GPU-Accelerated Construction of the Gabriel Graph: Parallelization Strategies and Applications to Large-Margin Classification and Graph-Based Clustering

The Gabriel graph is a proximity graph widely used in computational geometry and geometric machine learning, underlying large-margin classifiers and, as we show in this work, density-aware clustering methods. Its construction, however, has cubic time complexity $O(N^3)$ in the naive formulation, which severely limits its use on datasets beyond a few thousand samples. This paper presents a set of GPU parallelization strategies, implemented in CUDA, that make Gabriel graph construction practical at scale. We introduce (i) an implicit closed-form mapping from a linear thread index to the corresponding vertex pair $(i,j)$, which removes the need for an explicit $O(N^2)$ adjacency structure and lets every thread locate its pair in constant time and constant memory; (ii) a two-pass, bitmask-based, atomically-compacted edge-list construction that produces a dense edge array proportional only to the number of edges actually found; and (iii) a thread-coarsening strategy in which each thread evaluates several candidate pairs, trading launch and scheduling overhead against load balance. We evaluate these strategies on the acceleration of a Gabriel-graph-based large-margin classifier (NN-clas) and on a novel Gabriel-graph clustering algorithm that identifies connected components on a residual graph and performs GPU-parallel Union-Find with hierarchical edge-based merging. On 14 public benchmark datasets, the GPU implementation reduces Gabriel graph construction time by two to nearly four orders of magnitude relative to a sequential CPU baseline (up to 8484x), without loss of classification accuracy, and the proposed clustering method is statistically competitive with DBSCAN on the topological validity index Dunn while eliminating the need for an explicit density radius parameter.