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

TL;DR

The paper tackles the Steiner Minimal Tree problem in hyperbolic space, where exact solutions are NP-hard and existing deterministic heuristics can be locally suboptimal. It introduces Randomized HyperSteiner (RHS), a stochastic DT-based framework that expands candidate Steiner configurations via randomized hyperbolic DTs and refines them with Riemannian gradient descent, balancing exploration and optimization. Empirical results on synthetic and real datasets show RHS consistently beats MST, NJ, and vanilla HS, with the largest gains in near-boundary configurations—up to roughly 43% reductions and approaching the hyperbolic upper bound of 50%. The work demonstrates that leveraging hyperbolic geometry for SMT enables robust, near-optimal tree reconstructions in hierarchical data, at the cost of higher computational effort, and outlines future PTAS directions to bridge the remaining gap while controlling runtime.

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.

Figures (12)

• Figure 1: Comparison of MST and heuristic SMTs using deterministic versus stochastic sampling over Delaunay triangles. Red ($\newmoon$) denotes terminals, blue ($\newmoon$) denotes Steiner points, and dashed lines correspond to the auxiliary hyperbolic DT. The deterministic (vanilla) HyperSteiner exhibits myopic behavior, whereas our randomized variant achieves superior global topology and reduces total tree length.
• Figure 2: Example of isoptic curves for $\alpha=2\pi/3$. Each color represents a different curve. Square ($\,\blacksquare\,$) denotes terminals while star ($\bigstar$) denotes Steiner points. Source: garcia2025hypersteiner.
• Figure 3: Performance comparison across data distributions. Randomized HyperSteiner (orange), vanilla HyperSteiner (red), and Neighbour Joining (green). Curves report mean tree-length reduction over MST (RED, %) with standard deviation (shaded), averaged across 10 runs.
• Figure 4: Convergence analysis for mixtures of $\mathcal{G}(\mu_{d, k}(t), 0.1)$, $k \in \{1, \ldots, d\}$ with $d \in \{3, \ldots, 10\}$ and varying radial parameter $t$, sampling 20 points per Gaussian. Values show percentage reduction in tree-length of RHS vs. HS (left) and NJ (right) as points approach the boundary.
• Figure 5: Tree-length reduction over the MST with $|P|$ terminals sampled near the boundary ($t=1-10^{-10}$). RHS (orange) and NJ (green) approach the theoretical bound, while HS (red) lags significantly.

Theorems & Definitions (13)

• Definition 1: Steiner ratio
• Definition 2
• Theorem 4.1: Nielsen and Nock, 2009
• Definition 3
• Proposition 4.2: García-Castellanos et al., 2025
• Theorem 4.3
• Definition 4: Retraction
• Lemma C.1
• proof
• Proposition C.2: Bridson1999