We consider the problem of constructing distributed overlay networks, where nodes in a reconfigurable system can create or sever connections with nodes whose identifiers they know. Initially, each node knows only its own and its neighbors' identifiers, forming a local channel, while the evolving structure is termed the global channel. The goal is to reconfigure any connected graph into a desired topology, such as a bounded-degree expander graph or a well-formed tree with a constant maximum degree and logarithmic diameter, minimizing the total number of rounds and message complexity. This problem mirrors real-world peer-to-peer network construction, where creating robust and efficient systems is desired. We study the overlay reconstruction problem in a network of $n$ nodes in two models: \textbf{GOSSIP-reply} and \textbf{HYBRID}. In the \textbf{GOSSIP-reply} model, each node can send a message and receive a corresponding reply message in one round. In the \textbf{HYBRID} model, a node can send $O(1)$ messages to each neighbor in the local channel and a total of $O(\log n)$ messages in the global channel. In both models, we propose protocols with $O\left(\log^2 n\right)$ round complexities and $O\left(n \log^2 n\right)$ message complexities using messages of $O(\log n)$ bits. Both protocols use $O\left(n \log^3 n\right)$ bits of communication, which we conjecture to be optimal. Additionally, our approach ensures that the total number of messages for node $v$, with degree $\deg(v)$ in the initial topology, is bounded by $O\left(\deg(v) + \log^2 n\right)$ with high probability.

翻译：本文研究分布式覆盖网络的构建问题，其中可重构系统中的节点可与已知标识符的节点建立或断开连接。初始时，每个节点仅知晓自身及其邻居节点的标识符，形成局部信道，而演化中的结构称为全局信道。目标是将任意连通图重构为期望的拓扑结构（例如有界度扩展图或具有常数最大度与对数直径的良构树），同时最小化总轮数与消息复杂度。该问题反映了现实中对等网络的构建需求，即期望建立鲁棒且高效的系统。我们在两种模型中研究$n$个节点网络的覆盖重构问题：\textbf{GOSSIP-reply}模型与\textbf{HYBRID}模型。在\textbf{GOSSIP-reply}模型中，每个节点每轮可发送一条消息并接收对应回复；在\textbf{HYBRID}模型中，节点可向局部信道中的每个邻居发送$O(1)$条消息，并在全局信道中总计发送$O(\log n)$条消息。针对两种模型，我们提出了轮复杂度为$O\left(\log^2 n\right)$、消息复杂度为$O\left(n \log^2 n\right)$的协议，所用消息长度为$O(\log n)$比特。两种协议均使用$O\left(n \log^3 n\right)$比特的通信量，我们推测该结果已达最优。此外，我们的方法能以高概率保证：对于初始拓扑中度为$\deg(v)$的节点$v$，其消息总数上界为$O\left(\deg(v) + \log^2 n\right)$。