Consensus with Bounded Space and Minimal Communication

Population protocols are a fundamental model in distributed computing, where many nodes with bounded memory and computational power have random pairwise interactions over time. This model has been studied in a rich body of literature aiming to understand the tradeoffs between the memory and time needed to perform computational tasks. We study the population protocol model focusing on the communication complexity needed to achieve consensus with high probability. When the number of memory states is s = O(log log n), the best upper bound known was given by a protocol with O(n log n) communication, while the best lower bound was Ω(n log(n)/s) communication. We design a protocol that shows the lower bound is sharp, solving an open problem from [FHPR20]. When each agent has s = O(log nθ) states of memory, with θ ∈ (0, 1/2), consensus can be reached in time O(log(n)) with O(n log (n)/s) communications with high probability. ∗Purdue University, USA. E-mail: simina.branzei@gmail.com. †E-mail: yuval@yuvalperes.com. 1 ar X iv :2 10 1. 00 02 5v 1 [ cs .D C ] 3 1 D ec 2 02 0

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

[2]  Paul G. Spirakis,et al.  On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols , 2015, ICALP.

[3]  David Eisenstat,et al.  The computational power of population protocols , 2006, Distributed Computing.

[4]  Dan Alistarh,et al.  Polylogarithmic-Time Leader Election in Population Protocols , 2015, ICALP.

[5]  David Soloveichik,et al.  Stable leader election in population protocols requires linear time , 2015, Distributed Computing.

[6]  Dan Alistarh,et al.  Space-Optimal Majority in Population Protocols , 2017, SODA.

[7]  James M. Bower,et al.  Computational modeling of genetic and biochemical networks , 2001 .

[8]  Luca Trevisan,et al.  Simple dynamics for plurality consensus , 2013, Distributed Computing.

[9]  Rosa D. Hernansaiz-Ballesteros,et al.  Efficient Switches in Biology and Computer Science , 2017, PLoS Comput. Biol..

[10]  Lutz Warnke,et al.  On Wormald's differential equation method , 2019, ArXiv.

[11]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

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

[13]  N. Wormald.  The differential equation method for random graph processes and greedy algorithms , 1999 .

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

[15]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[16]  Dan Alistarh,et al.  Recent Algorithmic Advances in Population Protocols , 2018, SIGA.

[17]  Luca Cardelli,et al.  The Cell Cycle Switch Computes Approximate Majority , 2012, Scientific Reports.

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

[19]  M..  A Course of Pure Mathematics , 1909, Nature.

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

[21]  Dan Alistarh,et al.  Robust Detection in Leak-Prone Population Protocols , 2017, DNA.

[22]  Yves Mocquard,et al.  Explicit and Tight Bounds of the Convergence Time of Average-Based Population Protocols , 2019, SIROCCO.

[23]  Yuval Peres,et al.  Communication cost of consensus for nodes with limited memory , 2020, Proceedings of the National Academy of Sciences.

[24]  George Giakkoupis,et al.  Optimal time and space leader election in population protocols , 2020, STOC.

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

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

[27]  Fukuhito Ooshita,et al.  Time-Optimal Leader Election in Population Protocols , 2020, IEEE Transactions on Parallel and Distributed Systems.