The effect of range and bandwidth on the round complexity in the congested clique model

Abstract

The congested clique model is a message-passing model of distributed computation where $k$ players communicate with each other over a complete network. Here we consider synchronous protocols in which communication happens in rounds (we allow them to be randomized with public coins). In the \emph{unicast} communication mode, each player $i$ has her own $n$-bit input $x_i$ and may send $k-1$ different $b$-bit messages through each of her $k-1$ communication links in each round. On the other end is the \emph{broadcast} communication mode, where each player can only broadcast a single message over all her links in each round. The goal of this paper is to complete our Brief Announcement at PODC 2015, where we initiated the study of the space that lies between the two extremes. For that purpose, we parametrize the congested clique model by two values: the \emph{range} $r$, which is the maximum number of different messages a player is allowed to send in each round, and the \emph{bandwidth} $b$, which is the maximum size of these messages. We show that the space between the unicast and broadcast congested clique models is very rich and interesting. For instance, we show that the round complexity of the pairwise set-disjointness function $\textsc{pwdisj}$ is completely sensitive to the range $r$. This translates into a $\Omega(k)$ gap between the unicast ($r=k-1$) and the broadcast ($r=1$) modes. Moreover, provided that $r \geq 2$ and $rb/\log r = O(k)$, the round complexity of $\textsc{pwdisj}$ is $\Theta(n/ k \log r )$. On the other hand, we also prove that the behavior of $\textsc{pwdisj}$ is exceptional: almost every boolean function $f$ has maximal round complexity $\Theta(n/b)$. Finally, we prove that $\min \left(\left\lceil \frac{b'}{\lfloor \log r \rfloor} \right\rceil, \left\lceil\frac{r'}{r-1}\right\rceil\left\lceil\frac{b'}{b}\right\rceil\right)$ is an upper bound for the gap between the round complexities with parameters $(b,r)$ and parameters $(b',r')$ of any boolean function.