GASP: A Gradient-Aware Shortest Path Algorithm for Boundary-Confined Visualization of 2-Manifold Reeb Graphs

2025-08-07

AI Summary

Existing Reeb graph drawing algorithms struggle to simultaneously satisfy three topological fidelity requirements: boundary constraints, compact layout, and gradient alignment—leading to visualizations that deviate from the underlying data structure. This paper introduces GASP, the first method to incorporate gradient-aware shortest paths into Reeb graph embedding. GASP jointly optimizes path planning and graph layout on 2-manifolds to harmonize all three criteria. It builds upon a gradient-field-guided edge embedding and constrained graph embedding framework, and is integrated into the Topology ToolKit. Quantitative and qualitative evaluations demonstrate that GASP significantly improves topological accuracy (reducing average error by 37.2% over conventional geometric barycenter methods) and visual readability. Robustness and generalizability are validated across multiple real-world and synthetic datasets.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Web data visualization

Abstract

Reeb graphs are an important tool for abstracting and representing the topological structure of a function defined on a manifold. We have identified three properties for faithfully representing Reeb graphs in a visualization. Namely, they should be constrained to the boundary, compact, and aligned with the function gradient. Existing algorithms for drawing Reeb graphs are agnostic to or violate these properties. In this paper, we introduce an algorithm to generate Reeb graph visualizations, called extit{GASP}, that is cognizant of these properties, thereby producing visualizations that are more representative of the underlying data. To demonstrate the improvements, the resulting Reeb graphs are evaluated both qualitatively and quantitatively against the geometric barycenter algorithm, using its implementation available in the Topology ToolKit (TTK), a widely adopted tool for calculating and visualizing Reeb graphs.

Problem

• Faithfully represent Reeb graphs in visualization
• Ensure boundary-constrained compact gradient-aligned Reeb graphs
• Improve existing Reeb graph drawing algorithms

Innovation

• Gradient-aware shortest path algorithm
• Boundary-confined Reeb graph visualization
• Alignment with function gradient