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arXiv · 2510.09328

Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree

Aniss Aiman Medbouhi·Alejandro García-Castellanos·Giovanni Luca Marchetti·Daniel Pelt·Erik J Bekkers·Danica Kragic
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Abstract

We study the problem of constructing Steiner Minimal Trees (SMTs) in hyperbolic space. Exact SMT computation is NP-hard, and existing hyperbolic heuristics such as HyperSteiner are deterministic and often get trapped in locally suboptimal configurations. We introduce Randomized HyperSteiner (RHS), a stochastic Delaunay triangulation heuristic that incorporates randomness into the expansion process and refines candidate trees via Riemannian gradient descent optimization. Experiments on synthetic data sets and a real-world single-cell transcriptomic data show that RHS outperforms Minimum Spanning Tree (MST), Neighbour Joining, and vanilla HyperSteiner (HS). In near-boundary configurations, RHS can achieve a 32% reduction in total length over HS, demonstrating its effectiveness and robustness in diverse data regimes.

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Aniss Aiman Medbouhi, Alejandro García-Castellanos, Giovanni Luca Marchetti, Daniel Pelt, Erik J Bekkers, Danica Kragic. 2025-10-10. Randomized HyperSteiner: A Stochastic Delaunay Triangulation Heuristic for the Hyperbolic Steiner Minimal Tree. https://arxiv.org/abs/2510.09328

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