arXiv 2008.041482020-08cs.DCcs.DS

改进的分布式负载均衡边界

Improved Bounds for Distributed Load Balancing

Sepehr Assadi, Aaron Bernstein, Zachary Langley

摘要：

在负载均衡问题中，输入是一个$n$ -顶点二分图$G = (C \cup S, E)$和每个客户端$c \in C$的正权重.算法必须将每个客户端$c \in C$分配给相邻的服务器$s \in S$。然后，服务器的负载是分配给它的所有客户端的加权和，并且目标是计算使服务器负载的某个函数最小化的分配，通常是最大服务器负载（即，$\ell_{\infty}$ -范数）或服务器负载的$\ell_p$ -范数。我们研究分布式环境下的负载均衡。在CONGEST模型中有两个现有的结果。Czygrinow等人[DISC 2012]展示了具有轮复杂度$O(Δ^5)$的未加权客户端的2-近似，其中$Δ$是输入图的最大度。Halldórsson等人[SPAA 2015]显示了未加权客户端的$O(\log{n}/\log\log{n})$近似和加权客户端的$O(\log^2\!{n}/\log\log{n})$近似，具有轮复杂度polylog $(n)$。在本文中，我们展示了第一个分布式算法来计算$O(1)$ -近似的负载均衡问题的polylog $(n)$轮。在CONGEST模型中，我们给出了一个$O(1)$ -近似算法在polylog $(n)$轮未加权的客户端。对于加权客户端，近似比率为$O(\log{n})$。在约束较少的情况下，给出了加权客户端在polylog $(n)$轮中的$O(1)$ -近似算法。我们的方法也有影响的标准顺序设置中，我们获得的第一个$O(1)$ -近似这个问题，在近线性时间运行。2-近似是已知的，但它需要求解线性规划，因此要慢得多。最后，我们注意到，我们所有的结果同时逼近所有的$\ell_p$ -范数，包括$\ell_{\infty}$ -范数。

英文摘要：

In the load balancing problem, the input is an $n$-vertex bipartite graph $G = (C \cup S, E)$ and a positive weight for each client $c \in C$. The algorithm must assign each client $c \in C$ to an adjacent server $s \in S$. The load of a server is then the weighted sum of all the clients assigned to it, and the goal is to compute an assignment that minimizes some function of the server loads, typically either the maximum server load (i.e., the $\ell_{\infty}$-norm) or the $\ell_p$-norm of the server loads. We study load balancing in the distributed setting. There are two existing results in the CONGEST model. Czygrinow et al. [DISC 2012] showed a 2-approximation for unweighted clients with round-complexity $O(Δ^5)$, where $Δ$ is the maximum degree of the input graph. Halldórsson et al. [SPAA 2015] showed an $O(\log{n}/\log\log{n})$-approximation for unweighted clients and $O(\log^2\!{n}/\log\log{n})$-approximation for weighted clients with round-complexity polylog$(n)$. In this paper, we show the first distributed algorithms to compute an $O(1)$-approximation to the load balancing problem in polylog$(n)$ rounds. In the CONGEST model, we give an $O(1)$-approximation algorithm in polylog$(n)$ rounds for unweighted clients. For weighted clients, the approximation ratio is $O(\log{n})$. In the less constrained LOCAL model, we give an $O(1)$-approximation algorithm for weighted clients in polylog$(n)$ rounds. Our approach also has implications for the standard sequential setting in which we obtain the first $O(1)$-approximation for this problem that runs in near-linear time. A 2-approximation is already known, but it requires solving a linear program and is hence much slower. Finally, we note that all of our results simultaneously approximate all $\ell_p$-norms, including the $\ell_{\infty}$-norm.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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