Near-Optimal Scheduling in the Congested Clique
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This paper provides three nearly-optimal algorithms for scheduling t jobs in the π’π«π¨π°π΄π€ model. First, we present a deterministic scheduling algorithm that runs in O(π¦π ππ»πΊπ π’ππππΎπππππ + π½ππ πΊππππ) rounds for jobs that are sufficiently efficient in terms of their memory. The π½ππ πΊππππ is the maximum round complexity of any of the given jobs, and the π¦π ππ»πΊπ π’ππππΎπππππ is the total number of messages in all jobs divided by the per-round bandwidth of n^2 of the π’π«π¨π°π΄π€ model. Both are inherent lower bounds for any scheduling algorithm. Then, we present a randomized scheduling algorithm which runs t jobs in O(π¦π ππ»πΊπ π’ππππΎπππππ + π½ππ πΊππππΒ·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 π’π«π¨π°π΄π€ model and obtain an algorithm that schedules any t jobs in O(t / n + π«ππΌπΊπ π’ππππΎπππππ + π½ππ πΊππππΒ·logn) rounds, where the π«ππΌπΊπ π’ππππΎπππππ relates to the congestion at a single node of the π’π«π¨π°π΄π€. 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 + loglogΞlogn) rounds, that is, in O(1) amortized time, for tβ₯loglogΞlogn.
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