Optimal time and space leader election in population protocols

Population protocols are a model of distributed computing, where n agents with limited computational power and memory perform randomly scheduled pairwise interactions. A fundamental problem in this setting is that of leader election, where all agents start from the same state, and they seek to reach and maintain a global state where exactly one agent is in a dedicated leader state. A significant amount of work has been devoted to the study of the time and space complexity of this problem. Alistarh et al. (SODA’17) have shown that Ω(loglogn) states per agent are needed in order to elect a leader in fewer than Θ(n2) expected interactions. Moreover, Ω(nlogn) expected interactions are required regardless of the number of states (Sudo and Masuzawa, 2019). On the upper bound side, Gasieniec and Stachowiak (SODA’18) have presented the first protocol that uses an optimal, Θ(loglogn), number or states and elects a leader in O(n log2 n) expected interactions. This running time was subsequently improved to O(n lognloglogn) (Gasieniec et al., SPAA’19). In this paper we provide the first leader election population protocol that is both time and space optimal: it uses Θ(loglogn) states per agent, and elects a leader in O(nlogn) interactions in expectation. A key novel component of our approach is a simple protocol that efficiently selects a small set of agents, of poly(logn) size, given an initial set of s = O(nє) selected agents. Unlike existing approaches, which proceed by shrinking the initial set monotonically over time, our protocol first increases the set in a controlled way to a specific size (which is independent of s), before it shrinks the set to a poly(logn) size.

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

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

[3]  Robert Elsässer,et al.  Recent Results in Population Protocols for Exact Majority and Leader Election , 2018, Bull. EATCS.

[4]  Leszek Gasieniec,et al.  Fast Space Optimal Leader Election in Population Protocols , 2017, SODA.

[5]  Adrian Kosowski,et al.  Population Protocols Made Easy , 2018, ArXiv.

[6]  Paul G. Spirakis,et al.  Exact size counting in uniform population protocols in nearly logarithmic time , 2018, ArXiv.

[7]  Ho-Lin Chen,et al.  Speed faults in computation by chemical reaction networks , 2014, Distributed Computing.

[8]  Rafail Ostrovsky,et al.  Population Stability: Regulating Size in the Presence of an Adversary , 2018, PODC.

[9]  Petra Berenbrink,et al.  On Counting the Population Size , 2019, PODC.

[10]  Toshimitsu Masuzawa,et al.  Leader Election Requires Logarithmic Time in Population Protocols , 2020, Parallel Process. Lett..

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

[12]  Petra Berenbrink,et al.  Simple and Efficient Leader Election , 2018, SOSA.

[13]  George Giakkoupis,et al.  Brief Announcement: Optimal Time and Space Leader Election in Population Protocols , 2020, PODC.

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

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

[16]  David Doty,et al.  Efficient Size Estimation and Impossibility of Termination in Uniform Dense Population Protocols , 2018, PODC.

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

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

[19]  Fukuhito Ooshita,et al.  Logarithmic Expected-Time Leader Election in Population Protocol Model , 2019, PODC.

[20]  David Eisenstat,et al.  Fast computation by population protocols with a leader , 2006, Distributed Computing.

[21]  Petra Berenbrink,et al.  A Population Protocol for Exact Majority with O(log5/3 n) Stabilization Time and Θ(log n) States , 2018 .

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

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

[24]  Leszek Gasieniec,et al.  Almost Logarithmic-Time Space Optimal Leader Election in Population Protocols , 2018, SPAA.

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

[26]  David Doty,et al.  Timing in chemical reaction networks , 2013, SODA.

[27]  Colin Cooper,et al.  Population protocols for leader election and exact majority with O(log^2 n) states and O(log^2 n) convergence time , 2017, ArXiv.

[28]  Adrian Kosowski,et al.  Universal protocols for information dissemination using emergent signals , 2018, STOC.