Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering (2104.01126v1)

Abstract: This paper presents new parallel algorithms for generating Euclidean minimum spanning trees and spatial clustering hierarchies (known as HDBSCAN$^*$). Our approach is based on generating a well-separated pair decomposition followed by using Kruskal's minimum spanning tree algorithm and bichromatic closest pair computations. We introduce a new notion of well-separation to reduce the work and space of our algorithm for HDBSCAN$^*$. We also present a parallel approximate algorithm for OPTICS based on a recent sequential algorithm by Gan and Tao. Finally, we give a new parallel divide-and-conquer algorithm for computing the dendrogram and reachability plots, which are used in visualizing clusters of different scale that arise for both EMST and HDBSCAN$^*$. We show that our algorithms are theoretically efficient: they have work (number of operations) matching their sequential counterparts, and polylogarithmic depth (parallel time). We implement our algorithms and propose a memory optimization that requires only a subset of well-separated pairs to be computed and materialized, leading to savings in both space (up to 10x) and time (up to 8x). Our experiments on large real-world and synthetic data sets using a 48-core machine show that our fastest algorithms outperform the best serial algorithms for the problems by 11.13--55.89x, and existing parallel algorithms by at least an order of magnitude.

• The paper introduces fast parallel algorithms for Euclidean Minimum Spanning Tree (EMST) and Hierarchical Spatial Clustering (HDBSCAN*) leveraging a novel well-separated pair decomposition (WSPD).
• It proposes a novel well-separation concept for HDBSCAN* and a parallel divide-and-conquer strategy for dendrogram construction to reduce complexity and memory.
• Optimized implementation yields significant memory (up to 10x) and time (up to 8x) savings, outperforming existing algorithms by orders of magnitude.

Overview of "Fast Parallel Algorithms for Euclidean Minimum Spanning Tree and Hierarchical Spatial Clustering"

The paper presents a set of innovative parallel algorithms designed to efficiently compute the Euclidean Minimum Spanning Tree (EMST) and facilitate Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN$^*$). The authors employ a sophisticated approach grounded in a well-separated pair decomposition (WSPD) methodology, which integrates Kruskal's algorithm and bichromatic closest pair computations to enhance both computational and memory efficiency. This work introduces pioneering concepts in the field of parallel algorithms, particularly with respect to spatial clustering and graph-based operations.

Key Contributions

1. Parallel Algorithms for EMST and HDBSCAN$^*$: The paper introduces parallel algorithms capable of generating EMSTs and HDBSCAN$^*$ clustering hierarchies efficiently. The core technique revolves around leveraging a well-separated pair decomposition, which simplifies the construction of the EMST and facilitates parallel execution of Kruskal's algorithm.
2. New Concept of Well-Separation: The authors propose a novel notion of well-separation specifically tailored for the HDBSCAN$^*$ problem. This refined definition allows the algorithm to reduce overall computational complexity and memory requirements by avoiding unnecessary calculations and focusing on critical operations.
3. Divide-and-Conquer Approach for Dendrogram Construction: The paper introduces a robust parallel divide-and-conquer strategy to create dendrograms and reachability plots, which are instrumental in visualizing clusters of varying scales in both EMST and HDBSCAN$^*$ scenarios. This approach offers significant improvements over traditional methods by maintaining theoretical efficiency and scalability.
4. Implementation and Optimization Techniques: The implementation emphasizes memory optimization by limiting the calculation and storage of well-separated pairs. This results in substantial savings in both space usage (up to 10x) and processing time (up to 8x) when applied to large data sets. Experimental results validate the proposed algorithms' superiority over existing serial and parallel solutions.

Experimental Evaluation

The experimental analysis conducted on a 48-core machine encompasses both synthetic and sizable real-world data sets. The results demonstrate that the fastest algorithms developed in the research outperform existing serial methods by 11.13--55.89x and parallel algorithms by an order of magnitude, reinforcing the paper's contributions to advancing the state of the art in parallel computations for spatial clustering and minimum spanning tree calculations.

Implications and Future Developments

The research has significant implications for both theoretical and practical applications in large-scale spatial data analysis. The reduction in computational complexity and memory usage offered by the new algorithms facilitates real-time processing of vast data volumes encountered in fields such as geospatial analysis, network optimization, and large-scale machine learning tasks. Future developments may explore further refinements of the separation criteria introduced, as well as applications of these algorithms to a broader range of graph-based problems in parallel computing environments.

The advancements presented in this paper set a foundation for continued exploration into parallel algorithms for complex clustering tasks, potentially influencing a broad spectrum of data-intensive applications. The theoretical insights, coupled with demonstrated practical improvements, mark a significant progression in the field of computational geometry and spatial data analysis.