Population protocols are a popular model of distributed computing, in which randomlyinteracting agents with little computational power cooperate to jointly perform computational tasks. Inspired by developments in molecular computation, and in particular DNA computing, recent algorithmic work has focused on the complexity of solving simple yet fundamental tasks in the population model, such as leader election (which requires convergence to a single agent in a special “leader” state), and majority (in which agents must converge to a decision as to which of two possible initial states had higher initial count). Known results point towards an inherent tradeoff between the time complexity of such algorithms, and the space complexity, i.e. size of the memory available to each agent.
In this paper, we explore this tradeoff and provide new upper and lower bounds for majority and leader election. First, we prove a unified lower bound, which relates the space available per node with the time complexity achievable by a protocol: for instance, our result implies that any protocol solving either of these tasks for n agents using O(log log n) states must take Ω(n/polylogn) expected time. This is the first result to characterize time complexity for protocols which employ superconstant number of states per node, and proves that fast, polylogarithmic running times require protocols to have relatively large space costs.
On the positive side, we give algorithms showing that fast, polylogarithmic convergence time can be achieved using O (log2 n) space per node, in the case of both tasks. Overall, our results highlight a time complexity separation between O (log log n) and Θ(log2 n) state space size for both majority and leader election in population protocols, and introduce new techniques, which should be applicable more broadly.
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Message Complexity of Population Protocols
The standard population protocol model assumes that when two agents interact, each observes the entire state of the other agent. We initiate the study of the message complexity for population protocols, where the state of an agent is divided into an externallyvisible message and an internal component, where only the message can be observed by the other agent in an interaction.
We consider the case of O(1) message complexity. When time is unrestricted, we obtain an exact characterization of the stably computable predicates based on the number of internal states s(n): If s(n) = o(n) then the protocol computes a semilinear predicate (unlike the original model, which can compute nonsemilinear predicates with s(n) = O(log n)), and otherwise it computes a predicate decidable by a nondeterministic O(n log s(n))spacebounded Turing machine. We then consider time complexity, introducing novel O(polylog(n)) expected time protocols for junta/leader election and general purpose broadcast correct with high probability, and approximate and exact population size counting correct with probability 1. Finally, we show that the main constraint on the power of boundedmessagesize protocols is the size of the internal states: with unbounded internal states, any computable function can be computed with probability 1 in the limit by a protocol that uses only onebit messages.
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 NSFPAR ID:
 10209181
 Editor(s):
 Attiya, Hagit
 Date Published:
 Journal Name:
 34th International Symposium on Distributed Computing, DISC 2020, October 1216, 2020, Virtual Conference
 Volume:
 179
 Page Range / eLocation ID:
 6:16:18
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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