Abstract
We study the fundamental enumeration problem in asynchronous message-passing networks. Anonymous processes have to eventually decide on pairwise distinct identifiers, despite all starting in the same initial state. It is known since Angluin’s seminal result [2] that some grain of salt is required for distributed algorithms to solve the problem, e.g., the system needs to have a non-symmetrical topology or unbiased independent random bits.
The starting point of this paper is the observation that these approaches demand too strong assumptions. In short, by adding time to the picture, we show that the enumeration problem can be solved with far less. The idea is to consider a schedule of events in a distributed system as a space-time structure that is gradually learnt by the processes. We introduce the notion of divergence time which essentially measures the time by which the causal order induced by the system schedule has differentiated all the processes.
We prove lower bounds on the running time of any algorithm solving enumeration in terms of divergence time. In particular, we show that any adversary scheduler against which the enumeration problem can be solved necessarily selects schedules with finite divergence time.
We prove that this last condition is sufficient: we present the Torche algorithm which solves enumeration for all schedules with finite divergence time. In this sense, having finite divergence time is the smallest grain of salt required to solve the enumeration problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The bits used by the processes may, to some extent, be correlated, across the network and through time.
- 2.
More precisely, they have pairwise non-isomorphic causal pasts. See details below.
- 3.
References
Afek, Y., Matias, Y.: Elections in anonymous networks. Inf. Comput. 113(2), 312–330 (1994)
Angluin, D.: Local and global properties in networks of processors. In: 12th Symposium on the Theory of Computing, pp. 82–93. ACM (1980)
Awerbuch, B., Goldberg, A.V., Luby, M., Plotkin, S.A.: Network decomposition and locality in distributed computation. In: 30th Annual Symposium on Foundations of Computer Science, Research Triangle Park, North Carolina, USA, 30 October–1 November 1989, pp. 364–369 (1989)
Babai, L.: Graph isomorphism in quasipolynomial time [extended abstract]. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, 18–21 June 2016, pp. 684–697 (2016)
Barenboim, L., Elkin, M., Kuhn, F.: Distributed \((\Delta +1)\)-coloring in linear (in \(\Delta \)) time. SIAM J. Comput. 43(1), 72–95 (2014)
Boldi, P., Shammah, S., Vigna, S., Codenotti, B., Gemmell, P., Simon, J.: Symmetry breaking in anonymous networks: characterizations. In: Israel Symposium on Theory of Computing and Systems, pp. 16–26 (1996)
Boldi, P., Vigna, S.: Fibrations of graphs. Discrete Math. 243(1–3), 21–66 (2002)
Chalopin, J., Métivier, Y.: An efficient message passing election algorithm based on Mazurkiewicz’s algorithm. Fundamenta Informaticae 80(1–3), 221–246 (2007)
Chalopin, J., Métivier, Y., Morsellino, T.: Enumeration and leader election in partially anonymous and multi-hop broadcast networks. Fundamenta Informaticae 120(1), 1–27 (2012)
Hanckowiak, M., Karonski, M., Panconesi, A.: On the distributed complexity of computing maximal matchings. SIAM J. Discrete Math. 15(1), 41–57 (2001)
Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. In: 22nd Annual Symposium on Foundations of Computer Science, Nashville, Tennessee, USA, 28–30 October 1981, pp. 150–158 (1981)
Johnson, R.E., Schneider, F.B.: Symmetry and similarity in distributed systems. In: Proceedings of the Fourth Annual ACM Symposium on Principles of Distributed Computing, PODC 1985, pp. 13–22. ACM, New York (1985)
Linial, N.: Locality in distributed graph algorithms. SIAM J. Comput. 21(1), 193–201 (1992)
Lynch, N.: Distributed Algorithms. Morgan Kaufmann Publishers Inc., Burlington (1997)
Massey, W.S.: A Basic Course in Algebraic Topology. Springer, New York (1991)
Mazurkiewicz, A.: Distributed enumeration. Inf. Process. Lett. 61(5), 233–239 (1997)
Panconesi, A., Srinivasan, A.: On the complexity of distributed network decomposition. J. Algorithms 20(2), 356–374 (1996)
Schneider, J., Wattenhofer, R.: An optimal maximal independent set algorithm for bounded-independence graphs. Distrib. Comput. 22(5–6), 349–361 (2010)
Szegedy, M., Vishwanathan, S.: Locality based graph coloring. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, San Diego, CA, USA, 16–18 May 1993, pp. 201–207 (1993)
Tani, S.: Compression of view on anonymous networks - folded view -. IEEE Trans. Parallel Distrib. Syst. 23(2), 255–262 (2012)
Yamashita, M., Kameda, T.: Computing on anonymous networks: part I - characterizing the solvable cases. IEEE Trans. Parallel Distrib. Syst. 7(1), 69–89 (1996)
Yamashita, M., Kameda, T.: Computing on anonymous networks: part II - decision and membership problems. IEEE Trans. Parallel Distrib. Syst. 7(1), 90–96 (1996)
Yamashita, M., Kameda, T.: Leader election problem on networks in which processor identity numbers are not distinct. IEEE Trans. Parallel Distrib. Syst. 10(9), 878–887 (1999)
Acknowledgment
This work has been supported in part by the European ERC Grant 339539 - AOC.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Blanchard, P., Guerraoui, R. (2017). On the Smallest Grain of Salt to Get a Unique Identity. In: Das, S., Tixeuil, S. (eds) Structural Information and Communication Complexity. SIROCCO 2017. Lecture Notes in Computer Science(), vol 10641. Springer, Cham. https://doi.org/10.1007/978-3-319-72050-0_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-72050-0_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-72049-4
Online ISBN: 978-3-319-72050-0
eBook Packages: Computer ScienceComputer Science (R0)