One-Way Multi-Party Communication Lower Bound for Pointer Jumping with Applications

In this paper we study the one-way multi-party communication model, in which even party speaks exactly once in its turn. For every fixed k, we prove a tight lower hound of Omega (n^1/(k-1)) on the probabilistic communication complexity of pointer jumping in a k-layered tree, where the pointers of the i-lh layer reside on the forehead of the i-th party to speak. The lower bound remains nontrivial even for k = (log n)^1/2-Omega(1) parties. Previous to our work a lower bound was known only for k = 3 , and in very restricted models for k > 3. 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 (non one-wav) multi-party protocols of bounded rounds. Thus we generalize to this multi-party model results on two directions studied in the classical 2-party model. The first is a mund hierarchy: We give an exponential separation between the power of r and 2r rounds in general probabilistic k-party protocols, for any fixed 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 anvfixed 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 Omega (n^1/(k-1)) on the probabilistic complexity of k-set disjointness in the oneway 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. Finally, we infer an exponential separation between the power of different orders in which parties send messages in the one-way model, for every fixed k. Previous to our work such a separation was only known for k = 3. Our lower bound technique, which handles- functions of high discrepancy, may be of independent interest. It provides a "party-elimination " induction, based on a restricted form of a direct-product result, specific to the pointer jumping function.