Space-Optimal Counting in Population Protocols [Extended Version]
Abstract
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, prove correct and analyze the time complexities
of 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. The convergence time of the protocol is exponential, but we prove that
this is necessary (within a constant factor) for obtaining a space-optimal and semi-uniform solution.
Origin : Files produced by the author(s)
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