Some bounds on multiparty communication complexity of pointer jumping

We introduce the model of conservative one-way multiparty complexity and pr ove lower and upper bounds on the complexity of pointer jumping. The pointer jumping function takes as its input a directed layered graph wi th a starting node and Ic layers of n nodes, and a single edge from each no de to one node from the next layer. The output is the node reached by follo wing Ic edges from the starting node. In a conservative protocol, the ith p layer can see only the node reached by following the first i - 1 edges and the edges on the jth layer for each j > i. This is a restriction of the gen eral model where the ith player sees edges of all layers except for the ith one. In a one-way protocol, each player communicates only once in a prescr ibed order: first Player I writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total numb er of bits written on the blackboard. Our main results are the following bounds on k-party conservative one-way c ommunication complexity of pointer jumping with k layers: (1) A lower bound of order Omega(n/k(2)) for any k = O(n(1/3-epsilon)). (2) Matching upper and lower bounds of order Theta(n log((k-1)) n) for k le ss than or equal to log* n.