Lower Bounds on the NIH-Multi-Party Information Complexity of the AND-Function and Disjointness. In Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), pages 505-516. Schloss Dagstuhl, 2009. URL: https://doi.org/10.4230/LIPIcs.STACS.2009.1846.
  42. Johan Håstad and Avi Wigderson. The Randomized Communication Complexity of Set Disjointness. Theory of Computing, 3(1):211-219, 2007. URL: https://doi.org/10.4086/toc.2007.v003a011.
  43. Peter Høyer and Ronald de Wolf. Improved Quantum Communication Complexity Bounds for Disjointness and Equality. In Proceedings of the 19th Symposium on Theoretical Aspects of Computer Science (STACS), pages 299-310. Springer, 2002. URL: https://doi.org/10.1007/3-540-45841-7_24.
  44. Rahul Jain and Hartmut Klauck. The Partition Bound for Classical Communication Complexity and Query Complexity. In Proceedings of the 25th Conference on Computational Complexity (CCC), pages 247-258. IEEE, 2010. URL: https://doi.org/10.1109/CCC.2010.31.
  45. Rahul Jain, Hartmut Klauck, and Ashwin Nayak. Direct Product Theorems for Classical Communication Complexity via Subdistribution Bounds. In Proceedings of the 40th Symposium on Theory of Computing (STOC), pages 599-608. ACM, 2008. URL: https://doi.org/10.1145/1374376.1374462.
  46. Rahul Jain, Jaikumar Radhakrishnan, and Pranab Sen. A Lower Bound for the Bounded Round Quantum Communication Complexity of Set Disjointness. In Proceedings of the 44th Symposium on Foundations of Computer Science (FOCS), pages 220-229. IEEE, 2003. URL: https://doi.org/10.1109/SFCS.2003.1238196.
  47. Rahul Jain, Yaoyun Shi, Zhaohui Wei, and Shengyu Zhang. Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States. IEEE Transactions on Information Theory, 59(8):5171-5178, 2013. URL: https://doi.org/10.1109/TIT.2013.2258372.
  48. T.S. Jayram. Hellinger strikes back: A note on the multi-party information complexity of AND. In Proceedings of the 13th International Workshop on Randomization and Computation (RANDOM), pages 562-573. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_42.
  49. Bala Kalyanasundaram and Georg Schnitger. The Probabilistic Communication Complexity of Set Intersection. SIAM Journal on Discrete Mathematics, 5(4):545-557, 1992. URL: https://doi.org/10.1137/0405044.
  50. Hartmut Klauck. Rectangle Size Bounds and Threshold Covers in Communication Complexity. In Proceedings of the 18th Conference on Computational Complexity (CCC), pages 118-134. IEEE, 2003. URL: https://doi.org/10.1109/CCC.2003.1214415.
  51. Hartmut Klauck. A Strong Direct Product Theorem for Disjointness. In Proceedings of the 42nd Symposium on Theory of Computing (STOC), pages 77-86. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806702.
  52. Hartmut Klauck, Ashwin Nayak, Amnon Ta-Shma, and David Zuckerman. Interaction in Quantum Communication. IEEE Transactions on Information Theory, 53(6):1970-1982, 2007. URL: https://doi.org/10.1109/TIT.2007.896888.
  53. Hartmut Klauck and Supartha Podder. New Bounds for the Garden-Hose Model. In Proceedings of the 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS), pages 481-492. Schloss Dagstuhl, 2014. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2014.481.
  54. Hartmut Klauck, Robert Spalek, and Ronald de Wolf. Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs. SIAM Journal on Computing, 36(5):1472-1493, 2007. URL: https://doi.org/10.1137/05063235X.
  55. Gillat Kol, Shay Moran, Amir Shpilka, and Amir Yehudayoff. Approximate Nonnegative Rank Is Equivalent to the Smooth Rectangle Bound. Computational Complexity, 28(1):1-25, 2019. URL: https://doi.org/10.1007/s00037-018-0176-4.
  56. Eyal Kushilevitz and Enav Weinreb. The Communication Complexity of Set-Disjointness with Small Sets and 0-1 Intersection. In Proceedings of the 50th Symposium on Foundations of Computer Science (FOCS), pages 63-72. IEEE, 2009. URL: https://doi.org/10.1109/FOCS.2009.15.
  57. Troy Lee and Adi Shraibman. Disjointness is Hard in the Multiparty Number-on-the-Forehead Model. Computational Complexity, 18(2):309-336, 2009. URL: https://doi.org/10.1007/s00037-009-0276-2.
  58. Shachar Lovett and Emanuele Viola. Bounded-Depth Circuits Cannot Sample Good Codes. Computational Complexity, 21(2):245-266, 2012. URL: https://doi.org/10.1007/s00037-012-0039-3.
  59. Mihai Patrascu. Unifying the Landscape of Cell-Probe Lower Bounds. SIAM Journal on Computing, 40(3):827-847, 2011. URL: https://doi.org/10.1137/09075336X.
  60. Vladimir Podolskii and Alexander Sherstov. Inner Product and Set Disjointness: Beyond Logarithmically Many Parties. Technical Report abs/1711.10661, arXiv, 2017. URL: http://arxiv.org/abs/1711.10661.
  61. Anup Rao and Amir Yehudayoff. Simplified Lower Bounds on the Multiparty Communication Complexity of Disjointness. In Proceedings of the 30th Computational Complexity Conference (CCC), pages 88-101. Schloss Dagstuhl, 2015. URL: https://doi.org/10.4230/LIPIcs.CCC.2015.88.
  62. Alexander Razborov. On the Distributional Complexity of Disjointness. Theoretical Computer Science, 106(2):385-390, 1992. URL: https://doi.org/10.1016/0304-3975(92)90260-M.
  63. Alexander Razborov. Quantum Communication Complexity of Symmetric Predicates. Izvestiya: Mathematics, 67(1):145-159, 2003. URL: https://doi.org/10.1070/IM2003v067n01ABEH000422.
  64. Aviad Rubinstein. Hardness of Approximate Nearest Neighbor Search. In Proceedings of the 50th Symposium on Theory of Computing (STOC), pages 1260-1268. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188916.
  65. Mert Saglam and Gábor Tardos. On the Communication Complexity of Sparse Set Disjointness and Exists-Equal Problems. In Proceedings of the 54th Symposium on Foundations of Computer Science (FOCS), pages 678-687. IEEE, 2013. URL: https://doi.org/10.1109/FOCS.2013.78.
  66. Alexander Sherstov. The Pattern Matrix Method. SIAM Journal on Computing, 40(6):1969-2000, 2011. URL: https://doi.org/10.1137/080733644.
  67. Alexander Sherstov. Strong Direct Product Theorems for Quantum Communication and Query Complexity. SIAM Journal on Computing, 41(5):1122-1165, 2012. URL: https://doi.org/10.1137/110842661.
  68. Alexander Sherstov. Communication Lower Bounds Using Directional Derivatives. Journal of the ACM, 61(6):1-71, 2014. URL: https://doi.org/10.1145/2629334.
  69. Alexander Sherstov. The Multiparty Communication Complexity of Set Disjointness. SIAM Journal on Computing, 45(4):1450-1489, 2016. URL: https://doi.org/10.1137/120891587.
  70. Yaoyun Shi and Yufan Zhu. Quantum Communication Complexity of Block-Composed Functions. Quantum Information and Computation, 9(5-6):444-460, 2009. Google Scholar
  71. Pascal Tesson. Computational Complexity Questions Related to Finite Monoids and Semigroups. PhD thesis, McGill University, 2003. Google Scholar
  72. Emanuele Viola. Extractors for Turing-Machine Sources. In Proceedings of the 16th International Workshop on Randomization and Computation (RANDOM), pages 663-671. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32512-0_56.
  73. Emanuele Viola. The Complexity of Distributions. SIAM Journal on Computing, 41(1):191-218, 2012. URL: https://doi.org/10.1137/100814998.
  74. Emanuele Viola. Extractors for Circuit Sources. SIAM Journal on Computing, 43(2):655-672, 2014. URL: https://doi.org/10.1137/11085983X.
  75. Emanuele Viola. Quadratic Maps Are Hard to Sample. ACM Transactions on Computation Theory, 8(4):18:1-18:4, 2016. URL: https://doi.org/10.1145/2934308.
  76. Emanuele Viola. Sampling Lower Bounds: Boolean Average-Case and Permutations. Technical Report TR18-060, Electronic Colloquium on Computational Complexity (ECCC), 2018. URL: https://eccc.weizmann.ac.il/report/2018/060.
  77. Thomas Watson. Time Hierarchies for Sampling Distributions. SIAM Journal on Computing, 43(5):1709-1727, 2014. URL: https://doi.org/10.1137/120898553.
  78. Thomas Watson. Nonnegative Rank vs. Binary Rank. Chicago Journal of Theoretical Computer Science, 2016(2):1-13, 2016. URL: https://doi.org/10.4086/cjtcs.2016.002.
  79. Thomas Watson. Communication Complexity with Small Advantage. In Proceedings of the 33rd Computational Complexity Conference (CCC), pages 9:1-9:17. Schloss Dagstuhl, 2018. URL: https://doi.org/10.4230/LIPIcs.CCC.2018.9.
  80. Omri Weinstein and David Woodruff. The Simultaneous Communication of Disjointness with Applications to Data Streams. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP), pages 1082-1093. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_88.
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