Efficient Assignment of Identities in Anonymous Populations

Authors Leszek Gąsieniec , Jesper Jansson , Christos Levcopoulos , Andrzej Lingas



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Author Details

Leszek Gąsieniec
  • Department of Computer Science, University of Liverpool, UK
Jesper Jansson
  • Graduate School of Informatics, Kyoto University, Japan
Christos Levcopoulos
  • Department of Computer Science, Lund University, Sweden
Andrzej Lingas
  • Department of Computer Science, Lund University, Sweden

Acknowledgements

The authors are thankful to the anonymous referees for their valuable comments.

Cite AsGet BibTex

Leszek Gąsieniec, Jesper Jansson, Christos Levcopoulos, and Andrzej Lingas. Efficient Assignment of Identities in Anonymous Populations. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.OPODIS.2021.12

Abstract

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. Their efficiency is expressed in terms of the number of states utilized by agents, the size of the range from which the labels are drawn, and the expected number of interactions required by our solutions. Our primary goal is to provide efficient protocols for this fundamental problem complemented with tight lower bounds in all the three aspects. W.h.p. (with high probability), our labeling protocols are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never update the label assigned to any single agent. We first present a silent w.h.p. and safe labeling protocol that draws labels from the range [1,2n]. Both the number of interactions required and the number of states used by the protocol are asymptotically optimal, i.e., O(n log n) w.h.p. and O(n), respectively. Next, we present a generalization of the protocol, where the range of assigned labels is [1,(1+ε) n]. The generalized protocol requires O(n log n / ε) interactions in order to complete the assignment of distinct labels from [1,(1+ε) n] to the n agents, w.h.p. It is also silent w.h.p. and safe, and uses (2+ε)n+O(n^c) states, for any positive c < 1. On the other hand, we consider the so-called pool labeling protocols that include our fast protocols. We show that the expected number of interactions required by any pool protocol is ≥ (n²)/(r+1), when the labels range is 1,… , n+r < 2n. Furthermore, we provide a protocol which uses only n+5√ n +O(n^c) states, for any c < 1, and draws labels from the range 1,… ,n. The expected number of interactions required by the protocol is O(n³). Once a unique leader is elected it produces a valid labeling and it is silent and safe. On the other hand, we show that (even if a unique leader is given in advance) any silent protocol that produces a valid labeling and is safe with probability > 1-(1/n), uses ≥ n+√{(n-1)/2}-1 states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. Finally, we show that for any silent and safe labeling protocol utilizing n+t < 2n states, the expected number of interactions required to achieve a valid labeling is ≥ (n²)/(t+1).

Subject Classification

ACM Subject Classification
  • Theory of computation
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Distributed computing models
Keywords
  • population protocol
  • state efficiency
  • time efficiency
  • one-way epidemics
  • leader election
  • agent identities

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References

  1. D. Alistarh, H. Attiya, S. Gilbert, A. Giurgiu, and R. Guerraoui. Fast randomized test-and-set and renaming. In Proc. of the International Symposium on Distributed Computing , DISC, volume 6343 of Lect. Notes in Comput. Sci., pages 94-108. Springer, 2010. Google Scholar
  2. D. Alistarh, O. Denysyuk, L. Rodrigues, and N. Shavit. Balls-into-leaves: Sub-logarithmic renaming in synchronous message-passing systems. In Proc. of the 2014 ACM Symposium on Principles of Distributed Computing, PODC, pages 232-241. ACM, 2014. Google Scholar
  3. D. Alistarh and R. Gelashvili. Recent algorithmic advances in population protocols. ACM SIGACT News, 49(3):63-73, 2018. Google Scholar
  4. D. Angluin, J. Aspnes, Z. Diamadi, M.J. Fischer, and R. Peralta. Computation in networks of passively mobile finite-state sensors. Distributed Computing, 18(4):235-253, 2006. Google Scholar
  5. D. Angluin, J. Aspnes, and D. Eisenstat. Fast computation by population protocols with a leader. In Proc. of the International Symposium on Distributed Computing , DISC, volume 4167 of Lect. Notes in Comput. Sci., pages 61-75. Springer, 2006. Google Scholar
  6. J. Aspnes, J. Beauquier, J. Burman, and D. Sohier. Time and space optimal counting in population protocols. In Proc. of the 20th International Conference on Principles of Distributed Systems,OPODIS, volume 70 of LIPIcs, pages 13:1-13:17. Dagstuhl - LZI, 2016. Google Scholar
  7. J. Beauquier, J. Burman, L. Rosaz, and B. Rozoy. Non-deterministic population protocols. In Proc. of the 20th International Conference on Principles of Distributed Systems, OPODIS, Lect. Notes in Comput. Sci., pages 61-75. Springer, 2012. Google Scholar
  8. P. Berenbrink, A. Brinkmann, R. Elsässer, T. Friedetzky, and L. Nagel. Randomized renaming in shared memory systems. In Proc. of the EEE Int. Parallel and Distributed Processing Symp., IPDPS, pages 542-549. IEEE Computer Society, 2015. Google Scholar
  9. P. Berenbrink, G. Giakkoupis, and P. Kling. Optimal time and space leader election in population protocols. In Proc. of the 52nd ACM-SIGACT Symposium on Theory of Computing, STOC, pages 119-129. ACM, 2020. Google Scholar
  10. P. Berenbrink, D. Kaaser, and T. Radzik. On counting the population size. In Proc. of the ACM Symposium on Principles of Distributed Computing, PODC, pages 43-52. ACM, 2019. Google Scholar
  11. J. Burman, J. Beauquier, and D. Sohier. Space-optimal naming in population protocols. In Proc. of the 33rd International Symposium on Distributed Computing, DISC, volume 146 of LIPIcs, pages 9:1-9:16. Dagstuhl - LZI, 2019. Google Scholar
  12. J. Burman, H. Chen, H. Chen, D. Doty, T. Nowak, E. Severson, and C. Xu. Time-optimal self-stabilizing leader election in population protocols. In Proc. of the ACM Symposium on Principles of Distributed Computing, PODC, pages 33-44. ACM, 2021. Google Scholar
  13. S. Cai, T. Izumi, and K. Wada. How to prove impossibility under global fairness: On space complexity of self-stabilizing leader election on a population protocol model. Theory of Computing Systems, 50:433-445, 2012. Google Scholar
  14. A. Castañeda, S. Rajsbaum, and M. Raynal. The renaming problem in shared memory systems: An introduction. Comput. Sci. Rev., 5(3):229-251, 2011. Google Scholar
  15. S. Dolev, M.G. Gouda, and M. Schneider. Memory requirements for silent stabilization. Acta Inf., 36(6):447-462, 1999. Google Scholar
  16. D. Doty and M. Eftekhari. A survey of size counting in population protocols. Theoretical Computer Science, 894:91-102, 2021. Google Scholar
  17. D. Doty, M. Eftekhari, O. Michail, P. G. Spirakis, and M. Theofilatos. Exact size counting in uniform population protocols in nearly logarithmic time. ArXiv, 2018. preprint. URL: http://arxiv.org/abs/1805.04832.
  18. R. Elsässer and T. Radzik. Recent results in population protocols for exact majority and leader election. Bull. EATCS, 126, 2018. Google Scholar
  19. L. Gąsieniec and G. Stachowiak. Enhanced phase clocks, population protocols, and fast space optimal leader election. J. ACM, 68(1):2:1-2:21, 2021. Google Scholar
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