2021/12/30 by Aya Bernstine, Bernstine, Aya, Yehonatan Mizrahi +1
Computer Science · Engineering · #3D Shape Modeling and Analysis #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.2112.14980
openalex publication_date 2021/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Erdős' 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√(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 Ω(n√(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(n1+1/d) and space O(n), also matching the corresponding lower bound due to Elekes and Erdős.