Carsten Damm

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, the ith player 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 general 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 prescribed order: 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 order n

Automata That Take Advice

Omega(n/k^2)

Some Bounds on Multiparty Communication Complexity of Pointer Jumping

for any n

On the average sensitivity of testing square-free numbers

k = O(n^{1/3-varepsilon})

On covering and rank problems for boolean matrices and their applications

.¶(2) Matching upper and lower bounds of order n

Inductive Counting for Width-Restricted Branching Programs

Theta(n,{rm log}^{(k-1)}n)

Inductive Counting Below LOGSPACE

for n

On Boolean vs. Modular Arithmetic for Circuits and Communication Protocols

k,{rmle,log^ast},n

Expressing uniformity via oracles

.

Depth-efficient simulation of Boolean semi-unbounded circuits by arithmetic ones

Karp and Lipton introduced advice-taking Turing machines to capture nonuniform complexity classes. We study this concept for automata-like models and compare it to other nonuniform models studied in connection with formal languages in the literature. Based on this we obtain complete separations of the classes of the Chomsky hierarchy relative to advices.