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Register a new userNear-Optimal Scheduling in the Congested Clique
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Volodymyr Polosukhin
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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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 $O(mathsf{GlobalCongestion} + mathsf{dilation}cdotlog{n}+t)$ 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 emph{random-delay-based} scheduling algorithm [Ghaffari, PODC15] from the $mathsf{CLIQUE}$ model and obtain an algorithm that schedules any $t$ jobs in $O(t / n + mathsf{LocalCongestion} + mathsf{dilation}cdotlog{n})$ 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, PODC18], and we show that we can solve $t$ instances of MIS in $O(t + loglogDeltalog{n})$ rounds, that is, in $O(1)$ amortized time, for $tgeq loglogDeltalog{n}$.
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In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/omega})$ round matrix multiplication algorithm, where $omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.
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The coflow scheduling problem has emerged as a popular abstraction in the last few years to study data communication problems within a data center. In this basic framework, each coflow has a set of communication demands and the goal is to schedule many coflows in a manner that minimizes the total weighted completion time. A coflow is said to complete when all its communication needs are met. This problem has been extremely well studied for the case of complete bipartite graphs that model a data center with full bisection bandwidth and several approximation algorithms and effective heuristics have been proposed recently. In this work, we study a slightly different model of coflow scheduling in general graphs (to capture traffic between data centers) and develop practical and efficient approximation algorithms for it. Our main result is a randomized 2 approximation algorithm for the single path and free path model, significantly improving prior work. In addition, we demonstrate via extensive experiments that the algorithm is practical, easy to implement and performs well in practice.
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