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International Symposium on Mathematical Foundations of Computer Science

MFCS 2014: Mathematical Foundations of Computer Science 2014 pp 24–43Cite as

Communication Complexity Theory: Thirty-Five Years of Set Disjointness

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8634))

Abstract

The set disjointness problem features k communicating parties and k subsets S 1,S 2,…,S k  ⊆ {1,2,…,n}. No single party knows all k subsets, and the objective is to determine with minimal communication whether the k subsets have nonempty intersection. The important special case k = 2 corresponds to two parties trying to determine whether their respective sets intersect. The study of the set disjointness problem spans almost four decades and offers a unique perspective on the remarkable evolution of communication complexity theory. We discuss known results on the communication complexity of set disjointness in the deterministic, nondeterministic, randomized, unbounded-error, and multiparty models, emphasizing the variety of mathematical techniques involved.

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References

  1. Aaronson, S., Ambainis, A.: Quantum search of spatial regions. Theory of Computing 1(1), 47–79 (2005)

    Article  MathSciNet  Google Scholar 

  2. Aho, A.V., Ullman, J.D., Yannakakis, M.: On notions of information transfer in VLSI circuits. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (STOC), pp. 133–139 (1983)

    Google Scholar 

  3. Alon, N., Frankl, P., Rödl, V.: Geometrical realization of set systems and probabilistic communication complexity. In: Proceedings of the Twenty-Sixth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 277–280 (1985)

    Google Scholar 

  4. Alon, N., Matias, Y., Szegedy, M.: The space complexity of approximating the frequency moments. J. Comput. Syst. Sci. 58(1), 137–147 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Babai, L., Frankl, P., Simon, J.: Complexity classes in communication complexity theory. In: Proceedings of the Twenty-Seventh Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 337–347 (1986)

    Google Scholar 

  6. Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci. 45(2), 204–232 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bar-Yossef, Z., Jayram, T.S., Kumar, R., Sivakumar, D.: An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci. 68(4), 702–732 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Beame, P., Huynh-Ngoc, D.T.: Multiparty communication complexity and threshold circuit complexity of AC0. In: Proceedings of the Fiftieth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 53–62 (2009)

    Google Scholar 

  9. Beame, P., Pitassi, T., Segerlind, N., Wigderson, A.: A strong direct product theorem for corruption and the multiparty communication complexity of disjointness. Computational Complexity 15(4), 391–432 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ben-Aroya, A., Regev, O., de Wolf, R.: A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs. In: Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 477–486 (2008)

    Google Scholar 

  11. Ben-David, S., Eiron, N., Simon, H.U.: Limitations of learning via embeddings in Euclidean half spaces. J. Mach. Learn. Res. 3, 441–461 (2003)

    MATH  MathSciNet  Google Scholar 

  12. Braverman, M., Ellen, F., Oshman, R., Pitassi, T., Vaikuntanathan, V.: A tight bound for set disjointness in the message-passing model. In: Proceedings of the Fifty-Fourth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 668–677 (2013)

    Google Scholar 

  13. Braverman, M., Garg, A., Pankratov, D., Weinstein, O.: From information to exact communication. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC, pp. 151–160 (2013)

    Google Scholar 

  14. Buhrman, H., Cleve, R., Wigderson, A.: Quantum vs. classical communication and computation. In: Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, STOC, pp. 63–68 (1998)

    Google Scholar 

  15. Chakrabarti, A., Khot, S., Sun, X.: Near-optimal lower bounds on the multi-party communication complexity of set disjointness. In: Proceedings of the Eighteenth Annual IEEE Conference on Computational Complexity, CCC, pp. 107–117 (2003)

    Google Scholar 

  16. Chandra, A.K., Furst, M.L., Lipton, R.J.: Multi-party protocols. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, STOC, pp. 94–99 (1983)

    Google Scholar 

  17. Chattopadhyay, A.: Circuits, Communication, and Polynomials. Ph.D. thesis, McGill University (2008)

    Google Scholar 

  18. Chattopadhyay, A., Ada, A.: Multiparty communication complexity of disjointness. In: Electronic Colloquium on Computational Complexity (ECCC), report TR08-002 (January 2008)

    Google Scholar 

  19. Forster, J.: A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci. 65(4), 612–625 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Forster, J., Krause, M., Lokam, S.V., Mubarakzjanov, R., Schmitt, N., Simon, H.U.: Relations between communication complexity, linear arrangements, and computational complexity. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 171–182. Springer, Heidelberg (2001)

    Google Scholar 

  21. Gavinsky, D., Sherstov, A.A.: A separation of NP and coNP in multiparty communication complexity. Theory of Computing 6(10), 227–245 (2010)

    Article  MathSciNet  Google Scholar 

  22. Grolmusz, V.: The BNS lower bound for multi-party protocols in nearly optimal. Inf. Comput. 112(1), 51–54 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jain, R., Klauck, H., Nayak, A.: Direct product theorems for classical communication complexity via subdistribution bounds. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC, pp. 599–608 (2008)

    Google Scholar 

  24. Kalyanasundaram, B., Schnitger, G.: The probabilistic communication complexity of set intersection. SIAM J. Discrete Math. 5(4), 545–557 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Klauck, H.: Rectangle size bounds and threshold covers in communication complexity. In: Proceedings of the Eighteenth Annual IEEE Conference on Computational Complexity, CCC, pp. 118–134 (2003)

    Google Scholar 

  26. Klauck, H.: A strong direct product theorem for disjointness. In: Proceedings of the Forty-Second Annual ACM Symposium on Theory of Computing, STOC, pp. 77–86 (2010)

