Geometric Algorithms for Topological Data Analysis in Complex Networks

Abstract

Efficient algorithms for dynamic geometric data structures in high-dimensional spaces are increasingly critical in fields such as machine learning, computer graphics, and spatial databases, where large-scale, dynamic data is prevalent. This research explores the development of optimized geometric data structures capable of supporting dynamic operations—such as insertion, deletion, and querying—while maintaining performance and scalability in high-dimensional settings. By addressing challenges like the curse of dimensionality and computational complexity, the project aims to enhance the performance of algorithms used in high-dimensional geometric computations. Additionally, the integration of approximation techniques, parallel computing, and distributed algorithms will be explored to ensure scalability for large datasets. Practical applications of the research include real-time rendering, nearest neighbor searches, and spatial data querying in dynamic environments.