Abstract:
We consider the $k$-layer pointer jumping problem in the one-way multi-party number-on-the-forehead communication model. In this problem, the input is a layered directed graph with each vertex having outdegree $1$, shared amongst $k$ players: Player~$i$ knows all layers {\em except} the $i$th. The players must communicate, in the order $1,2,\ldots,k$, to determine the vertex reached by following edges from a special start vertex. This problem has been considered by a number of researchers in the past because sufficiently strong lower bounds for it would have major consequences in circuit complexity. We take an information complexity approach to this problem and obtain three lower bounds that improve upon earlier work. For myopic protocols (where players may see only one layer ahead but arbitrarily far behind), we greatly improve a lower bound due to Gronemeier (2006). Our new lower bound is $\Omega(n/k)$, where $n$ is the number of vertices per layer. For conservative protocols (where players may see arbitrarily far ahead but not behind, instead seeing only the vertex reached by following the pointers up to their layer), we extend an $\Omega(n/k^2)$ lower bound due to Damm, Jukna and Sgall (1998) so that it applies for all $k$. The above two bounds apply even to the Boolean version of pointer jumping. Our third lower bound is for the non-Boolean case and for $k\le\log^*n$. We obtain an $\Omega(n\log^{(k-1)}n)$ bound for myopic protocols. Damm et al.~had obtained a similar bound for deterministic conservative protocols. All our lower bounds apply directly to randomised protocols.