An anonymous dynamic network is a network of indistinguishable processes whose communication links may appear or disappear unpredictably over time. Previous research has shown that deterministically computing an arbitrary function of a multiset of input values given to these processes takes only a linear number of communication rounds (Di Luna-Viglietta, FOCS 2022). However, fast algorithms for anonymous dynamic networks rely on the construction and transmission of large data structures called "history trees", whose size is polynomial in the number of processes. This approach is unfeasible if the network is congested, and only messages of logarithmic size can be sent through its links. Observe that sending a large message piece by piece over several rounds is not in itself a solution, due to the anonymity of the processes combined with the dynamic nature of the network. Moreover, it is known that certain basic tasks such as all-to-all token dissemination (by means of single-token forwarding) require $\Omega(n^2/\log n)$ rounds in congested networks (Dutta et al., SODA 2013). In this work, we develop a series of practical and efficient techniques that make it possible to use history trees in congested anonymous dynamic networks. Among other applications, we show how to compute arbitrary functions in such networks in $O(n^3)$ communication rounds, greatly improving upon previous state-of-the-art algorithms for congested networks.

翻译：匿名动态网络是由不可区分进程组成的网络，其通信链路可能随时间不可预测地出现或消失。先前研究表明，确定性计算这些进程给定输入值多重集的任意函数仅需线性轮数的通信回合（Di Luna-Viglietta, FOCS 2022）。然而，匿名动态网络的快速算法依赖于构建和传输称为"历史树"的大型数据结构，其规模是进程数的多项式函数。若网络发生拥塞，且仅能通过链路发送对数级大小的消息时，该方法不可行。值得注意的是，由于进程的匿名性与网络的动态特性，将大消息分轮次逐片发送本身并非解决方案。此外，已知某些基本任务（如通过单令牌转发实现的全对全令牌分发）在拥塞网络中需要$\Omega(n^2/\log n)$轮次（Dutta 等，SODA 2013）。本文开发了一系列实用且高效的技术，使历史树能够应用于拥塞匿名动态网络。在其他应用中，我们展示了如何在$O(n^3)$通信轮次内计算此类网络中的任意函数，显著改进了先前拥塞网络的最优算法。