Starting with with work of Michail et al., the problem of Counting the number of nodes in Anonymous Dynamic Networks has attracted a lot of attention. The problem is challenging because nodes are indistinguishable (they lack identifiers and execute the same program), and the topology may change arbitrarily from round to round of communication, as long as the network is connected in each round. The problem is central in distributed computing, as the number of participants is frequently needed to make important decisions, including termination, agreement, synchronization, among others. A variety of distributed algorithms built on top of mass-distribution techniques have been presented, analyzed, and experimentally evaluated; some of them assumed additional knowledge of network characteristics, such as bounded degree or given upper bound on the network size. However, the question of whether Counting can be solved deterministically in sub-exponential time remained open. In this work, we answer this question positively by presenting M ethodical C ounting , which runs in polynomial time and requires no knowledge of network characteristics. Moreover, we also show how to extend M ethodical C ounting to compute the sum of input values and more complex functions without extra cost. Our analysis leverages previous work on random walks in evolving graphs, combined with carefully chosen alarms in the algorithm that control the process and its parameters. To the best of our knowledge, our Counting algorithm and its extensions to other algebraic and Boolean functions are the first that can be implemented in practice with worst-case guarantees.

中文翻译：

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匿名动态网络中的多项式计数及其在匿名动态代数计算中的应用

从 Michail 等人的工作开始，问题数数匿名动态网络中的节点数量引起了很多关注。这个问题具有挑战性，因为节点是不可区分的（它们缺少标识符并执行相同的程序），并且拓扑可能在每一轮通信中任意变化，只要网络在每一轮中都连接。这个问题是分布式计算的核心，因为参与者的数量经常需要做出重要的决定，包括终止、协议、同步等。多种分布式算法建立在大规模分布技术已经被介绍、分析和实验评估；他们中的一些人假设了网络特征的额外知识，例如有界度或给定网络大小的上限。然而，计数是否可以在亚指数时间内确定性地解决的问题仍然悬而未决。在这项工作中，我们通过呈现 M道德的C计数，它在多项式时间内运行，不需要了解网络特征。此外，我们还展示了如何扩展 M道德的C计数无需额外成本即可计算输入值和更复杂函数的总和。我们的分析利用了先前在演化图中随机游走的工作，并结合了算法中精心挑选的控制过程及其参数的警报。据我们所知，我们的计数算法及其对其他代数和布尔函数的扩展是第一个可以在最坏情况保证的情况下在实践中实现的算法。