带有催化输入的近似多数
Approximate Majority With Catalytic Inputs
AI总结：
本文研究催化输入（CI）模型下群体协议中的近似多数问题，证明计算精确多数需 $Ω(n^2)$ 次交互，并提出一种常量状态协议在 $O(n \log n)$ 步内以高概率解决近似多数，且对瞬态泄漏事件具有鲁棒性。
摘要：
群体协议是一类用于对通过成对交互进行通信的有限状态智能体网络中的分布式计算进行建模的算法。它们非常适合分析大量化学过程，这促使了对原始群体协议框架进行调整以更好地对这些化学系统进行建模。在本文中，我们进一步研究在解决近似多数问题背景下的两种此类调整：持久状态智能体（或催化剂）和自发状态变化（或泄漏）。\n 基于近期针对具有持久状态智能体的群体的协议所考虑的模型，我们假设一个群体具有 $n$ 个催化输入智能体和 $m$ 个工作智能体，并且工作智能体的目标是计算关于催化输入状态的某个谓词。我们将此模型称为催化输入（CI）模型。对于 $m = Θ(n)$，我们证明以高概率计算输入群体的精确多数至少需要 $Ω(n^2)$ 次总交互，展示了 CI 模型与标准群体协议模型之间的强分离。另一方面，我们证明 Angluin 等人用于标准模型中近似多数的简单第三状态动力学可以自然地调整到 CI 模型：我们为 CI 模型提出了此类常量状态协议，当输入裕度为 $Ω(\sqrt{n \log n})$ 时，该协议以高概率在 $O(n \log n)$ 总步数内解决近似多数问题。\n 然后我们证明第三状态动力学协议对 Alistarh 等人引入的瞬态泄漏事件的鲁棒性。在原始模型和 CI 模型中，当每步以概率 $β\leq O\left(\sqrt{n \log n}/n\right)$ 发生泄漏时，这些协议均能以高概率成功计算近似多数，展现出对泄漏的适应力，类似于先前工作中对拜占庭智能体的适应力。
英文摘要：
Population protocols are a class of algorithms for modeling distributed computation in networks of finite-state agents communicating through pairwise interactions. Their suitability for analyzing numerous chemical processes has motivated the adaptation of the original population protocol framework to better model these chemical systems. In this paper, we further the study of two such adaptations in the context of solving approximate majority: persistent-state agents (or catalysts) and spontaneous state changes (or leaks). Based on models considered in recent protocols for populations with persistent-state agents, we assume a population with $n$ catalytic input agents and $m$ worker agents, and the goal of the worker agents is to compute some predicate over the states of the catalytic inputs. We call this model the Catalytic Input (CI) model. For $m = Θ(n)$, we show that computing the exact majority of the input population with high probability requires at least $Ω(n^2)$ total interactions, demonstrating a strong separation between the CI model and the standard population protocol model. On the other hand, we show that the simple third-state dynamics of Angluin et al. for approximate majority in the standard model can be naturally adapted to the CI model: we present such a constant-state protocol for the CI model that solves approximate majority in $O(n \log n)$ total steps w.h.p. when the input margin is $Ω(\sqrt{n \log n})$. We then show the robustness of third-state dynamics protocols to the transient leaks events introduced by Alistarh et al. In both the original and CI models, these protocols successfully compute approximate majority with high probability in the presence of leaks occurring at each step with probability $β\leq O\left(\sqrt{n \log n}/n\right)$, exhibiting a resilience to leaks similar to that of Byzantine agents in previous works.
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