We focus on the well-studied problem of distributed overlay network construction. We consider a synchronous gossip-based communication model where in each round a node can send a message of small size to another node whose identifier it knows. The network is assumed to be reconfigurable, i.e., a node can add new connections (edges) to other nodes whose identifier it knows or drop existing connections. Each node initially has only knowledge of its own identifier and the identifiers of its neighbors. The overlay construction problem is, given an arbitrary (connected) graph, to reconfigure it to obtain a bounded-degree expander graph as efficiently as possible. The overlay construction problem is relevant to building real-world peer-to-peer network topologies that have desirable properties such as low diameter, high conductance, robustness to adversarial deletions, etc. Our main result is that we show that starting from any arbitrary (connected) graph $G$ on $n$ nodes and $m$ edges, we can construct an overlay network that is a constant-degree expander in polylog $n$ rounds using only $\tilde{O}(n)$ messages. Our time and message bounds are both essentially optimal (up to polylogarithmic factors). Our distributed overlay construction protocol is very lightweight as it uses gossip (each node communicates with only one neighbor in each round) and also scalable as it uses only $\tilde{O}(n)$ messages, which is sublinear in $m$ (even when $m$ is moderately dense). To the best of our knowledge, this is the first result that achieves overlay network construction in polylog $n$ rounds and $o(m)$ messages. Our protocol uses graph sketches in a novel way to construct an expander overlay that is both time and communication efficient. A consequence of our overlay construction protocol is that distributed computation can be performed very efficiently in this model.

翻译：我们关注已广泛研究的分布式覆盖网络构建问题。考虑一个基于同步闲聊的通信模型，在该模型中，每一轮节点可以向其已知标识符的另一个节点发送一个小尺寸消息。假设网络是可重构的，即节点可以添加已知标识符的其他节点的新连接（边）或删除现有连接。每个节点最初仅知道自身标识符及其邻居的标识符。覆盖网络构建问题是指，给定任意（连通）图，尽可能高效地将其重构为有界度扩展图。该问题与构建具有低直径、高电导、抗恶意删除等理想特性的实际点对点网络拓扑相关。我们的主要结果是：从任意（连通）图 $G$（含 $n$ 个节点和 $m$ 条边）出发，我们可以在 polylog $n$ 轮内仅使用 $\tilde{O}(n)$ 条消息构建一个常度扩展覆盖网络。我们的时间与消息复杂度（忽略多对数因子）均达到理论上界。该分布式覆盖构建协议非常轻量，因为它使用闲聊机制（每轮每个节点仅与一个邻居通信），同时具有可扩展性，仅使用 $\tilde{O}(n)$ 条消息，这相对于 $m$ 是次线性的（即使 $m$ 为中密度）。据我们所知，这是首个在 polylog $n$ 轮和 $o(m)$ 条消息下实现覆盖网络构建的结果。我们的协议以新颖方式使用图摘要来构建兼具时间与通信效率的扩展覆盖网络。该覆盖构建协议的一个推论是，在该模型中分布式计算可以非常高效地执行。