We study the problem of deterministically exploring an undirected and initially unknown graph with n vertices either by a single agent equipped with a set of pebbles or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles Θ(log n ) bits of memory are necessary and sufficient to explore any graph with at most n vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory Θ(log log n ) pebbles are necessary and sufficient for exploration. We further prove that using collaborating agents instead of pebbles does not help as Θ(log log n ) agents with constant memory each are necessary and sufficient for exploration. For the upper bounds, we devise an algorithm for a single agent with constant memory that explores any n -vertex graph using O (log log n ) pebbles, even when n is not known a priori . The algorithm terminates after polynomial time and returns to the starting vertex. We further show that the algorithm can be realized with additional constant-memory agents rather than pebbles, implying that O (log log n ) agents with constant memory can explore any n -vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph with at most n vertices is already Ω(log log n ) when we allow each agent to have at most O ((log n ) 1 -ε) bits of memory for any ε > 0. Our argument also implies that a single agent with sublogarithmic memory needs Θ(log log n ) pebbles to explore any n -vertex graph.

中文翻译：

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Pebbles 和多个智能体的无向图探索的紧密边界

我们研究确定性地探索无向且最初未知的图的问题n顶点由配备一组鹅卵石的单个代理或一组协作代理。图的顶点是未标记的，并且无法被代理区分，但与顶点相关的边具有局部不同的标签。当至少一个代理访问了所有顶点时，将探索该图。在这种情况下，已知对于没有鹅卵石的单个智能体 Θ(logn) 内存位对于探索任何最多包含n顶点。我们感兴趣的是，当代理可能通过丢弃和检索可区分的鹅卵石来标记顶点时，或者当多个代理共同探索图形时，内存需求如何降低。我们对这两个问题给出了严格的结果，表明对于具有恒定记忆 Θ(log logn) 鹅卵石对于探索来说是必要和充分的。我们进一步证明，使用协作代理而不是鹅卵石并没有帮助，因为 Θ(log logn) 具有恒定记忆的智能体对于探索来说都是必要和充分的。对于上限，我们为具有恒定内存的单个代理设计了一种算法，该算法可以探索任何n-顶点图使用○（日志日志n) 鹅卵石，即使当n不知道先验. 该算法在多项式时间后终止并返回到起始顶点。我们进一步表明，该算法可以用额外的常数记忆代理而不是鹅卵石来实现，这意味着○（日志日志n) 具有恒定记忆的智能体可以探索任何n-顶点图。对于下界，我们表明探索任何图所需的代理数量最多n顶点已经是 Ω(log logn) 当我们允许每个代理最多有○（（日志n)1-ε) 任何 ε > 0 的内存位。我们的论点还意味着具有次对数内存的单个代理需要 Θ(log logn) 鹅卵石探索任何n-顶点图。