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
Polynomial Counting in Anonymous Dynamic Networks with Applications to Anonymous Dynamic Algebraic Computations
摘要：
从Michail、Chatzigiannakis和Spirakis的工作开始，匿名动态网络中节点数的计算问题就引起了人们的广泛关注。这个问题是具有挑战性的，因为节点是不可区分的（它们缺乏标识符并执行相同的程序），并且只要网络在每一轮中都是连接的，拓扑结构就可以从一轮通信到另一轮通信任意变化。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策，如终止、协议、同步等。建立在质量分布技术之上的各种算法已经被提出、分析和实验评估;其中一些算法假设了网络特性的额外知识，例如有界度或给定的网络大小上限。然而，计数是否可以在次指数时间内确定性地解决的问题仍然是开放的。在这项工作中，我们回答了这个问题，积极提出方法计数，它运行在多项式时间，不需要网络特性的知识。此外，我们还展示了如何扩展方法计数来计算输入值和更复杂的函数的总和，而无需额外的成本。我们的分析利用了以前在进化图中随机游走的工作，结合了控制过程及其参数的算法中精心选择的警报。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实践中实现最坏情况保证的算法。
英文摘要：
Starting with Michail, Chatzigiannakis, and Spirakis work, 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, such as termination, agreement, synchronization, and many others. A variety of algorithms built on top of mass-distribution techniques have been presented, analyzed, and also 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 Methodical Counting, which runs in polynomial time and requires no knowledge of network characteristics. Moreover, we also show how to extend Methodical Counting 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.
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