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

Authors Dariusz R. Kowalski, Miguel A. Mosteiro

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Author Details

Dariusz R. Kowalski
  • Computer Science Department, University of Liverpool, Liverpool, UK
Miguel A. Mosteiro
  • Computer Science Department, Pace University, New York, NY, USA

Funding

  • Kowalski, Dariusz R.: Supported by Polish National Science Center grant UMO-2015/17/B/ST6/01897.
  • Mosteiro, Miguel A.: Research work partially supported by Pace University Scholarly Research Award P1258 and the UK Royal Society Grant IES\textbackslash{}R3\textbackslash{}170293.

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Dariusz R. Kowalski and Miguel A. Mosteiro. Polynomial Counting in Anonymous Dynamic Networks with Applications to Anonymous Dynamic Algebraic Computations. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 156:1-156:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.156

Abstract

Starting with Michail, Chatzigiannakis, and Spirakis work [Michail et al., 2013], 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • Anonymous Dynamic Networks
  • Counting
  • Boolean functions
  • distributed algorithms
  • deterministic algorithms

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