Efficient Counting with Optimal Resilience

  • Christoph Lenzen
  • Joel Rybicki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9363)


In the synchronous c-counting problem, we are given a synchronous system of n nodes, where up to f of the nodes may be Byzantine, that is, have arbitrary faulty behaviour. The task is to have all of the correct nodes count modulo c in unison in a self-stabilising manner: regardless of the initial state of the system and the faulty nodes’ behavior, eventually rounds are consistently labelled by a counter modulo c at all correct nodes.

We provide a deterministic solution with resilience \(f<n/3\) that stabilises in O(f) rounds and every correct node broadcasts \(O(\log ^2 f)\) bits per round. We build and improve on a recent result offering stabilisation time O(f) and communication complexity \(O(\log ^2 f /\log \log f)\) but with sub-optimal resilience \(f = n^{1-o(1)}\) (PODC 2015). Our new algorithm has optimal resilience, asymptotically optimal stabilisation time, and low communication complexity. Finally, we modify the algorithm to guarantee that after stabilisation very little communication occurs. In particular, for optimal resilience and polynomial counter size \(c=n^{O(1)}\), the algorithm broadcasts only O(1) bits per node every \(\Theta (n)\) rounds without affecting the other properties of the algorithm; communication-wise this is asymptotically optimal.


Stabilisation Time Counting Problem Faulty Node Consensus Protocol Consecutive Round 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Max Planck Institute for InformaticsSaarbrückenGermany
  2. 2.Helsinki Institute for Information Technology HIIT, Department of Computer ScienceAalto UniversityEspooFinland

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