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 GAFG, 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 GAFG, 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 GAFG, 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 GAF Zt2, k. In particular, when there are three players (k = 3), we give an upper bound of O(n0.73), where n = 2t. This improves a bound of O(n0.92) from [BKL]. The lower bound in this case is ω (√n) [BKL, PRS]. More generally, for the k player case, we prove an upper bound of O(nH (1/(2k-2))) improving a bound of O(nH(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(n1/(k-1)/(k - 1)) [BKL, PRS]. Our proof extends some algebraic techniques from [BKL] and employs a greedy construction of covering codes. © Springer-Verlag Berlin Heidelberg 2000.