    Google Scholar 

  27. Klauck, H., Špalek, R., de Wolf, R.: Quantum and classical strong direct product theorems and optimal time-space tradeoffs. SIAM J. Comput. 36(5), 1472–1493 (2007)

    Article  MATH  Google Scholar 

  28. Klivans, A.R., Servedio, R.A.: Learning DNF in time \(2^{\tilde O(n^{1/3})}\). J. Comput. Syst. Sci. 68(2), 303–318 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kushilevitz, E., Nisan, N.: Communication complexity. Cambridge University Press (1997)

    Google Scholar 

  30. Lee, T., Shraibman, A.: Disjointness is hard in the multiparty number-on-the-forehead model. Computational Complexity 18(2), 309–336 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  31. Linial, N., Mendelson, S., Schechtman, G., Shraibman, A.: Complexity measures of sign matrices. Combinatorica 27(4), 439–463 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lovász, L., Saks, M.E.: Lattices, Möbius functions and communication complexity. In: Proceedings of the Twenty-Ninth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 81–90 (1988)

    Google Scholar 

  33. Mehlhorn, K., Schmidt, E.M.: Las Vegas is better than determinism in VLSI and distributed computing. In: Proceedings of the Fourteenth Annual ACM Symposium on Theory of Computing, STOC, pp. 330–337 (1982)

    Google Scholar 

  34. Newman, I.: Private vs. common random bits in communication complexity. Inf. Process. Lett. 39(2), 67–71 (1991)

    Article  MATH  Google Scholar 

  35. Nisan, N., Segal, I.: The communication requirements of efficient allocations and supporting prices. J. Economic Theory 129(1), 192–224 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Nisan, N., Szegedy, M.: On the degree of Boolean functions as real polynomials. Computational Complexity 4, 301–313 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  37. Nisan, N., Wigderson, A.: On rank vs. communication complexity. Combinatorica 15(4), 557–565 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  38. Paturi, R., Simon, J.: Probabilistic communication complexity. J. Comput. Syst. Sci. 33(1), 106–123 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  39. Razborov, A.A.: Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica 10(1), 81–93 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  40. Razborov, A.A.: On the distributional complexity of disjointness. Theor. Comput. Sci. 106(2), 385–390 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  41. Razborov, A.A.: Quantum communication complexity of symmetric predicates. Izvestiya: Mathematics 67(1), 145–159 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  42. Razborov, A.A., Sherstov, A.A.: The sign-rank of AC0. SIAM J. Comput. 39(5), 1833–1855 (2010); preliminary version in Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, FOCS (2008)

    Google Scholar 

  43. Sherstov, A.A.: Halfspace matrices. Computational Complexity 17(2), 149–178 (2008); preliminary version in Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity, CCC (2007)

    Google Scholar 

  44. Sherstov, A.A.: Separating AC0 from depth-2 majority circuits. SIAM J. Comput. 38(6), 2113–2129 (2009); preliminary version in Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC (2007)

    Google Scholar 

  45. Sherstov, A.A.: The pattern matrix method. SIAM J. Comput. 40(6), 1969–2000 (2011); preliminary version in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, STOC) (2008)

    Google Scholar 

  46. Sherstov, A.A.: The unbounded-error communication complexity of symmetric functions. Combinatorica 31(5), 583–614 (2011); preliminary version in Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, FOCS (2008)

    Google Scholar 

  47. Sherstov, A.A.: The multiparty communication complexity of set disjointness. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, STOC, pp. 525–544 (2012)

    Google Scholar 

  48. Sherstov, A.A.: Strong direct product theorems for quantum communication and query complexity. SIAM J. Comput. 41(5), 1122–1165 (2012); preliminary version in Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC (2011)

    Google Scholar 

  49. Sherstov, A.A.: Communication lower bounds using directional derivatives. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, STOC, pp. 921–930 (2013)

    Google Scholar 

  50. Shi, Y., Zhu, Y.: Quantum communication complexity of block-composed functions. Quantum Information & Computation 9(5-6), 444–460 (2009)

    MATH  MathSciNet  Google Scholar 

  51. Tesson, P.: Computational complexity questions related to finite monoids and semigroups. Ph.D. thesis, McGill University (2003)

    Google Scholar 

  52. Yao, A.C.C.: Some complexity questions related to distributive computing. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, STOC, pp. 209–213 (1979)

    Google Scholar 

  53. Yao, A.C.C.: Lower bounds by probabilistic arguments. In: Proceedings of the Twenty-Fourth Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 420–428 (1983)

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of California, Los Angeles, USA

    Alexander A. Sherstov

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  1. Alexander A. Sherstov
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Editors and Affiliations

  1. Faculty of Informatics, Eötvös Loránd University, Pázmány Péter sétány 1/C, 1117, Budapest, Hungary

    Erzsébet Csuhaj-Varjú

  2. Fakultät für Informatik und Automatisierung, Technische Universität Ilmenau, Helmholtzplatz, 598693, Ilmenau, Germany

    Martin Dietzfelbinger

  3. Institute of Informatics, Szeged University, Tisza Lajos krt. 103, 6720, Szeged, Hungary

    Zoltán Ésik

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Sherstov, A.A. (2014). Communication Complexity Theory: Thirty-Five Years of Set Disjointness. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds) Mathematical Foundations of Computer Science 2014. MFCS 2014. Lecture Notes in Computer Science, vol 8634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44522-8_3

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  • DOI: https://doi.org/10.1007/978-3-662-44522-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-44521-1

  • Online ISBN: 978-3-662-44522-8

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