Population protocols are a popular model of distributed computing, in which randomly-interacting 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 trade-off 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 trade-off 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 super-constant number of states per node, and proves that fast, poly-logarithmic running times require protocols to have relatively large space costs.
On the positive side, we give algorithms showing that fast, poly-logarithmic 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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Approximate Majority With Catalytic Inputs
Third-state dynamics (Angluin et al. 2008; Perron et al. 2009) is a well-known process for quickly and robustly computing approximate majority through interactions between randomly-chosen pairs of agents. In this paper, we consider this process in a new model with persistent-state catalytic inputs, as well as in the presence of transient leak faults.
Based on models considered in recent protocols for populations with persistent-state agents (Dudek et al. 2017; Alistarh et al. 2017; Alistarh et al. 2020), we formalize a Catalytic Input (CI) model comprising n input agents and m worker agents. For m = Θ(n), we show that computing the parity of the input population with high probability requires at least Ω(n2) total interactions, demonstrating a strong separation between the CI and standard population protocol models. On the other hand, we show that the third-state dynamics can be naturally adapted to this new model to solve approximate majority in O(n log n) total steps with high probability when the input margin is Ω(√(n log n)), which preserves the time and space efficiency of the corresponding protocol in the original model.
We then show the robustness of third-state dynamics protocols to the transient leak faults considered by (Alistarh et al. 2017; Alistarh et al 2020). In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each time step with probability β ≤ O(√(n log n}/n). The resilience of these dynamics to adversarial leaks exhibits a subtle connection to previous results involving Byzantine agents.
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- Award ID(s):
- 1650596
- PAR ID:
- 10209182
- Date Published:
- Journal Name:
- OPODIS 2020
- Page Range / eLocation ID:
- 19:1-19:16
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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