超越环:1区间连通图的聚集

O. Michail, P. Spirakis, Michail Theofilatos
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引用次数: 0

摘要

我们研究了在每一轮可能变化的动态图中收集[公式:见文本]代理(或多代理集合)的问题。我们考虑了[公式:见文本]-区间连接模型[9]的一种变体,其中所有实例(快照)总是连接底层图的生成子图,而不一定是一个团。代理是相同的,没有配备显式通信功能,并且最初在图上是任意定位的。问题是代理聚集在同一节点,而不是事先固定的。我们首先证明,如果底层图有一个循环,这个问题就不可能解决。鉴于此,我们研究了该问题的一个宽松版本,称为弱聚集,其中允许代理在同一节点或两个相邻节点上聚集。我们的目标是描述一类1间隔连接图和初始配置,其中问题是可解的,无论是否有本垒。在消极方面,我们表明当底层图包含生成双环子图并满足附加连通性时,弱聚集是不可解的,因此我们主要集中在单环图上。正如我们所展示的,在初始代理配置的大多数实例中,代理必须在循环中相遇。这给问题增加了额外的难度,因为他们需要探索图并识别形成循环的节点。我们为这个问题的可解情况提供了一个确定性算法,该算法运行在[公式:见文本]轮数中。
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Beyond Rings: Gathering in 1-Interval Connected Graphs
We examine the problem of gathering [Formula: see text] agents (or multi-agent rendezvous) in dynamic graphs which may change in every round. We consider a variant of the [Formula: see text]-interval connectivity model [9] in which all instances (snapshots) are always connected spanning subgraphs of an underlying graph, not necessarily a clique. The agents are identical and not equipped with explicit communication capabilities, and are initially arbitrarily positioned on the graph. The problem is for the agents to gather at the same node, not fixed in advance. We first show that the problem becomes impossible to solve if the underlying graph has a cycle. In light of this, we study a relaxed version of this problem, called weak gathering, where the agents are allowed to gather either at the same node, or at two adjacent nodes. Our goal is to characterize the class of 1-interval connected graphs and initial configurations in which the problem is solvable, both with and without homebases. On the negative side we show that when the underlying graph contains a spanning bicyclic subgraph and satisfies an additional connectivity property, weak gathering is unsolvable, thus we concentrate mainly on unicyclic graphs. As we show, in most instances of initial agent configurations, the agents must meet on the cycle. This adds an additional difficulty to the problem, as they need to explore the graph and recognize the nodes that form the cycle. We provide a deterministic algorithm for the solvable cases of this problem that runs in [Formula: see text] number of rounds.
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