Improved Upper Bounds on the Simultaneous Messages Complexity of the Generalized Addressing Function

Abstract

We study communication complexity in the model of Simultaneous Messages (SM). The SM model is a restricted version of the well-known multiparty communication complexity model [CFL,KN]. Motivated by connections to circuit complexity, lower and upper bounds on the SM complexity of several explicit functions have been intensively investigated in [PR,PRS,BKL,Am1,BGKL].

A class of functions called the Generalized Addressing Functions (GAF), denoted GAF_{G, k}, where G is a finite group and k denotes the number of players, plays an important role in SM complexity. In particular, lower bounds on SM complexity of GAF_G,k were used in [PRS] and [BKL] to show that the SM model is exponentially weaker than the general communication model [CFL] for sufficiently small number of players. Moreover, certain unexpected upper bounds from [PRS] and [BKL] on SM complexity of GAF_{G, k} have led to refined formulations of certain approaches to circuit lower bounds.

In this paper, we show improved upper bounds on the SM complexity of ${\rm GAF}_{{\mathbb Z}_2^t,k}$. In particular, when there are three players (k = 3), we give an upper bound of O(n ^0.73), where n = 2^t. This improves a bound of O(n ^0.92) from [BKL]. The lower bound in this case is $\Omega(\sqrt{n})$ [BKL,PRS]. More generally, for the k player case, we prove an upper bound of O(n ^{H(1/(2 k− − 2))}) improving a bound of O(n ^{H(1/ k)}) from [BKL], where H(.) denotes the binary entropy function. For large enough k, this is nearly a quadratic improvement. The corresponding lower bound is Ω(n ^{1/( k− − 1)}/(k − 1)) [BKL,PRS]. Our proof extends some algebraic techniques from [BKL] and employs a greedy construction of covering codes.