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Starting with Michail, Chatzigiannakis, and Spirakis work, the problem ofCounting the number of nodes in Anonymous Dynamic Networks has attracted a lotof attention. The problem is challenging because nodes are indistinguishable(they lack identifiers and execute the same program) and the topology maychange arbitrarily from round to round of communication, as long as the networkis connected in each round. The problem is central in distributed computing asthe number of participants is frequently needed to make important decisions,such as termination, agreement, synchronization, and many others. A variety ofalgorithms built on top of mass-distribution techniques have been presented,analyzed, and also experimentally evaluated; some of them assumed additionalknowledge of network characteristics, such as bounded degree or given upperbound on the network size. However, the question of whether Counting can besolved deterministically in sub-exponential time remained open. In this work,we answer this question positively by presenting Methodical Counting, whichruns in polynomial time and requires no knowledge of network characteristics.Moreover, we also show how to extend Methodical Counting to compute the sum ofinput values and more complex functions without extra cost. Our analysisleverages previous work on random walks in evolving graphs, combined withcarefully chosen alarms in the algorithm that control the process and itsparameters. To the best of our knowledge, our Counting algorithm and itsextensions to other algebraic and Boolean functions are the first that can beimplemented in practice with worst-case guarantees.

Polynomial Counting in Anonymous Dynamic Networks with Applications to Anonymous Dynamic Algebraic Computations