Complexity and Quasi - randomness 1

Many problems arising in interactive and distributive computation share the general framework that a number of processors wish to collaboratively evaluate a Boolean function while each processor has only partial information. The question of interest is to determine the minimum amount of information transfer required under the assumption that each processor has unlimited computational power and the messages are transferred by a “blackboard”, viewed by all processors. One of the most interesting examples is the round-table model, proposed by Chandra, Furst and Lipton [CFL], involving k players each having a number Xi on his/her forehead; (so that the i-th player knows all numbers except for Xi). For k = 3, they proved a tight lower bound for the minimum number of bits to be exchanged to compute the sum of Xi’s. For general k, the lower bounds were further improved by Babai, Nisan and Szegedy [BNS] who gave a lower bound of Ω(m2−k) for computing some explicit functions on k strings m-bits each. When only two players are involved, it is just the usual model for communication complexity, which was first proposed by Yao [Y1] and has been studied extensively by many researchers [HMT, LS, MS, PS, Th]. In this paper we consider the following model generalizing both the round-table model and Yao’s model: