Fast convergence of k-undecided state dynamics in the population protocol model

Talley Amir, James Aspnes, Petra Berenbrink, Felix Biermeier, Christopher Hahn, Dominik Kaaser, and John Lazarsfeld. Fast convergence of k-undecided state dynamics in the population protocol model. 2023 ACM Symposium on Principles of Distributed Computing, June 2023, pp. 13–23. Submitted to Distributed Computing (PODC 2022 special issue).

Abstract

We analyze the convergence of the k-opinion Undecided State Dynamics (USD) in the population protocol model. For k=2 opinions it is well known that the USD reaches consensus with high probability within O(n log n) interactions. Proving that the process also quickly solves the consensus problem for k > 2 opinions has remained open, despite analogous results for larger k in the related parallel gossip model. In this paper we prove such convergence: under mild assumptions on k and on the initial number of undecided agents we prove that the USD achieves plurality consensus within O(k n log n) interactions with high probability, regardless of the initial bias. Moreover, if there is an initial additive bias of at least Ω(√n log n), we prove that the initial plurality opinion wins with high probability, and if there is a multiplicative bias the convergence time is further improved. Note that this is the first result for k>2 for the USD in the population protocol model. Furthermore, it is the first result for the unsynchronized variant of the USD with k>2 which does not need any initial bias.

• PODC 2023 proceedings version: PDF.
• ArXiv version: PDF.

BibTeX

@inproceedings{DBLP:conf/podc/AmirABBHKL23,
  author       = {Talley Amir and
                  James Aspnes and
                  Petra Berenbrink and
                  Felix Biermeier and
                  Christopher Hahn and
                  Dominik Kaaser and
                  John Lazarsfeld},
  editor       = {Rotem Oshman and
                  Alexandre Nolin and
                  Magn{\'{u}}s M. Halld{\'{o}}rsson and
                  Alkida Balliu},
  title        = {Fast Convergence of k-Opinion Undecided State Dynamics in the Population
                  Protocol Model},
  booktitle    = {Proceedings of the 2023 {ACM} Symposium on Principles of Distributed
                  Computing, {PODC} 2023, Orlando, FL, USA, June 19-23, 2023},
  pages        = {13--23},
  publisher    = {{ACM}},
  year         = {2023},
  url          = {https://doi.org/10.1145/3583668.3594589},
  doi          = {10.1145/3583668.3594589},
  timestamp    = {Mon, 19 Jun 2023 08:58:36 +0200},
  biburl       = {https://dblp.org/rec/conf/podc/AmirABBHKL23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}