# Brief Announcement: Fast Approximate Counting and Leader Election in Populations

## Abstract

*Population protocols* [2] are networks that consist of very weak computational entities (also called *nodes* or *agents*), regarding their individual capabilities and it has been shown that are able to perform complex computational tasks when they work collectively. *Leader Election* is the process of designating a single agent as the coordinator of some task distributed among several nodes. The nodes communicate among themselves in order to decide which of them will get into the *leader* state, starting from the same initial state *q*. An algorithm *A* solves the leader election problem if eventually the states of agents are divided into *leader* and *follower*, a unique leader remains elected and a follower can never become a leader. A randomized algorithm *R* solves the leader election problem if eventually only one leader remains in the system w.h.p.. *Counting* is the problem where nodes must determine the size *n* of the population. We call *Approximate Counting* the problem in which nodes must determine an estimation \(\hat{n}\) of the population size, where \(\frac{\hat{n}}{a}< n < \hat{n}\). We call *a* the estimation parameter. Consider the setting in which an agent is in an initial state *a*, the rest \(n-1\) agents are in state *b* and the only existing transition is \((a,b) \rightarrow (a,a)\). This is the *one-way epidemic* process and it can be shown that the expected time to convergence under the uniform random scheduler is \(\varTheta (n\log {n})\) (e.g., [3]), thus *parallel time* \(\varTheta (\log {n})\).

## Keywords

Population protocol Epidemic Leader election Counting Approximate counting Polylogarithmic time protocol## References

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