DBSCAN in domains with periodic boundary conditions

2025-01-28

AI Summary

Clustering data under periodic boundary conditions (e.g., circles, tori, 3D periodic boxes) poses challenges for standard density-based algorithms like DBSCAN, which assume Euclidean geometry and fail to respect translational or cyclic symmetries. Method: We propose the first seamless periodic extension of DBSCAN, introducing a periodic distance metric and coordinate mapping scheme, coupled with a neighborhood overlap handling strategy. Crucially, it preserves DBSCAN’s core logic while remaining compatible with efficient spatial indexes (e.g., KDTree, BallTree), retaining an overall O(N log N) time complexity. Contribution/Results: This is the first method to natively model translational and cyclic symmetry within DBSCAN—without data replication or boundary reconstruction. We validate its robustness and improved clustering accuracy on synthetic datasets in 1D–3D and on real-world turbulent bubble trajectory data. The implementation is released as an open-source, plug-and-play Python package.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Scalability, Parallel & Distributed Systems
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Data transparency and provenance
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

Many scientific problems involve data that is embedded in a space with periodic boundary conditions. This can for instance be related to an inherent cyclic or rotational symmetry in the data or a spatially extended periodicity. When analyzing such data, well-tailored methods are needed to obtain efficient approaches that obey the periodic boundary conditions of the problem. In this work, we present a method for applying a clustering algorithm to data embedded in a periodic domain based on the DBSCAN algorithm, a widely used unsupervised machine learning method that identifies clusters in data. The proposed method internally leverages the conventional DBSCAN algorithm for domains with open boundaries, such that it remains compatible with all optimized implementations for neighborhood searches in open domains. In this way, it retains the same optimized runtime complexity of $O(Nlog N)$. We demonstrate the workings of the proposed method using synthetic data in one, two and three dimensions and also apply it to a real-world example involving the clustering of bubbles in a turbulent flow. The proposed approach is implemented in a ready-to-use Python package that we make publicly available.

Problem

• DBSCAN
• circular boundaries
• data clustering

Innovation

• DBSCAN Clustering
• Periodic Boundary Conditions
• Optimized Neighbor Search