James Aspnes, Joffroy Beauquier, Janna Burman, and Devan Sohier.
Time and space optimal counting in population protocols.
*20th International Conference on Principles of Distributed Systems,
OPODIS 2016, December 13–16, 2016, Madrid, Spain*, December 2016, pp. 13:1–13:17.

This work concerns the general issue of combined optimality in terms of time and space complexity. In this context, we study the problem of *counting* resource-limited and passively mobile nodes in the model of *population protocols*, in which the space complexity is crucial. The counted nodes are memory-limited anonymous devices (called agents) communicating asynchronously in pairs (according to a *fairness* condition).
Moreover, we assume that these agents are prone to failures so that they cannot be correctly initialized.

This study considers two classical fairness conditions, and for each we investigate the issue of time optimality of (exact) counting given optimal space. First, with randomly interacting agents (*probabilistic fairness*), we present a ``non-guessing'' time optimal protocol of O(n log n) expected interactions given an optimal space of only one bit (for a population of size n). We prove the time optimality of such protocol.

Then, under *weak fairness* (where every pair of agents interacts infinitely often), we show that a space optimal (*semi-uniform*) solution cannot converge faster than in Ω(2^{n}) time, in terms of *non-null transitions* (i.e, the transitions that affect the states of the interacting agents). This result together with the time complexity analysis of an already known space optimal protocol shows that it is also optimal in time (given the optimal space constraints).

- OPODIS 2016 proceedings version:
**PDF**.

@inproceedings{AspnesBBS2016, author = {James Aspnes and Joffroy Beauquier and Janna Burman and Devan Sohier}, title = {{Time and Space Optimal Counting in Population Protocols}}, booktitle = {20th International Conference on Principles of Distributed Systems (OPODIS 2016)}, pages = {13:1--13:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-031-6}, ISSN = {1868-8969}, year = {2017}, volume = {70}, editor = {Panagiota Fatourou and Ernesto Jim{\'e}nez and Fernando Pedone}, publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7082}, URN = {urn:nbn:de:0030-drops-70828}, doi = {10.4230/LIPIcs.OPODIS.2016.13}, annote = {Keywords: networks of passively mobile agents/sensors, population protocols, counting, anonymous non-initialized agents, time and space complexity, lower bounds} }

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