The Maximum Communication Complexity of Multi-Party Pointer Jumping

We study the one-way number-on-the-forhead (NOF) communication complexity of the $k$-layer pointer jumping problem. Strong lower bounds for this problem would have important implications in circuit complexity. All of our results apply to myopic protocols (where players see only one layer ahead, but can still see arbitrarily far behind them.) Furthermore, our results apply to the maximum communication complexity, where a protocol is charged for the maximum communication sent by a single player rather than the total communication sent by all players. Our main result is a lower bound of $n/2$ bits for deterministic protocols, independent of the number of players. We also provide a matching upper bound, as well as an $\Omega(n/k\log n)$ lower bound for randomized protocols, improving on the bounds of Chakrabarti. In the non-Boolean version of the problem, we give a lower bound of $n (\log^{(k-1)} n)(1-o(1))$ bits, essentially matching the upper bound from Damm et al.