On the Number of Binary-Minded Individuals Required To Compute $\sqrt\frac12$

Olivier Bournez (1) , Guillaume Aupy (1, 2, 3, 4)

1 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
2 LIP - Laboratoire de l'Informatique du Parallélisme
3 ROMA - Optimisation des ressources : modèles, algorithmes et ordonnancement
4 ENS de Lyon - École normale supérieure de Lyon

We recently obtained partial results on the computational power of population protocols when population is assumed to be huge. We studied in particular a particular protocol that we proved to converge towards $\sqrt{\frac12}$, using weak-convergence methods for stochastic processes. In this note, we prove that this is possible to compute $\sqrt{\frac12}$ with precision $\epsilon>0$ in a time polynomial in $\frac1\epsilon$ using a number of agents polynomial in $\frac1\epsilon$, with individuals that can have only two states. This is established through a general result on approximation of stochastic differential equations by a stochastic Euler like discretization algorithm, of general interest.

Domaines: Complexité [cs.CC]

Olivier Bournez, Guillaume Aupy. On the Number of Binary-Minded Individuals Required To Compute $\sqrt\frac12$. Theoretical Computer Science, 2011, 411 (22), pp.2262--2267. ⟨10.1016/j.tcs.2011.01.003⟩. ⟨hal-00760928⟩