确定性高维网格探索的紧边界
Tight Bounds for Deterministic High-Dimensional Grid Exploration
摘要：
我们研究的问题，探索一个有向网格与有限自动机管理的自主智能体。在二维网格的情况下，需要多少代理来探索网格，或者等价地，在网格中找到隐藏的宝藏，在同步和半同步设置中都是完全理解的。对于更高的维度，Dobrev、Narayanan、Opatrny和Pankratov [ICALP'19]最近表明，令人惊讶的是，（少量）恒定数量的主体就足以找到宝藏，与维度的数量无关，从而反驳了Cohen、Emek、Louidor和Uitto的猜想[SODA'17]。Dobrev等人留下了一个悬而未决的问题，即他们对代理数量的限制是否可以改进。我们肯定地回答这个问题的确定性有限自动机：我们表明，3同步和4半同步代理足以探索任何常数$n$的$n$维网格。的界限是最佳的，值得注意的是，匹配的下限已经在2维的情况下。我们的技术也可以用于解决Dobrev等人提出的其他开放性问题：我们证明了4个同步和5个半同步代理足以进行多项式时间的探索，并且我们表明，在自然的假设下，3个同步和4个半同步代理足以探索任意维度的无方向网格（这又是紧的）。
英文摘要：
We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an $n$-dimensional grid for any constant $n$. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case. Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
