## Abstract

A population protocol *stably elects a leader* if, for all *n*, starting from an initial configuration with *n* agents each in an identical state, with probability 1 it reaches a configuration \(\mathbf {y}\) that is *correct* (exactly one agent is in a special leader state \(\ell \)) and *stable* (every configuration reachable from \(\mathbf {y}\) also has a single agent in state \(\ell \)). We show that any population protocol that stably elects a leader requires \(\varOmega (n)\) expected “parallel time”—\(\varOmega (n^2)\) expected total pairwise interactions—to reach such a stable configuration. Our result also informs the understanding of the time complexity of chemical self-organization by showing an essential difficulty in generating exact quantities of molecular species quickly.

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## Notes

Some work allows “non-deterministic” transitions, in which the transition function maps to subsets of \(\varLambda \times \varLambda \). Our results are independent of whether the transition function is deterministic or nondeterministic in this manner.

In the most generic model, there is no restriction on which agents are permitted to interact. If one prefers to think of the agents as existing on nodes of a graph, then it is the complete graph \(K_n\) for a population of

*n*agents.The set of valid initial configurations for a “self-stabilizing” PP is \(\mathbb {N}^\varLambda \), where leader election is provably impossible [9]. We don’t require the PP to work if started in any possible configuration, but rather allow potentially “helpful” initial configurations as long as they don’t already have small count states (see “\(\alpha \)-dense” below).

What “correct” means depends on the task. For computing a predicate, for example, \(\varLambda \) is partitioned into “yes” and “no” voters, and a “correct” configuration is one in which every state present has the correct vote.

See “Open questions” for the distinction between time to

*converge*and time to*stabilize*. In this paper, the time lower bound we prove is on stabilization.If it were possible to detect when a leader election protocol such as \(\ell ,\ell \rightarrow \ell ,f\) has stabilized, in the sense that each agent carries a bit \(\{g,s\}\) in which all agents start with

*g*and only transition to*s*after a single \(\ell \) exists, then one could consider the product state \((\ell ,s)\) to be the “true” leader state, and \((\ell ,g)\) is considered only a “candidate” leader state that may have count \(>1\) prior to the stabilization to a single \(\ell \).We give a slightly different formalism than that of [8] for population protocols. The main difference is that since we are not deciding a predicate, there is no notion of inputs being mapped to states or states being mapped to outputs. Another difference is that we assume the transition function is symmetric (so there is no notion of a “sender” and “receiver” agent as in [8]; the unordered pair of states completely determines the next pair of states). However, the results of this paper hold even if we allow the transition function to be non-symmetric or even to be non-deterministic (allowing transitions such as \(a,b \rightarrow c,d\) and \(a,b \rightarrow x,y\) to coexist).

When the initial configuration to which a transition sequence is applied is clear from context, we may overload terminology and refer to \((\mathbf {c}_0, \mathbf {c}_1, \ldots )\) as a transition sequence or path.

Since PP’s have a finite reachable configuration space, this is equivalent to requiring that for all \(\mathbf {x}\) reachable from \(\mathbf {c}\), there is a \(\mathbf {c}' \in C\) reachable from \(\mathbf {x}\).

Recall that the condition \(\mathsf {Pr}[\mathbf {i}\mathop {\Longrightarrow }\nolimits Y]=1\) is equivalent to \([(\forall \mathbf {c}\in \mathbb {N}^\varLambda )\ \mathbf {i}\mathop {\Longrightarrow }\nolimits \mathbf {c}\) implies \((\exists \mathbf {y}\in Y)\ \mathbf {c}\mathop {\Longrightarrow }\nolimits \mathbf {y}]\).

If \(r_1 \ne r_2\) and \(\mathbf {c}(r_1)=\mathbf {c}(r_2)=b\), then the probability to pick the first agent in one of the states \(r_1\) or \(r_2\) is \(\frac{2b}{n}\), and the probability to pick the second agent in the other state is \(\frac{b}{n-1}\), so the total probability of both is \(\frac{2b^2}{n(n-1)}.\) The case for \(r_1=r_2\) gives \(\frac{b}{n}\) for the first times \(\frac{b-1}{n-1}\) for the second, resulting in lower total probability \(\frac{b^2-b}{n(n-1)}\).

With a nondeterministic transition function, the total number of transitions would replace the quantity \(|\varLambda |^2\) in the conclusion, but it would remain a constant independent of the size of the initial configuration.

Theorem 4.3 was proven in a more general model for Chemical Reaction Networks (CRNs) that obey a certain technical condition [17]. As observed in that paper, the class of CRNs obeying that condition includes all PPs, so the theorem holds unconditionally for PPs. The theorem proved in [17] is more general than Theorem 4.3, but we have stated a corollary of it here. A similar statement is implicit in the proof sketch of Lemma 5 of a technical report on a variant model called “urn automata” that has PPs as a special case [5].

Note the need for \(\mathbf {p}\) to ensure that the total count never goes negative. In writing “\(a=7, b=6, c=-3\)”, we are examining the effect only on \(\mathbf {y}_m\) of modifying \(p_m\), but in applying the lemma, the starting configuration is \(\mathbf {x}_m+\mathbf {p}\), not merely \(\mathbf {x}_m\), so the actual count of each state

*s*will be \(\mathbf {p}(s)\) larger than just stated.

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## Acknowledgments

The authors thank Anne Condon and Monir Hajiaghayi for several insightful discussions. We also thank the attendees of the 2014 Workshop on Programming Chemical Reaction Networks at the Banff International Research Station, where the first incursions were made into the solution of the problem of PP stable leader election. We are also grateful to anonymous reviewers whose comments have significantly improved the presentation. David Doty was supported by NSF Grants CCF-1619343, CCF-1219274, and CCF-1162589 and the Molecular Programming Project under NSF Grant 1317694. David Soloveichik was supported by an NIGMS Systems Biology Center Grant P50 GM081879 and NSF Grant CCF-1618895.

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A preliminary version of this article appeared as [18]; the current version has been revised for clarity, and includes several omitted proofs.

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Doty, D., Soloveichik, D. Stable leader election in population protocols requires linear time.
*Distrib. Comput.* **31**, 257–271 (2018). https://doi.org/10.1007/s00446-016-0281-z

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DOI: https://doi.org/10.1007/s00446-016-0281-z

### Keywords

- Population protocols
- Leader election
- Time lower bound
- Chemical reaction network