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**Published in:** LIPIcs, Volume 146, 33rd International Symposium on Distributed Computing (DISC 2019)

The distributed naming problem, assigning unique names to the nodes in a distributed system, is a fundamental task. This problem is nontrivial, especially when the amount of memory available for the task is low, and when requirements for fault-tolerance are added.
The considered distributed communication model is population protocols. In this model, a priori anonymous and indistinguishable mobile nodes (called agents), communicate in pairs and in an asynchronous manner (according to a fairness condition). Fault-tolerance is addressed through self-stabilization, in terms of arbitrary initialization of agents.
This work comprises a comprehensive study of the necessary and sufficient state space conditions for naming. The problem is studied under various combinations of model assumptions: weak or global fairness, arbitrary or uniform initialization of agents, existence or absence of a distinguishable agent (arbitrarily initialized or not), possibility of breaking symmetry in pair-wise interactions (symmetric or asymmetric transitions). For each possible combination of these assumptions, either an impossibility is proven or the necessary exact number of states (per mobile agent) is determined and an appropriate space-optimal naming protocol is presented.

Janna Burman, Joffroy Beauquier, and Devan Sohier. Space-Optimal Naming in Population Protocols. In 33rd International Symposium on Distributed Computing (DISC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 146, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{burman_et_al:LIPIcs.DISC.2019.9, author = {Burman, Janna and Beauquier, Joffroy and Sohier, Devan}, title = {{Space-Optimal Naming in Population Protocols}}, booktitle = {33rd International Symposium on Distributed Computing (DISC 2019)}, pages = {9:1--9:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-126-9}, ISSN = {1868-8969}, year = {2019}, volume = {146}, editor = {Suomela, Jukka}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2019.9}, URN = {urn:nbn:de:0030-drops-113161}, doi = {10.4230/LIPIcs.DISC.2019.9}, annote = {Keywords: networks of passively mobile agents, population protocols, deterministic naming, self-stabilization, exact space complexity, tight lower bounds, global and weak fairness} }

Document

**Published in:** LIPIcs, Volume 70, 20th International Conference on Principles of Distributed Systems (OPODIS 2016)

This work concerns the general issue of combined optimality in terms of time and space complexity. In this context, we study the problem of (exact) 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 counting given the optimal space per agent. In the case of randomly interacting agents (probabilistic fairness), as usual, the convergence time is measured in terms of parallel time (or parallel interactions), which is defined as the number of pairwise interactions until convergence, divided by n (the number of agents). In case of weak fairness, where it is only required that every pair of agents interacts infinitely often, the convergence time is defined in terms of non-null transitions, i.e, the transitions that affect the states of the interacting agents.
First, assuming probabilistic fairness, we present a "non-guessing" time optimal protocol of O(n log n) expected time given an optimal space of only one bit, and we prove the time optimality of this protocol. Then, for weak fairness, we show that a space optimal (semi-uniform) solution cannot converge faster than in big-omega (2^n) time (non-null transitions). 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 constrains).

James Aspnes, Joffroy Beauquier, Janna Burman, and Devan Sohier. Time and Space Optimal Counting in Population Protocols. In 20th International Conference on Principles of Distributed Systems (OPODIS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 70, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{aspnes_et_al:LIPIcs.OPODIS.2016.13, author = {Aspnes, James and Beauquier, Joffroy and Burman, Janna and Sohier, Devan}, 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 = {Fatourou, Panagiota and Jim\'{e}nez, Ernesto and Pedone, Fernando}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.OPODIS.2016.13}, 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} }