Simultaneous Multiparty Communication Complexity of Composed Functions

The Number On the Forehead (NOF) model is a multiparty communication model introduced by Chandra, Furst and Lipton [CFL83] that generalizes the two player communication game of Yao [Yao79]. In this model, $k$ players are given $k$ inputs $x_{1}\in\mathcal{X}_{1},\dots,x_{k}\in\mathcal{X}_{k}$ on which they want to compute some function $F:\mathcal{X}_{1}\times\cdots\times\mathcal{X}_{k}\rightarrow\mathcal{Y}$. Each player $i$ sees all of the input $(x_{1},\dots,x_{k})$, except $x_{i}$. The situation is as if input $x_{i}$ is written on the forehead of player $i$.

In order to collaboratively evaluate $F(x_{1},\dots,x_{k})$, the players communicate by broadcasting bits according to a predetermined protocol. This protocol specifies whose turn it is to speak, and which bit is to be sent given the information exchanged so far and the input seen by the speaking player. It also determines when communication stops. At the end, all the players must be able to recover $F(x_{1},\dots,x_{k})$ from the input they see and the transcript of the exchange. The cost of the protocol on input $(x_{1},\dots,x_{k})$ is the number of exchanged bits, and the total cost is the worst case cost on all inputs. The $k$-party deterministic communication complexity of $F$, denoted $\mathsf{D}_{k}(F)$, is the cost of the most efficient protocol computing $F$.

In most of the settings, the $x_{i}$’s are $\operatorname{poly}n$-bits long (for some parameter $n$) and $\mathcal{Y}=\{0,1\}$. In this case, the naive protocol is to broadcast first the entire input $x_{1}$ (this can be done by player 2), and then player 1 computes $F(x_{1},\dots,x_{k})$ and sends the result to the other players. This protocol has cost $m+1$ (where $m=\operatorname{poly}(n)$ is the number of bits required for sending $x_{1}$), which proves $\mathsf{D}_{k}(F)=\mathcal{O}\left(\operatorname{poly}n\right)$. Consequently, a protocol will be said to be efficient if it has cost $\mathcal{O}\left(\operatorname{polylog}n\right)$ (i.e. we seek for exponential speed-up over the naive protocol).

Among the many variants of the previous framework (randomized, quantum, etc.), we will be interested in the simultaneous (or Simultaneous Message Passing - SMP) model [Yao79, NW93, BKL95, PRS97] in which the players cannot speak to each other but instead send information to a referee. The referee does not know the players’ inputs, and cannot give any information back. At the end, the referee must be able to recover $F(x_{1},\dots,x_{k})$ from what she obtained from the players. The simultaneous deterministic communication complexity is denoted $\mathsf{D}_{k}^{||}(F)$, and it always satisfies $\mathsf{D}_{k}(F)\leq\mathsf{D}_{k}^{||}(F)$. It has often been easier to reason first in this weaker model for proving lower bounds [BGKL04, PRS97, BPSW05, BYJKS02]. It is also more suitable and fruitful for studying certain functions, such as Equality in the two party setting [Yao79, Amb96, NS96, BK97, BCWdW01, GRd08, BGK15]. We will show in the next section that the simultaneous deterministic communication model is also closely connected to lower bound results in the complexity class $\mathsf{ACC}^{0}$.

The NOF model has proved to be of value in the study of many areas of computer science, such as branching programs [CFL83], Ramsey theory [CFL83], circuit complexity [HG91, BT94], quasirandom graphs [CT93], proof complexity [BPS07], etc. One of the most interesting connections, pointed out by Håstad and Goldmann [HG91] and refined in [BGKL04], is a way to derive lower bounds for the complexity class¹¹1$\mathsf{ACC}_{0}$ refers to the functions computable by constant-depth poly-size circuits with unbounded fan-in And, OR, Not and $\textsc{Mod}_{m}$ gates (where $\textsc{Mod}_{m}$ outputs $0$ iff the sum of its inputs is divisible by $m$). $\mathsf{ACC}^{0}$ from lower bounds in the simultaneous NOF model. More precisely, according to a result from Yao, Beigel and Tarui [Yao90, BT94], any function $f\in\mathsf{ACC}^{0}$ can be expressed as a depth-2 circuit whose top gate is a symmetric gate of fan-in $2^{\log^{c}n}$, and each bottom gate is an And gate of fan-in $\log^{d}n$ (for some constants $c,d$). Consequently, for any partition of the input of $f$ between $k=\log^{d}n-1$ players in the simultaneous NOF model, there exists a partition of the And gates between the players such that each of them sees all the input bits she needs to evaluate the gates she received. The players can then send to the referee the number of gates that evaluate to 1, which enables the referee to compute $f$. The total cost of this protocol is $\mathcal{O}\left(k\log\left(2^{\log^{c}n}\right)\right)=\mathcal{O}\left(\log^{c+d}n\right)$. Conversely, any super-polylogarithmic lower bound in the simultaneous NOF model for a function $f$ and a partition of its input between $\operatorname{polylog}(n)$ players would imply $f\notin\mathsf{ACC}^{0}$.

