标题： O(1)时间内双曲镶嵌邻域 显示英文标题

标题： Hyperbolic tiling neighborhoods in O(1) time

摘要： 双曲平面的镶嵌在许多数学、物理和计算机科学领域中具有重要意义。然而，它们的构建仍然是一个非平凡的任务。当前的方法主要使用基于树的递归算法，这些方法存在根本性的限制：它们不能直接生成表示单元邻接关系的邻域图，而这对许多应用来说却是必需的。我们引入了一种新方法，可以同时构建双曲镶嵌及其相关图结构，仅使用组合规则，而不需要显式的坐标表示。这使得能够生成任意大小的精确双曲图，其算法复杂度不依赖于晶格尺寸。我们提供了一个易于使用的实现，其性能显著优于现有方法，从而使这些几何结构上的超大规模数值模拟对科学界变得可行。 显示英文摘要

摘要： Tilings of the hyperbolic plane are of significant interest among many branches of mathematics, physics and computer science. Yet, their construction remains a non-trivial task. Current approaches primarily use tree-based recursive algorithms, which are fundamentally limited: they do not readily yield the neighborhood graph representing cell adjacencies, which is however required for many applications. We introduce a novel approach that allows to build hyperbolic tilings and their associated graph structure simultaneously, using only combinatoric rules without requiring an explicit coordinate representation. This allows to generate arbitrarily large, exact hyperbolic graphs, with an algorithmic complexity that does not depend on the lattice size. We provide an easy-to-use implementation which substantially outperforms existing methods, hence rendering ultra large-scale numerical simulations on these geometric structures accessible for the scientific community.