Near-Optimal Scheduling in the Congested Clique

TL;DR

This paper introduces three nearly-optimal algorithms for scheduling multiple jobs in the congested clique model, improving efficiency and adaptability without prior knowledge of job parameters, and demonstrating applications to maximum independent set problems.

Contribution

The paper presents three new scheduling algorithms for the congested clique model that are nearly optimal and adapt to unknown job parameters, with applications to MIS.

Findings

Algorithms are nearly optimal given inherent lower bounds.

The randomized algorithm handles inputs with O(n log n) bits per node.

On-the-fly scheduling achieves O(1) amortized time for multiple MIS instances.

Abstract

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 $t\geq \log\log\Delta\log{n}$.