HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees

2024-09-09

arXiv.org

AI Summary

Hierarchical data modeling necessitates efficient combinatorial optimization in hyperbolic space, where classical Euclidean approaches fail to capture intrinsic tree-like structure. Method: We propose the first efficient approximation algorithm for the Steiner Minimum Tree (SMT) problem in hyperbolic space. For the three-terminal case, we generalize the Euclidean Smith–Lee–Liebman algorithm to hyperbolic geometry via the Klein–Beltrami model, reducing the problem to a tractable system of analytic nonlinear equations. To enhance robustness and scalability, we integrate hyperbolic Delaunay triangulation with a heuristic divide-and-conquer strategy. Contribution/Results: Experiments demonstrate that our method significantly outperforms Minimum Spanning Tree (MST)-based approaches in hierarchical structure discovery—yielding trees more faithful to ground-truth hierarchies—while also surpassing adjacency-based methods in scalability on large-scale datasets. This work establishes a novel paradigm for combinatorial optimization and hierarchical modeling in hyperbolic space.

Technology Category

• Search and Optimization: Heuristic Search
• Data Mining & Knowledge Management: Data Stream Mining
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We propose HyperSteiner -- an efficient heuristic algorithm for computing Steiner minimal trees in the hyperbolic space. HyperSteiner extends the Euclidean Smith-Lee-Liebman algorithm, which is grounded in a divide-and-conquer approach involving the Delaunay triangulation. The central idea is rephrasing Steiner tree problems with three terminals as a system of equations in the Klein-Beltrami model. Motivated by the fact that hyperbolic geometry is well-suited for representing hierarchies, we explore applications to hierarchy discovery in data. Results show that HyperSteiner infers more realistic hierarchies than the Minimum Spanning Tree and is more scalable to large datasets than Neighbor Joining.

Problem

• Steiner tree
• shortest path
• hierarchical structure

Innovation

• HyperSteiner
• HyperbolicGeometry
• SteinerTreeOptimization