Separating $\mathsf{ACC}^{0}$ from other complexity classes is a central question in complexity theory. It is conjectured that $\mathsf{ACC}^{0}$ does not contain the majority function Maj, but the only result known so far is $\mathsf{NEXP}\not\subset\mathsf{ACC}^{0}$ [Wil14]. The aforementioned connection with communication complexity has motivated the search for a function which is hard to compute for $k\geq\log n$ players in the simultaneous NOF model. This problem is called the $\log n$ barrier.

Obtaining lower bounds in the NOF model is a challenging task, as the current methods become very weak when $k\geq\log n$. The only general lower bound technique known so far is the discrepancy method and its variants [BNS92, CT93, Raz00, She11]. One of the early application of it was an $\Omega(n/4^{k})$ lower bound on the randomized complexity of the Generalized Inner Product (Gip) function [BNS92]. A long series of generalizations and improvements of the discrepancy method subsequently led to an $\Omega\left(\frac{\sqrt{n}}{k2^{k}}\right)$ (resp. $\Omega(n/4^{k})$) lower bound on the randomized (resp. deterministic) complexity of the Disjointness (Disj) function [Tes03, BPSW06, CA08, LS09, BH12, She16, She14, RY15]. It might seem like other lower bound arguments could prove that Gip and Disj remain hard for $k\geq\log n$ players. However, surprising non-simultaneous [Gro94, ACFN15] and simultaneous [BGKL04, ACFN15] protocols proved that the aforementioned lower bounds are nearly optimal, and that these two functions cannot break the $\log n$ barrier. Very recently, Podolskii and Sherstov [PS17] showed that the randomized complexity of Gip and Disj is exactly $\Theta\left(\frac{\log n}{\left\lceil 1+k/\log n\right\rceil}+1\right)$ when $k\geq\log n$, and built a function having complexity $\Omega(\log n)$ independently of $k$. Although these last results do not break the $\log n$ barrier, they are the first superconstant lower bounds proved for explicit functions when $k\geq\log n$.

An input $x_{1},\dots,x_{k}\in\{0,1\}^{n}$ to $k$ players in the NOF model can be visualized as a $k\times n$ boolean matrix $M$ where row $i$ is the number $x_{i}$ on the forehead of player $i$. The protocols known so far for Gip and Disj strongly rely on the particular way these functions act on matrix $M$. They both consist in applying the $g=\textsc{And}$ function on each of the $n$ columns of $M$, followed by the $f=\textsc{Mod}_{2}$ (for Gip) or $f=\textsc{Nor}$ (for Disj) function on the $n$ resulting bits. Since Gip and Disj do not break the $\log n$ barrier, a natural move has been to try other $f$ and $g$ functions, and to increase the number $t$ of columns on which each $g$ function applies. These are called the composed functions, formally defined below and depicted in Figure 1.

Both $\textsc{Gip}=\textsc{Mod}_{2}\circ\textsc{And}$ and $\textsc{Disj}=\textsc{Nor}\circ\textsc{And}$ are composed functions for $t=1$, with the additional property that $\textsc{Mod}_{2}$, Nor and And are symmetric functions (i.e. invariant under any permutation of their input). Since the majority function Maj is conjectured to be outside of $\mathsf{ACC}^{0}$, Babai et. al. [BKL95, BGKL04] suggested to look at $\textsc{Maj}\circ\textsc{Maj}_{t}$ and $\textsc{Maj}\circ\textsc{Thr}^{s}_{t}$ for breaking the $\log n$ barrier (where $\textsc{Maj}_{t}$ outputs 1 if at least $kt/2$ bits of the input block are set to 1, and $\textsc{Thr}^{s}_{t}(r_{1},\dots,r_{k})=1$ if $r_{1}+\dots+r_{k}\geq s$ for $r_{1},\dots,r_{k}$ seen as $t$-bits numbers).

Another way to look at composed functions of block-width $t$ is to interpret each sub-row $r\in\{0,1\}^{t}$ of each block as a number in $\mathbb{Z}_{d}$, where $d=2^{t}$. This representation of the input as a $k\times n$ matrix $M$ over some set $\mathbb{Z}_{d}$ is sometimes more convenient to use. Below, we reformulate Definition 1 using this point of view.

