我们研究并行\emph{Load Balancing}协议的客户-服务器的【形】分布式模型定义如下。存在一组$n$客户端$\sC$和一组$n$服务器$\sS$，其中每个客户端具有（最多）必须分配给某个服务器的恒定数量的请求$d \geq 1$。客户端集合和服务器集合通过固定的二分图相互连接：客户端$v$的请求只能发送到其邻居$N(v)$中的服务器。目标是分配每个客户端请求，以便最大限度地减少服务器的最大负载。在这种情况下，高效的并行协议只适用于密集拓扑。特别是，一个简单的对称，非自适应协议实现恒定的最大负载最近已被Becchetti等人\cite{BCNPT18}定期密集二分图。并行完成的时间是$\bigO(\log n)$和整体工作是$\bigO(n)$，WP。激发在一些客户-服务器的【形】系统中出现的邻近约束，我们设计了一个简单的变种Becchetti等人的协议\cite{BCNPT18}，我们分析它几乎经常二分图的节点可能有小尺寸的邻域。在细节上，我们证明，w. h. p.，这个新版本在每个度为$Ω(\log^2n)$的几乎正则二分图上具有与Becchetti等人的协议（分别在最大负载、完成时间和工作复杂度方面）相等的成本。我们的分析显着偏离，在\cite{BCNPT18}的原始协议，并需要应付非平凡的随机依赖问题的随机选择的算法过程中，这是由于最坏的情况下，稀疏的拓扑结构的基础图形。
英文摘要：
We study parallel \emph{Load Balancing} protocols for a client-server distributed model defined as follows.
There is a set $\sC$ of $n$ clients and a set $\sS$ of $n$ servers where each client has
(at most) a constant number $d \geq 1$ of requests that must be assigned to some server. The client set and the server one are connected to each other via a fixed bipartite graph: the requests of client $v$ can only be sent to the servers in its neighborhood $N(v)$. The goal is to assign every client request so as to minimize the maximum load of the servers.
In this setting, efficient parallel protocols are available only for dense topolgies. In particular, a simple symmetric, non-adaptive protocol achieving constant maximum load has been recently introduced by Becchetti et al \cite{BCNPT18} for regular dense bipartite graphs. The parallel completion time is $\bigO(\log n)$ and the overall work is $\bigO(n)$, w.h.p.
Motivated by proximity constraints arising in some client-server systems, we devise a simple variant of Becchetti et al's protocol \cite{BCNPT18} and we analyse it over almost-regular bipartite graphs where nodes may have neighborhoods of small size. In detail, we prove that, w.h.p., this new version has a cost equivalent to that of Becchetti et al's protocol (in terms of maximum load, completion time, and work complexity, respectively) on every almost-regular bipartite graph with degree $Ω(\log^2n)$.
Our analysis significantly departs from that in \cite{BCNPT18} for the original protocol and requires to cope with non-trivial stochastic-dependence issues on the random choices of the algorithmic process which are due to the worst-case, sparse topology of the underlying graph.
Balancing Similarity and Complementarity for Federated Learning
Institute for Artificial Intelligence, Tsinghua University（清华大学人工智能研究院）；Tsinghua University（清华大学）；The University of Auckland（奥克兰大学）；RIKEN（理化学研究所）；Hong Kong Baptist University（香港浸会大学）；The University of Tokyo（东京大学）
