Efficient Assignment of Identities in Anonymous Populations

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. They also rely on a unique leader which can be precomputed with a negligible impact on our upper bounds. Among other things, we consider silent labeling protocols, where eventually each agent reaches its final state and remains in it forever, as well as safe labeling protocols which can produce a valid agent labeling in a finite number of interactions, and guarantee that at any step of the protocol no two agents have the same label. We first provide a silent and safe protocol which uses only $n+5\sqrt n +4$ states and draws labels from the range $1,\dots,n.$ . The expected number of interactions required by the protocol is $O(n^3).$ On the other hand, we show that any safe protocol, as well as any silent protocol which provides a valid labeling with probability $>1-\frac 1n$, uses $\ge n+\sqrt n-1$ states. Hence, our protocol is almost state-optimal. We also present a generalization of the protocol to include a trade-off between the number of states and the expected number of interactions. We show also that for any safe labeling protocol utilizing $n+t<2n$ states the expected number of interactions required to achieve a valid labeling is $\ge \frac{n^2}{t+1}$. Next, we present a fast, silent and safe labeling protocol for which the required number of interactions is asymptotically optimal, i.e., $O(n \log n/\epsilon)$ w.h.p. It uses $(2+\epsilon)n+O(\log n)$ states and the label range $1,\dots,(1+\epsilon)n.$ Finally, we consider the so-called pool labeling protocols that include our fast protocol. We show that the expected number of interactions required by any pool protocol is $\ge \frac{n^2}{r+1}$, when the labels range is $1,\dots, n+r<2n.$