The set of all composed functions $f\circ\vec{g}$ (resp. $f\circ g$) over $\mathbb{Z}_{d}$ is denoted $\mathsf{ANY}\circ\overrightarrow{\mathsf{ANY}}_{\mathbb{Z}_{d}}$ (resp. $\mathsf{ANY}\circ\mathsf{ANY}_{\mathbb{Z}_{d}}$). Similarly, $\mathsf{SYM}\circ\mathsf{SYM}_{\mathbb{Z}_{d}}$ is the set of $f\circ g$ for symmetric $f$ and symmetric $g$ functions, $\mathsf{SYM}\circ\overrightarrow{\mathsf{ANY}}_{\mathbb{Z}_{d}}$ is the set of $f\circ\vec{g}$ for symmetric $f$ and any $\vec{g}$, etc. If $d=2$ (which corresponds to block-width $t=1$), we will drop the subscript and write $\mathsf{ANY}\circ\overrightarrow{\mathsf{ANY}}$, $\mathsf{SYM}\circ\mathsf{SYM}$, etc. We have for instance $\textsc{Gip},\textsc{Disj}\in\mathsf{SYM}\circ\mathsf{SYM}$ and $\textsc{Maj}\circ\textsc{Maj}_{t},\textsc{Maj}\circ\textsc{Thr}^{s}_{t}\in\mathsf{SYM}\circ\mathsf{SYM}_{\mathbb{Z}_{2^{t}}}$.

The first efficient protocol for composed functions with $\operatorname{polylog}(n)$ or more players was given by Grolmusz [Gro94]. It is a non-simultaneous protocol of cost $\mathcal{O}\left(\log^{2}n\right)$ for any composed function in $\mathsf{SYM}\circ\textsc{And}$ (the inner function is fixed to be And) when $k\geq\log n$. The study of composed functions with symmetric outer function $f$ was subsequently continued, as it captures many other interesting cases in communication complexity. Babai et. al. [BKL95] proposed first $\textsc{Maj}\circ\textsc{Maj}_{1}$ as a candidate to break the $\log n$ barrier. However, in a subsequent work [BGKL04], they found a simultaneous protocol that applies to $\mathsf{SYM}\circ\mathsf{COMP}^{c}$ (where $\mathsf{COMP}^{c}$ holds for $c$-compressible symmetric functions²²2A class $\mathscr{G}$ (parameterized by $k$) of symmetric functions $g:\{0,1\}^{k}\rightarrow\{0,1\}$ is $c$-compressible if for any function $g\in\mathscr{G}$, set $S\subsetneq\{1,\dots,k\}$ and input $(x_{i})_{i\in S}\in\{0,1\}^{|S|}$ there is a message $m_{S}$ of size $\mathcal{O}\left(1\right)+c\log(k-|S|)$ such that $g(x_{1},\dots,x_{k})$ can be computed for any $(x_{i})_{i\in\{1,\dots,k\}\backslash S}\in\{0,1\}^{k-|S|}$ from knowledge of $m_{S}$ and $(x_{i})_{i\in\{1,\dots,k\}\backslash S}$. The $\textsc{Maj}_{1}$ and $\textsc{Thr}^{s}_{1}$ functions are $1$-compressible [BGKL04]., a subset of $\mathsf{SYM}$ that contains Maj and And). It has cost $\mathcal{O}\left(\log^{2+c}n\right)$ when $k>1+\log n$. Later, Ada et. al. [ACFN15] generalized this result to $\mathsf{SYM}\circ\overrightarrow{\mathsf{ANY}}$, with a simultaneous protocol of cost $\mathcal{O}\left(\log^{3}n\right)$ for $k>1+2\log n$ players. The only protocol known so far for block-width $t>1$ has been discovered by Chattopadhyay and Saks [CS14]. It has cost $\mathcal{O}\left(d\log n\log(dn)\right)$ for $\mathsf{SYM}\circ\overrightarrow{\mathsf{ANY}}_{\mathbb{Z}_{d}}$ when $k>1+d\log(3n)$ (which is efficient for $d\leq\operatorname{polylog}n$). However, it is not simultaneous in the deterministic setting (the authors showed how to make it simultaneous using shared randomness between the players). Thus, none of these previous results prevents from breaking the $\log n$ barrier in the SMP model with composed functions of block-width as small as $t=2$. The goal of this paper is to rule out this possibility for all symmetric composed functions of constant block-width $t>1$.

Below, we describe our main results, and summarize in Table 2 the complexity of all the known protocols for composed functions. Then, we review the main ideas used in the previous literature, and we explain how we differ from them.

We extend the protocol for $\mathsf{SYM}\circ\mathsf{SYM}_{\mathbb{Z}_{2}}$ from [BGKL04] to any $d>1$. It applies to all functions in $\mathsf{SYM}\circ\mathsf{SYM}_{\mathbb{Z}_{d}}$ as long as $k\geq 4^{d-1}\log n$, but it is not efficient if $d$ is nonconstant or if the number $k$ of players is super-polylogarithmic (we will remove this last condition in the next section). For convenience in the proof, we state the result over $\mathbb{Z}_{d+1}$ instead of $\mathbb{Z}_{d}$: