Problems on repeated geometric patterns in finite point sets in Euclidean space are extensively studied in the literature of combinatorial and computational geometry. Such problems trace their inspiration to Erd\H{o}s' original work on that topic. In this paper, we investigate the particular case of finding scaled copies of any pattern within a set of $n$ points, that is, the algorithmic task of efficiently enumerating all such copies. We initially focus on one particularly simple pattern of axis-parallel squares, and present an algorithm with an $O(n\sqrt{n})$ running time and $O(n)$ space for this task, involving various bucket-based and sweep-line techniques. Our algorithm is worst-case optimal, as it matches the known lower bound of $\Omega(n\sqrt{n})$ on the maximum number of axis-parallel squares determined by $n$ points in the plane, thereby solving an open question for more than three decades of realizing that bound for this pattern. We extend our result to an algorithm that enumerates all copies, up to scaling, of any full-dimensional fixed set of points in $d$-dimensional Euclidean space, that works in time $O(n^{1+1/d})$ and space $O(n)$, also matching the corresponding lower bound due to Elekes and Erd\H{o}s.
翻译:Euclidean 空间定点数中重复几何模式的问题,已在组合和计算几何文献中进行了广泛研究。这些问题的灵感来源于Erd\H{o}s 最初关于该主题的工作。在本文中,我们调查在一组美元点内找到任何模式的缩放复制件的特殊案例,即高效计算所有此类复制件的算法任务。我们最初侧重于一个特别简单的轴-单方平方形模式,并提出一个包含美元运行时间和美元(n)的算法,用于这一任务,涉及各种桶基和扫描线技术。我们的算法是最坏的,因为它与已知的美元(n\sqrt{n})在最大数量轴-单方平方形中以美元点确定,从而解决了一个超过30年以上实现这一模式的开放问题。我们将结果扩展为一个对美元(nqouraxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx