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

• Authors: Emanuele Viola;Avi Wigderson
• Venue: Combinatorica
• Year: 2009

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 k3 [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.