The NOF Multiparty Communication Complexity of Composed Functions

Abstract

We study the k-party ‘number on the forehead’ communication complexity of composed functions f ∘ g, where f:{0,1}ⁿ → {±1}, g : {0,1}^k → {0,1} and for (x ₁,…,x _k ) ∈ ({0,1}ⁿ)^k, f ∘ g(x ₁,…,x _k ) = f(…,g(x _1,i ,…,x _k,i ), …). We show that there is an O(log³ n) cost simultaneous protocol for $\textnormal{\textsc{sym}} \circ g$ when k > 1 + logn, $\textnormal{\textsc{sym}}$ is any symmetric function and g is any function. Previously, an efficient protocol was only known for $\textnormal{\textsc{sym}} \circ g$ when g is symmetric and “compressible”. We also get a non-simultaneous protocol for $\textnormal{\textsc{sym}} \circ g$ of cost O((n/2^k) logn + k logn) for any k ≥ 2.

In the setting of k ≤ 1 + logn, we study more closely functions of the form $\textnormal{\textsc{majority}} \circ g$, $\textnormal{\textsc{mod}}_m \circ g$, and $\textnormal{\textsc{nor}} \circ g$, where the latter two are generalizations of the well-known and studied functions Generalized Inner Product and Disjointness respectively. We characterize the communication complexity of these functions with respect to the choice of g. As the main application of our results, we answer a question posed by Babai et al. (SIAM Journal on Computing, 33:137–166, 2004) and determine the communication complexity of $\textnormal{\textsc{majority}} \circ \textnormal{\textsc{qcsb}}_k$, where $\textnormal{\textsc{qcsb}}_k$ is the “quadratic character of the sum of the bits” function.

A full version can be found online [1].