# Fast and exact majority in population protocols

Alistarh D-A, Gelashvili R, Vojnović M. 2015. Fast and exact majority in population protocols. PODC: Principles of Distributed Computing vol. 2015–July, 47–56.

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Alistarh, Dan-Adrian

^{IST Austria}; Gelashvili, Rati; Vojnović, MilanAbstract

Population protocols, roughly defined as systems consisting of large numbers of simple identical agents, interacting at random and updating their state following simple rules, are an important research topic at the intersection of distributed computing and biology. One of the fundamental tasks that a population protocol may solve is majority: each node starts in one of two states; the goal is for all nodes to reach a correct consensus on which of the two states was initially the majority. Despite considerable research effort, known protocols for this problem are either exact but slow (taking linear parallel time to converge), or fast but approximate (with non-zero probability of error). In this paper, we show that this trade-off between preciasion and speed is not inherent. We present a new protocol called Average and Conquer (AVC) that solves majority ex-actly in expected parallel convergence time O(log n/(sε) + log n log s), where n is the number of nodes, εn is the initial node advantage of the majority state, and s = Ω(log n log log n) is the number of states the protocol employs. This shows that the majority problem can be solved exactly in time poly-logarithmic in n, provided that the memory per node is s = Ω(1/ε + lognlog1/ε). On the negative side, we establish a lower bound of Ω(1/ε) on the expected paraallel convergence time for the case of four memory states per node, and a lower bound of Ω(logn) parallel time for protocols using any number of memory states per node.per node, and a lower bound of (log n) parallel time for protocols using any number of memory states per node.

Publishing Year

Date Published

2015-07-21

Volume

2015-July

Page

47 - 56

Conference

PODC: Principles of Distributed Computing

IST-REx-ID

### Cite this

Alistarh D-A, Gelashvili R, Vojnović M. Fast and exact majority in population protocols. In: Vol 2015-July. ACM; 2015:47-56. doi:10.1145/2767386.2767429

Alistarh, D.-A., Gelashvili, R., & Vojnović, M. (2015). Fast and exact majority in population protocols (Vol. 2015–July, pp. 47–56). Presented at the PODC: Principles of Distributed Computing, ACM. https://doi.org/10.1145/2767386.2767429

Alistarh, Dan-Adrian, Rati Gelashvili, and Milan Vojnović. “Fast and Exact Majority in Population Protocols,” 2015–July:47–56. ACM, 2015. https://doi.org/10.1145/2767386.2767429.

D.-A. Alistarh, R. Gelashvili, and M. Vojnović, “Fast and exact majority in population protocols,” presented at the PODC: Principles of Distributed Computing, 2015, vol. 2015–July, pp. 47–56.

Alistarh, Dan-Adrian, et al.

*Fast and Exact Majority in Population Protocols*. Vol. 2015–July, ACM, 2015, pp. 47–56, doi:10.1145/2767386.2767429.