The graph exploration problem requires a group of mobile robots, initially placed arbitrarily on the nodes of a graph, to work collaboratively to explore the graph such that each node is eventually visited by at least one robot. One important requirement of exploration is the {\em termination} condition, i.e., the robots must know that exploration is completed. The problem of live exploration of a dynamic ring using mobile robots was recently introduced in [Di Luna et al., ICDCS 2016]. In it, they proposed multiple algorithms to solve exploration in fully synchronous and semi-synchronous settings with various guarantees when $2$ robots were involved. They also provided guarantees that with certain assumptions, exploration of the ring using two robots was impossible. An important question left open was how the presence of $3$ robots would affect the results. In this paper, we try to settle this question in a fully synchronous setting and also show how to extend our results to a semi-synchronous setting. In particular, we present algorithms for exploration with explicit termination using $3$ robots in conjunction with either (i) unique IDs of the robots and edge crossing detection capability (i.e., two robots moving in opposite directions through an edge in the same round can detect each other), or (ii) access to randomness. The time complexity of our deterministic algorithm is asymptotically optimal. We also provide complementary impossibility results showing that there does not exist any explicit termination algorithm for $2$ robots. The theoretical analysis and comprehensive simulations of our algorithm show the effectiveness and efficiency of the algorithm in dynamic rings. We also present an algorithm to achieve exploration with partial termination using $3$ robots in the semi-synchronous setting.

中文翻译：

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重新审视动态环中移动机器人的实时探索

图探索问题需要一组移动机器人，最初任意放置在图的节点上，协同工作来探索图，使得每个节点最终至少被一个机器人访问。探索的一个重要要求是 {\em 终止} 条件，即机器人必须知道探索已完成。最近在 [Di Luna et al., ICDCS 2016] 中介绍了使用移动机器人实时探索动态环的问题。在其中，当涉及 2 美元机器人时，他们提出了多种算法来解决完全同步和半同步设置中的探索问题，并提供各种保证。他们还保证，在某些假设下，使用两个机器人探索环是不可能的。一个悬而未决的重要问题是 $3$ 机器人的存在将如何影响结果。在本文中，我们尝试在完全同步的设置中解决这个问题，并展示如何将我们的结果扩展到半同步设置。特别是，我们提出了使用 $3$ 机器人结合 (i) 机器人的唯一 ID 和边缘交叉检测能力（即，两个机器人在同一轮中通过边缘以相反方向移动可以检测到）的显式终止探索算法彼此），或（ii）获得随机性。我们的确定性算法的时间复杂度是渐近最优的。我们还提供了补充的不可能结果，表明对于 $2$ 机器人不存在任何明确的终止算法。我们算法的理论分析和综合仿真表明了该算法在动态环中的有效性和效率。我们还提出了一种算法，以在半同步设置中使用 $3$ 机器人实现部分终止探索。