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 (WFT) 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: \textsf{GOSSIP-reply}{} and \textsf{HYBRID}{}. In the \textsf{GOSSIP-reply}{} model, each node can send a message and receive a corresponding reply message in one round. In the \textsf{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 for WFT construction with $O\left(n \log n\right)$ message complexities using messages of $O(\log n)$ bits. In the \textsf{GOSSIP-reply}{} model, our protocol takes $O(\log n)$ rounds while in the \textsf{HYBRID} model, our protocol takes $O(\log^2 n)$ rounds. Both protocols use $O\left(n \log^2 n\right)$ bits of communication.

翻译：我们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以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成期望的拓扑结构，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。我们在两种模型下研究包含 $n$ 个节点的网络中的覆盖重构问题：\textsf{GOSSIP-reply}{} 模型与 \textsf{HYBRID}{} 模型。在 \textsf{GOSSIP-reply}{} 模型中，每个节点每轮可发送一条消息并接收对应的回复消息。在 \textsf{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，并在全局信道中发送总计 $O(\log n)$ 条消息。针对两种模型，我们提出了构建良构树的协议，其消息复杂度为 $O\left(n \log n\right)$，消息长度为 $O(\log n)$ 比特。在 \textsf{GOSSIP-reply}{} 模型中，协议需 $O(\log n)$ 轮；而在 \textsf{HYBRID} 模型中，协议需 $O(\log^2 n)$ 轮。两种协议均使用 $O\left(n \log^2 n\right)$ 比特的通信量。