Abstract
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.
Highlights
How much information can one derive deterministically and distributedly from an arbitrarily evolving connected graph in polynomial time? Can we learn its size, or compute some simple Boolean functions, on its input? In this work, we answer this question, posed in [20], in the affirmative
The maximum length of an opportunistic shortest path between any pair of nodes over many time slots is called the chronopath [13] and it is denoted as D. 156:4 Polynomial Counting in Anonymous Dynamic Networks
We present a deterministic distributed algorithm, which we call Methodical Counting, to compute the number of nodes in an Anonymous Dynamic Network
Summary
How much information can one derive deterministically and distributedly from an arbitrarily evolving connected graph in polynomial time? Can we learn its size, or compute some simple Boolean functions, on its (distributed) input? In this work, we answer this question, posed in [20], in the affirmative. Previous papers have either weaken the objective (e.g., computing only upper bound, only stochastic guarantees, etc.), assumed availability of network information (e.g., maximum number of neighbors, size upper bound, etc.), relied on a stronger model of communication, or provided only superpolynomial time guarantees. An algorithm solves the Counting Problem if, after completing its execution, all nodes have obtained the exact size of the network and stop. At any given round the topology is such that there is a path, i.e., a sequence of links, between each pair of nodes, but the set of links may change arbitrarily from round to round This adversarial model of dynamics was called 1-interval connectivity in [19]. The maximum length of a shortest path between any pair of nodes at any given time is called the dynamic diameter and it is denoted as D.
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