Near-Optimal Scheduling in the Congested Clique

This paper provides three nearly-optimal algorithms for scheduling $t$ jobs in the $\mathsf{CLIQUE}$ model. First, we present a deterministic scheduling algorithm that runs in $O(\mathsf{GlobalCongestion} + \mathsf{dilation})$ rounds for jobs that are sufficiently efficient in terms of their memory. The $\mathsf{dilation}$ is the maximum round complexity of any of the given jobs, and the $\mathsf{GlobalCongestion}$ is the total number of messages in all jobs divided by the per-round bandwidth of $n^2$ of the $\mathsf{CLIQUE}$ model. Both are inherent lower bounds for any scheduling algorithm. Then, we present a randomized scheduling algorithm which runs $t$ jobs in $O(\mathsf{GlobalCongestion} + \mathsf{dilation}\cdot\log{n}+t)$ rounds and only requires that inputs and outputs do not exceed $O(n\log n)$ bits per node, which is met by, e.g., almost all graph problems. Lastly, we adjust the \emph{random-delay-based} scheduling algorithm [Ghaffari, PODC'15] from the $\mathsf{CLIQUE}$ model and obtain an algorithm that schedules any $t$ jobs in $O(t / n + \mathsf{LocalCongestion} + \mathsf{dilation}\cdot\log{n})$ rounds, where the $\mathsf{LocalCongestion}$ relates to the congestion at a single node of the $\mathsf{CLIQUE}$. We compare this algorithm to the previous approaches and show their benefit. We schedule the set of jobs on-the-fly, without a priori knowledge of its parameters or the communication patterns of the jobs. In light of the inherent lower bounds, all of our algorithms are nearly-optimal. We exemplify the power of our algorithms by analyzing the message complexity of the state-of-the-art MIS protocol [Ghaffari, Gouleakis, Konrad, Mitrovic and Rubinfeld, PODC'18], and we show that we can solve $t$ instances of MIS in $O(t + \log\log\Delta\log{n})$ rounds, that is, in $O(1)$ amortized time, for $t\geq \log\log\Delta\log{n}$.