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

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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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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Mathematics Subject Classification (2000)

  • 68Q99