Stable Leader Election in Population Protocols Requires Linear Time

A population protocol stably elects a leader if, for all n, starting from an initial configuration with n agents each in an identical state, with probability 1 it reaches a configuration y that is correct exactly one agent is in a special leader state $$\ell $$ and stable every configuration reachable from y also has a single agent in state $$\ell $$. We show that any population protocol that stably elects a leader requires $$\Omega n$$ expected "parallel time" -- $$\Omega n^2$$ expected total pairwise interactions -- to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.

[1]  Mónica Bettencourt-Dias,et al.  From Zero to Many: Control of Centriole Number in Development and Disease , 2009, Traffic.

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

[3]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

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

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

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

[7]  L. Dickson.  Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors , 1913 .

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

[9]  Tomoko Izumi,et al.  Space-efficient self-stabilizing counting population protocols on mobile sensor networks , 2014, Theor. Comput. Sci..

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

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

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

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

[14]  Ho-Lin Chen,et al.  Speed Faults in Computation by Chemical Reaction Networks , 2014, DISC.

[15]  James Aspnes,et al.  Time and Space Optimal Counting in Population Protocols , 2016, OPODIS.

[16]  Joffroy Beauquier,et al.  Space-Optimal Counting in Population Protocols , 2015, DISC.

[17]  G. Seelig,et al.  DNA as a universal substrate for chemical kinetics , 2010, Proceedings of the National Academy of Sciences.

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

[19]  Richard M. Karp,et al.  Parallel Program Schemata , 1969, J. Comput. Syst. Sci..

[20]  Michael J. Fischer,et al.  Self-stabilizing Population Protocols , 2005, OPODIS.

[21]  Ho-Lin Chen,et al.  Deterministic function computation with chemical reaction networks , 2012, Natural Computing.

[22]  Dan Alistarh,et al.  Time-Space Trade-offs in Molecular Computation , 2016 .

[23]  C. A. Petri.  Communication with automata , 1966 .