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One-way multiparty communication lower bound for pointer jumping with applications

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Abstract

In this paper we study the one-way multiparty communication model, in which every party speaks exactly once in its turn. For every k, we prove a tight lower bound of Ω(n 1/(k−1)}) on the probabilistic communication complexity of pointer jumping in a k-layered tree, where the pointers of the i-th layer reside on the forehead of the i-th party to speak. The lower bound remains nontrivial even for k = (logn)1/2−ɛ parties, for any constant ɛ > 0. Previous to our work a lower bound was known only for k =3 (Wigderson, see [7]), and in restricted models for k>3 [2},24,18,4,13]. Our results have the following consequences to other models and problems, extending previous work in several directions.

The one-way model is strong enough to capture general (not one-way) multiparty protocols with a bounded number of rounds. Thus we generalize two problem areas previously studied in the 2-party model (cf. [30,21,29]). The first is a rounds hierarchy: we give an exponential separation between the power of r and 2r rounds in general probabilistic k-party protocols, for any k and r. The second is the relative power of determinism and nondeterminism: we prove an exponential separation between nondeterministic and deterministic communication complexity for general k-party protocols with r rounds, for any k,r.

The pointer jumping function is weak enough to be a special case of the well-studied disjointness function. Thus we obtain a lower bound of Ω(n 1/(k−1)) on the probabilistic complexity of k-set disjointness in the one-way model, which was known only for k = 3 parties. Our result also extends a similar lower bound for the weaker simultaneous model, in which parties simultaneously send one message to a referee [12].

Finally, we infer an exponential separation between the power of any two different orders in which parties send messages in the one-way model, for every k. Previous results [29, 7] separated orders based on who speaks first.

Our lower bound technique, which handles functions of high discrepancy over cylinder intersections, provides a “party-elimination” induction, based on a restricted form of a direct-product result, specific to the pointer jumping function.

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References

  1. Noga Alon, Yossi Matias and Mario Szegedy: The space complexity of approximating the frequency moments, J. Comput. System Sci. 58(1, part 2) (1999), 137–147.

    Article  MATH  MathSciNet  Google Scholar 

  2. Alfred V. Aho, Jeffrey D. Ullman and Mihalis Yannakakis: On notions of information transfer in VLSI circuits, in: 15th ACM symposium on Theory of computing (STOC), pages 133–139, ACM Press, 1983.

  3. Avraham Ben-Aroya, Oded Regev and Ronald de Wolf: A hypercontractive inequality for matrix-valued functions with applications to quantum computing and LDCs, in: 49th IEEE Symposium on Foundations of Computer Science (FOCS), pages 477–486, IEEE Computer Society, 2008.

  4. Joshua Brody and Amit Chakrabarti: Sublinear communication protocols for multi-party pointer jumping and a related lower bound, in: 25th Symposium on Theoretical Aspects of Computer Science (STACS), pages 145–156, 2008.

  5. Paul Beame, Matei David, Toniann Pitassi and Philipp Woelfel: Separating deterministic from nondeterministic nof multiparty communication complexity, in: 34th International Colloquium on Automata, Languages and Programming (ICALP), pages 134–145, Springer, 2007.

  6. László Babai, Anna Gál, Peter G. Kimmel and Satyanarayana V Lokam: Communication complexity of simultaneous messages, SIAM J. Comput. 33(1) (2003), 137–166.

    Article  MATH  MathSciNet  Google Scholar 

  7. László Babai, Thomas P. Hayes and Peter G. Kimmel: The cost of the missing bit: communication complexity with help, Combinatorica21(4) (2001), 455–488.

    Article  MATH  MathSciNet  Google Scholar 

  8. Paul Beame and Dang-Trinh Huynh-Ngoc: Multiparty communication complexity and threshold circuit size of AC0, in: 50th Symposium on Foundations of Computer Science (FOCS), 2009, to appear.

  9. Paul Beame, Russell Impagliazzo and Toniann Pitassi: Improved depth lower bounds for small distance connectivity, Comput. Complexity7(4) (1998), 325–345.

    Article  MATH  MathSciNet  Google Scholar 

  10. László Babai, Noam Nisan and Márió Szegedy: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs; J. Comput. System Sci.45(2) (1992), 204–232.

    Article  MATH  MathSciNet  Google Scholar 

  11. Paul Beame, Toniann Pitassi and Nathan Segerlind: Lower bounds for Lovász-Schrijver systems and beyond follow from multiparty communication complexity, SIAM J. Comput.37(3) (2007), 845–869.

    Article  MathSciNet  Google Scholar 

  12. Paul Beame, Toniann Pitassi, Nathan Segerlind and Avi Wigderson: A direct sum theorem for corruption and the multiparty NOF communication complexity of set disjointness, in: 20th Annual Conference on Computational Complexity (CCC), pages 52–66, IEEE, 2005.

  13. Joshua Brody: The maximum communication complexity of multiparty pointer jumping, in: Proc. 24th Conference on Computational Complexity (CCC), IEEE, 2009.

  14. Richard Beigel and Jun Tarui: On ACC, Comput. Complexity4(4) (1994), 350–366. Special issue devoted to the 4th Annual McGill Workshop on Complexity Theory. Preliminary version in FOCS ′91.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  16. Arkadev Chattopadhyay and Anil Ada: Multiparty communication complexity of disjointness, Electronic Colloquium on Computational Complexity, Technical Report TR08-002, 2008.

  17. Ashok K. Chandra, Merrick L. Furst and Richard J. Lipton: Multi-party protocols, in: 15th Annual Symposium on Theory of Computing (STOC), pages 94–99, 1983.

  18. Amit Chakrabarti: Lower bounds for multi-player pointer jumping, in: Proceedings of the 22nd Annual Conference on Computational Complexity, IEEE, June 13–16 2007.

  19. Arkadev Chattopadhyay: Discrepancy and the power of bottom fan-in in depththree circuits, in: 48th Annual Symposium on Foundations of Computer Science (FOCS), pages 449–458, IEEE, 2007.

  20. Fan R. K. Chung and Prasad Tetali: Communication complexity and quasi randomness, SIAM J. Discrete Math. 6(1) (1993), 110–123.

    Article  MATH  MathSciNet  Google Scholar 

  21. Pavol Duris, Zvi Galil and Georg Schnitger: Lower bounds on communication complexity, Inf. Comput. 73(1) (1987), 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  22. Carsten Damm, Stasys Jukna and Jpirí Sgall: Some bounds on multiparty communication complexity of pointer jumping, Comput. Complexity7(2) (1998), 109–127.

    Article  MATH  MathSciNet  Google Scholar 

  23. Matei David, Toniann Pitassiand Emanuele Viola: Improved separations between nondeterministic and randomized multiparty communication, in: 12th Workshop on Randomization and Computation (RANDOM), Springer, 2008; Transactions on Computation Theory1 (2) (2009), 1–20 (Art. No. 5).

  24. Andre Gronemeier: NOF-multiparty information complexity bounds for pointer jumping, in: 31st Symposium on Mathematical Foundations of Computer Science (MFCS), volume 4162 of Lecture Notes in Computer Science, pages 459–470, Springer, 2006.

  25. Johan Håstad and Mikael Goldmann: On the power of small-depth threshold circuits, Comput. Complexity1(2) (1991), 113–129.

    Article  MATH  MathSciNet  Google Scholar 

  26. Eyal Kushilevitz and Noam Nisan: Communication complexity, Cambridge University Press, Cambridge, 1997.

    MATH  Google Scholar 

  27. Troy Lee and Adi Shraibman: Disjointness is hard in the multiparty number-on-the-forehead model, Computational Complexity18(2) (2009), 309–336.

    Article  MathSciNet  Google Scholar 

  28. Noam Nisan and Ziv Bar-Yossef: Pointer jumping requires concurrent read, in 29th ACM Symposium on Theory of Computing (STOC), pages 549–558 (electronic), 1997.

  29. Noam Nisan and Avi Wigderson: Rounds in communication complexity revisited, SIAM J. Comput. 22(1) (1993), 211–219.

    Article  MATH  MathSciNet  Google Scholar 

  30. Christos H. Papadimitriou and Michael Sipser: Communication complexity, J. Comput. System Sci. 28(2) (1984), 260–269.

    Article  MATH  MathSciNet  Google Scholar 

  31. Ran Raz: The BNS-Chung criterion for multi-party communication complexity, Comput. Complexity9(2) (2000), 113–122.

    Article  MATH  MathSciNet  Google Scholar 

  32. Alexander Razborov and Avi Wigderson: nΩ(log n) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom, Inform. Process. Lett.45(6) (1993), 303–307.

    Article  MATH  MathSciNet  Google Scholar 

  33. Alexander A. Sherstov: Communication lower bounds using dual polynomials, Bulletin of the EATCS, 95 (2008), 59–93.

    MATH  MathSciNet  Google Scholar 

  34. Alexander A. Sherstov: The pattern matrix method for lower bounds on quantum communication, in: 40th Annual Symposium on the Theory of Computing (STOC), pages 85–94, ACM, 2008.

  35. Alexander A. Sherstov: Separating AC0 from depth-2 majority circuits, SIAM J. Comput.38(6) (2009), 2113–2129.

    Article  MATH  MathSciNet  Google Scholar 

  36. Emanuele Viola: Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates, SIAM Journal on Computing36(5) (2007), 1387–1403.

    Article  MATH  MathSciNet  Google Scholar 

  37. Emanuele Viola and Avi Wigderson: One-way multiparty communication lower bound for pointer jumping with applications, in: 48th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 2007.

  38. Emanuele Viola and Avi Wigderson: Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols, Theory of Computing4 (2008), 137–168.

    Article  MathSciNet  Google Scholar 

  39. Andrew Chi-Chih Yao: Some complexity questions related to distributive computing, in: 11th ACM symposium on theory of computing (STOC), pages 209–213, ACM Press, 1979.

  40. Andrew Chi-Chih Yao: On ACC and threshold circuits, in: Proc. 31st Ann. IEEE Symp. Found. Comput. Sci., pages 619–627, 1990.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Emanuele Viola & Avi Wigderson

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  1. Emanuele Viola
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  2. Avi Wigderson
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Correspondence to Emanuele Viola.

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The author is supported by NSF grant CCF-0845003. This work was partially done while the author was a postdoctoral fellow at the Institute for Advanced Study, supported by NSF grant CCR-0324906.

The author is supported by NSF grant CCR-0324906.

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Viola, E., Wigderson, A. One-way multiparty communication lower bound for pointer jumping with applications. Combinatorica 29, 719–743 (2009). https://doi.org/10.1007/s00493-009-2667-z

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