Singular Arrange and Traverse Algorithm for Computing Reeb Spaces of Bivariate PL Maps

2026-02-24

AI Summary

This work addresses the inefficiency of existing methods for computing Reeb spaces of multivariate functions, which hinders their practical applicability. We propose a novel algorithm based on an “arrange-and-traverse” framework that leverages the key observation that, in piecewise-linear bivariate mappings, only singular edges influence the topology of the Reeb space. By precisely identifying these singular edges and integrating geometric and topological analysis, our approach drastically reduces computational overhead. The method maintains exactness while achieving speedups of up to four orders of magnitude over state-of-the-art algorithms on real-world datasets, substantially enhancing the practicality and scalability of Reeb space computation.

Technology Category

• Machine Learning: Learning with Manifolds
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We present an exact and efficient algorithm for computing the Reeb space of a bivariate PL map. The Reeb space is a topological structure that generalizes the Reeb graph to the setting of multiple scalar-valued functions defined over a shared domain, a situation that frequently arises in practical applications. While the Reeb graph has become a standard tool in computer graphics, shape analysis, and scientific visualization, the Reeb space is still in the early stages of adoption. Although several algorithms for computing the Reeb space have been proposed, none offer an implementation that is both exact and efficient, which has substantially limited its practical use. To address this gap, we introduce singular arrange and traverse, a new algorithm built upon the arrange and traverse framework. Our method exploits the fact that, in the bivariate case, only singular edges contribute to the structure of Reeb space, allowing us to ignore many regular edges. This observation results in substantial efficiency gains on datasets where most edges are regular, which is common in many numerical simulations of physical systems. We provide an implementation of our method and benchmark it against the original arrange and traverse algorithm, showing performance gains of up to four orders of magnitude on real-world datasets.

Problem

• Reeb space
• bivariate PL maps
• exact algorithm
• efficient computation
• topological structure

Innovation

• Reeb space
• bivariate PL maps
• singular edges
• efficient algorithm
• topological data analysis