Some bounds on multiparty communication complexity of pointer jumping

Abstract

We introduce the model of conservative one-way multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping.

The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node.

In a conservative protocol Player i can see only the node reached by following the first i−1 edges and the edges on the jth layer for each j>i (compared to the general model where he sees edges of all layers except for the ith one). In a one-way protocol, each player communicates only once: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard.

Our main results are the following bounds on k-party conservative one-way communication complexity of pointer jumping with k layers:

• (1) A lower bound of Ω(n/k ²) for any $k = O(n^{1/3 - \varepsilon } )$. This is the first lower bound on multiparty communication complexity that works for more than log n players.

• (2) Matching upper and lower bounds of Θ(n log^(k−1) n) for k≤log^* n. No better one-way protocols are known, even if we consider non-conservative ones.

On leave from Institute of Mathematics, Vilnius, Lithuania.

This work was done at Institute of Computer Science, Hebrew University, Jerusalem, Israel, supported in part by Golda Meir Postdoctoral Fellowship.