Near-Optimal Scheduling in the Congested Clique
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This paper provides three nearly-optimal algorithms for scheduling t jobs in the 𝖢𝖫𝖨𝖰𝖴𝖤 model. First, we present a deterministic scheduling algorithm that runs in O(𝖦𝗅𝗈𝖻𝖺𝗅𝖢𝗈𝗇𝗀𝖾𝗌𝗍𝗂𝗈𝗇 + 𝖽𝗂𝗅𝖺𝗍𝗂𝗈𝗇) rounds for jobs that are sufficiently efficient in terms of their memory. The 𝖽𝗂𝗅𝖺𝗍𝗂𝗈𝗇 is the maximum round complexity of any of the given jobs, and the 𝖦𝗅𝗈𝖻𝖺𝗅𝖢𝗈𝗇𝗀𝖾𝗌𝗍𝗂𝗈𝗇 is the total number of messages in all jobs divided by the per-round bandwidth of n^2 of the 𝖢𝖫𝖨𝖰𝖴𝖤 model. Both are inherent lower bounds for any scheduling algorithm. Then, we present a randomized scheduling algorithm which runs t jobs in O(𝖦𝗅𝗈𝖻𝖺𝗅𝖢𝗈𝗇𝗀𝖾𝗌𝗍𝗂𝗈𝗇 + 𝖽𝗂𝗅𝖺𝗍𝗂𝗈𝗇·logn+t) rounds and only requires that inputs and outputs do not exceed O(nlog n) bits per node, which is met by, e.g., almost all graph problems. Lastly, we adjust the random-delay-based scheduling algorithm [Ghaffari, PODC'15] from the 𝖢𝖫𝖨𝖰𝖴𝖤 model and obtain an algorithm that schedules any t jobs in O(t / n + 𝖫𝗈𝖼𝖺𝗅𝖢𝗈𝗇𝗀𝖾𝗌𝗍𝗂𝗈𝗇 + 𝖽𝗂𝗅𝖺𝗍𝗂𝗈𝗇·logn) rounds, where the 𝖫𝗈𝖼𝖺𝗅𝖢𝗈𝗇𝗀𝖾𝗌𝗍𝗂𝗈𝗇 relates to the congestion at a single node of the 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 + loglogΔlogn) rounds, that is, in O(1) amortized time, for t≥loglogΔlogn.
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