从简单到复杂基于共识：一个案例研究
From Simplicity to Complexity Based on Consensus: A Case Study
AI总结：
本文探讨了共识算法如何生成复杂性，通过扰动最小共识协议实现最短路径规划，并证明了图动态收敛与最短路径解的等价性。
摘要：
分布式共识近年来被广泛研究，作为缓解图上动态节点状态差异的手段。它已成功应用于多机器人编队控制、负载均衡和时钟同步等应用。然而，几乎所有现有应用都将共识视为一种简单的迭代达成一致的过程，而没有迹象表明可以产生更高级的复杂性，例如最短路径规划，已被证明为NP难问题。反直觉地，我们首次展示最短路径规划的复杂性可以从最小共识协议的扰动版本中产生，这可能为分布式控制领域的研究者重新思考复杂性和控制与智能之间的距离提供启示。此外，我们严格证明了图动态的收敛性及其等价于最短路径解。通过一个小型图的示例模拟展示了有偏最小共识动态收敛到图上的最短路径解。为了展示其在大规模问题中的可扩展性，考虑了一个包含43826个节点的图，对应于2D迷宫地图。除了可能应用于机器人路径规划外，该结果还扩展到机器人全覆盖，展示了其在实际应用如清洁机器人中的潜力。
英文摘要：
Distributed consensus has been intensively studied in recent years as a means to mitigate state differences among dynamic nodes on a graph. It has been successfully employed in various applications, e.g., formation control of multi-robots, load balancing, clock synchronization. However, almost all existing applications cast an impression of consensus as a simple process to iteratively reach agreement, without any clue on possibility to generate advanced complexity, say shortest path planning, which has been proved to be NP-hard. Counter-intuitively, we show for the first time that the complexity of shortest path planning can emerge from a perturbed version of min-consensus protocol, which as a case study may shed lights to researchers in the field of distributed control to re-think the nature of complexity and the distance between control and intelligence. Besides, we rigorously prove the convergence of graph dynamics and its equivalence to shortest path solutions. An illustrative simulation on a small scale graph is provided to show the convergence of the biased min-consensus dynamics to shortest path solution over the graph. To demonstrate the scalability to large scale problems, a graph with 43826 nodes, which corresponds to a map of a maze in 2D, is considered in the simulation study. Apart from possible applications in robot path planning, the result is further extended to robot complete coverage, showing its potential in real practice such as cleaning robots.
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