2023/02/21 by A. Karim Abu-Affash, Abu-Affash, A. Karim, Sujoy Bhore +3
Computer Science · Engineering · #3D Shape Modeling and Analysis #Advanced Numerical Analysis Techniques #Computational Geometry (cs.CG) #Computational Geometry and Mesh Generation #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.2302.10513
openalex publication_date 2023/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A fundamental question in computational geometry is for a set of input points in the Euclidean space, that is subject to discrete changes (insertion/deletion of points at each time step), whether it is possible to maintain an approximate bottleneck matching in sublinear update time. In this work, we answer this question in the affirmative for points on a real line and for points in the plane with a bounded geometric spread. For a set P of n points on a line, we show that there exists a dynamic algorithm that maintains a bottleneck matching of P and supports insertion and deletion in O(log n) time. Moreover, we show that a modified version of this algorithm maintains a minimum-weight matching with O(log n) update (insertion and deletion) time. Next, for a set P of n points in the plane, we show that a (6√(2))-factor approximate bottleneck matching of Pk, at each time step k, can be maintained in O(logΔ) amortized time per insertion and O(logΔ + |Pk|) amortized time per deletion, where Δ is the geometric spread of P.