Space-Optimal Proportion Consensus with Population Protocols

Population protocols provide a distributed computing model in which a set of finite-state identical agents cooperate through random interactions, between neighbors in the interaction graph, to collectively carry out a computation in a distributed setting. Population protocols have become very popular in various research areas, such as distributed computing, sensor or social networks, as well as chemistry and biology. A central task in this model is majority computation, in which agents need to reach an agreement on the leading one of two possible initial opinions. In this paper we consider a generalization of the majority problem, named proportion consensus, which asks for an agreement on the proportion of one opinion, between two possible views (say \(\mathcal A\) or \(\mathcal B\)). The objective is to reach a configuration where all the agents agree on a range \(\gamma _A \subseteq [0,1]\) which contains the value of the fraction \(\rho _A\) of agents that started with view \(\mathcal A\); the goal is to get the size of \(\gamma _A\) as small as possible while also minimizing the number of states adopted by agents. We provide a lower bound on the trade-off between precision \(\epsilon \) (the size of \(\gamma _A\)) and the number of states required by any population protocol that solves the proportion consensus problem. In particular, we show that in any population protocol that solves the proportion consensus problem with precision \(\epsilon \), any agent must have at least \(\lceil 2/\epsilon \rceil \) states. We also provide a population protocol that exactly solves the proportion consensus problem with precision \(\epsilon \) and \(6\lceil 1/(2\epsilon ) \rceil -1\) states. We show that in case of an arbitrary interaction graph our protocol requires \(O(n^6/\epsilon )\) interactions (which corresponds to the number of rounds in the sequential communication model) on any network with n agents. On complete interaction networks, the expected number of required interactions is \(O(n^2 \log n)\). Using the random matching communication model, the expected number of rounds, required to reach a consensus, decreases to \(O(\varDelta n^4/\epsilon )\) in case of arbitrary interaction networks (where \(\varDelta \) denotes the maximum degree among the agents in the network) and \(O(n \log n)\) for complete networks.

[1]  Yves Mocquard,et al.  Optimal proportion computation with population protocols , 2016, 2016 IEEE 15th International Symposium on Network Computing and Applications (NCA).

[2]  Paul G. Spirakis,et al.  Stably Computing Order Statistics with Arithmetic Population Protocols , 2016, MFCS.

[3]  Gennaro Cordasco,et al.  Label propagation algorithm: a semi-synchronous approach , 2012, Int. J. Soc. Netw. Min..

[4]  Masafumi Yamashita,et al.  On space complexity of self-stabilizing leader election in mediated population protocol , 2012, Distributed Computing.

[5]  Dan Alistarh,et al.  Time-Space Trade-offs in Population Protocols , 2016, SODA.

[6]  Luca Cardelli,et al.  Programmable chemical controllers made from DNA. , 2013, Nature nanotechnology.

[7]  Moez Draief,et al.  Convergence Speed of Binary Interval Consensus , 2010, 2010 Proceedings IEEE INFOCOM.

[8]  Rachid Guerraoui,et al.  When Birds Die: Making Population Protocols Fault-Tolerant , 2006, DCOSS.

[9]  Paul G. Spirakis,et al.  Determining majority in networks with local interactions and very small local memory , 2014, Distributed Computing.

[10]  Dan Alistarh,et al.  Fast and Exact Majority in Population Protocols , 2015, PODC.

[11]  James Aspnes,et al.  An Introduction to Population Protocols , 2007, Bull. EATCS.

[12]  David Eisenstat,et al.  Stably computable predicates are semilinear , 2006, PODC '06.

[13]  Yves Mocquard,et al.  Counting with Population Protocols , 2015, 2015 IEEE 14th International Symposium on Network Computing and Applications.

[14]  Milan Vojnovic,et al.  Using Three States for Binary Consensus on Complete Graphs , 2009, IEEE INFOCOM 2009.

[15]  Joffroy Beauquier,et al.  Self-stabilizing Leader Election in Population Protocols over Arbitrary Communication Graphs , 2013, OPODIS.

[16]  Michael J. Fischer,et al.  Computation in networks of passively mobile finite-state sensors , 2004, PODC '04.

[17]  Michael J. Fischer,et al.  Stabilizing Consensus in Mobile Networks , 2006, DCOSS.

[18]  Petra Berenbrink,et al.  Plurality Consensus in Arbitrary Graphs: Lessons Learned from Load Balancing , 2016, ESA.

[19]  Paul G. Spirakis,et al.  Deterministic Population Protocols for Exact Majority and Plurality , 2017, OPODIS.

[20]  David Eisenstat,et al.  A simple population protocol for fast robust approximate majority , 2007, Distributed Computing.