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**Published in:** LIPIcs, Volume 221, 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)

The population protocol model describes a network of anonymous agents that interact asynchronously in pairs chosen at random. Each agent starts in the same initial state s. We introduce the dynamic size counting problem: approximately counting the number of agents in the presence of an adversary who at any time can remove any number of agents or add any number of new agents in state s. A valid solution requires that after each addition/removal event, resulting in population size n, with high probability each agent "quickly" computes the same constant-factor estimate of the value log₂(n) (how quickly is called the convergence time), which remains the output of every agent for as long as possible (the holding time). Since the adversary can remove agents, the holding time is necessarily finite: even after the adversary stops altering the population, it is impossible to stabilize to an output that never again changes.
We first show that a protocol solves the dynamic size counting problem if and only if it solves the loosely-stabilizing counting problem: that of estimating log n in a fixed-size population, but where the adversary can initialize each agent in an arbitrary state, with the same convergence time and holding time. We then show a protocol solving the loosely-stabilizing counting problem with the following guarantees: if the population size is n, M is the largest initial estimate of log n, and s is the maximum integer initially stored in any field of the agents' memory, we have expected convergence time O(log n + log M), expected polynomial holding time, and expected memory usage of O(log²(s) + (log log n)²) bits. Interpreted as a dynamic size counting protocol, when changing from population size n_prev to n_next, the convergence time is O(log n_next + log log n_prev).

David Doty and Mahsa Eftekhari. Dynamic Size Counting in Population Protocols. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{doty_et_al:LIPIcs.SAND.2022.13, author = {Doty, David and Eftekhari, Mahsa}, title = {{Dynamic Size Counting in Population Protocols}}, booktitle = {1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022)}, pages = {13:1--13:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-224-2}, ISSN = {1868-8969}, year = {2022}, volume = {221}, editor = {Aspnes, James and Michail, Othon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2022.13}, URN = {urn:nbn:de:0030-drops-159558}, doi = {10.4230/LIPIcs.SAND.2022.13}, annote = {Keywords: Loosely-stabilizing, population protocols, size counting} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

The standard population protocol model assumes that when two agents interact, each observes the entire state of the other. We initiate the study of message complexity for population protocols, where an agent’s state is divided into an externally-visible message and externally-hidden local state.
We consider the case of O(1) message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states s(n): If s(n) = o(n) then the protocol computes semilinear predicates (unlike the original model, which can compute non-semilinear predicates with s(n) = O(log n)), and otherwise it computes a predicate decidable by a nondeterministic O(n log s(n))-space-bounded Turing machine. We then introduce novel O(polylog(n)) expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of bounded-message-size protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only 1-bit messages.

Talley Amir, James Aspnes, David Doty, Mahsa Eftekhari, and Eric Severson. Message Complexity of Population Protocols. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{amir_et_al:LIPIcs.DISC.2020.6, author = {Amir, Talley and Aspnes, James and Doty, David and Eftekhari, Mahsa and Severson, Eric}, title = {{Message Complexity of Population Protocols}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.6}, URN = {urn:nbn:de:0030-drops-130848}, doi = {10.4230/LIPIcs.DISC.2020.6}, annote = {Keywords: population protocol, message complexity, space-optimal} }

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Brief Announcement

**Published in:** LIPIcs, Volume 121, 32nd International Symposium on Distributed Computing (DISC 2018)

We study population protocols: networks of anonymous agents whose pairwise interactions are chosen uniformly at random. The size counting problem is that of calculating the exact number n of agents in the population, assuming no leader (each agent starts in the same state). We give the first protocol that solves this problem in sublinear time.
The protocol converges in O(log n log log n) time and uses O(n^60) states (O(1) + 60 log n bits of memory per agent) with probability 1-O((log log n)/n). The time to converge is also O(log n log log n) in expectation. Crucially, unlike most published protocols with omega(1) states, our protocol is uniform: it uses the same transition algorithm for any population size, so does not need an estimate of the population size to be embedded into the algorithm.

David Doty, Mahsa Eftekhari, Othon Michail, Paul G. Spirakis, and Michail Theofilatos. Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time. In 32nd International Symposium on Distributed Computing (DISC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 121, pp. 46:1-46:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{doty_et_al:LIPIcs.DISC.2018.46, author = {Doty, David and Eftekhari, Mahsa and Michail, Othon and Spirakis, Paul G. and Theofilatos, Michail}, title = {{Brief Announcement: Exact Size Counting in Uniform Population Protocols in Nearly Logarithmic Time}}, booktitle = {32nd International Symposium on Distributed Computing (DISC 2018)}, pages = {46:1--46:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-092-7}, ISSN = {1868-8969}, year = {2018}, volume = {121}, editor = {Schmid, Ulrich and Widder, Josef}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2018.46}, URN = {urn:nbn:de:0030-drops-98359}, doi = {10.4230/LIPIcs.DISC.2018.46}, annote = {Keywords: population protocol, counting, leader election, polylogarithmic time} }

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