Efficient assignment of identities in anonymous populations

Abstract

We consider the fundamental problem of assigning distinct labels to agents in the probabilistic model of population protocols. Our protocols operate under the assumption that the size n of the population is embedded in the transition function. W.h.p. (with high probability), they are silent, i.e., eventually each agent reaches its final state and remains in it forever, and they are safe, i.e., never change a label that has already been assigned to an agent. We provide efficient protocols for this problem complemented with tight lower bounds. Our fast labeling protocol uses only $O\left(\right(n\log n)/\epsilon )$ interactions w.h.p., $(2+\epsilon )n+O\left(n^a\right)$ states, and the label range $[1,(1+\epsilon \left)n\right]$, where $1\ge \epsilon >0$ and $0<a<1$, while our nearly state-optimal protocol uses only $n+5\sqrt{n}+O(\log \log n)$ states, the label range $[1,n]$, and w.h.p., $O\left(n^3\right)$ interactions.