Space-Optimal Counting in Population Protocols
Résumé
In this paper, we study the fundamental problem of counting, which consists in computing the size of a system. We consider the distributed communication model of population protocols of finite state,
anonymous and asynchronous mobile devices (agents) communicating
in pairs (according to a fairness condition). This work significantly improves
the previous results known for counting in this model, in terms of
(exact) space complexity. We present and prove correct the first space optimal
protocols solving the problem for two classical types of fairness,
global and weak. Both protocols require no initialization of the counted
agents.
The protocol designed for global fairness, surprisingly, uses only one bit of
memory (two states) per counted agent. The protocol, functioning under
weak fairness, requires the necessary log P bits (P states, per counted
agent) to be able to count up to P agents. Interestingly, this protocol
exploits the intriguing Gros sequence of natural numbers, which is also
used in the solutions to the Chinese Rings and the Hanoi Towers puzzles.
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