The NOF Multiparty Communication Complexity of Composed Functions

Abstract

We study the k-party “number on the forehead” communication complexity of composed functions ${f \circ \vec{g}}$, where ${f:\{0,1\}^n \to \{\pm 1\}}$, ${\vec{g} = (g_1,\ldots,g_n)}$, ${g_i : \{0,1\}^k \to \{0,1\}}$ and for ${(x_1,\ldots,x_k) \in (\{0,1\}^n)^k}$, ${f \circ \vec{g}(x_1,\ldots,x_k) = f(\ldots,g_i(x_{1,i},\ldots,x_{k,i}), \ldots)}$. When ${\vec{g} = (g, g,\ldots, g)}$, we denote ${f \circ \vec{g}}$ by ${f \circ g}$. We show that there is an ${O({\rm log}^3 n)}$ cost simultaneous protocol for SYM ${\circ g}$ when k >  1 + log  n, SYM is any symmetric function and g is any function. When k >  1 +  2 log  n, our simultaneous protocol applies to SYM ${\circ \, \vec{g}}$ with ${\vec{g}}$ being a vector of n arbitrary functions. We also get a non-simultaneous protocol for SYM ${\circ \, \vec{g}}$ of cost ${O(n/2^k \cdot {\rm log}\, n+ k {\rm log}\, n)}$ for any k ≥  2. In the setting of k ≤  1 + log  n, we study more closely functions of the form MAJORITY ${\circ g}$, MOD _m ${\circ g}$ and 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. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137–166, 2003) and determine the communication complexity of MAJORITY◦QCSB _k , where QCSB _k is the “quadratic character of the sum of the bits” function.

In the second part of our paper, we utilize the connection between the ‘number on the forehead’ model and Ramsey theory to construct a large set without a k-dimensional corner (k-dimensional generalization of a k-term arithmetic progression) in ${(\mathbb{F}_{2}^{n})^k}$, thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [N] ×  [N] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost ${O(\sqrt{n})}$ for the EXACT _N function. For x ₁,x ₂,x ₃ n-bit integers, EXACT _N (x ₁,x ₂,x ₃) = −1 iff x ₁ +  x ₂ +  x ₃ = N.