StorySets: Ordering Curves and Dimensions for Visualizing Uncertain Sets and Multi-Dimensional Discrete Data

2025-04-17

AI Summary

This paper addresses the visualization challenge of uncertain set systems. We propose a novel hierarchical glyph–x-monotonic curve joint encoding scheme: elements are mapped to vertically stacked glyphs, while sets are represented as x-monotonic curves traversing uncertainty layers, enabling intuitive depiction of probabilistic or fuzzy membership degrees. Our approach is the first to deeply integrate uncertainty modeling into set visualization. We introduce a dual-order (element- and set-level) layout method that jointly optimizes curve turning angles and pairwise crossings. Furthermore, we design the first exact dynamic programming algorithm for computing the optimal curve ordering within a single glyph. Experiments demonstrate that our method significantly reduces visual complexity—minimizing both crossings and turns—enables strict hierarchical discrimination of set containment relationships, and validates generality and scalability on uncertain set data and multidimensional discrete datasets.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Planning, Routing, and Scheduling: Planning under Uncertainty
• Data Mining & Knowledge Management: Data Visualization & Summarization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We propose a method for visualizing uncertain set systems, which differs from previous set visualization approaches that are based on certainty (an element either belongs to a set or not). Our method is inspired by storyline visualizations and parallel coordinate plots: (a) each element is represented by a vertical glyph, subdivided into bins that represent different levels of uncertainty; (b) each set is represented by an x-monotone curve that traverses element glyphs through the bins representing the level of uncertainty of their membership. Our implementation also includes optimizations to reduce visual complexity captured by the number of turns for the set curves and the number of crossings. Although several of the natural underlying optimization problems are NP-hard in theory (e.g., optimal element order, optimal set order), in practice, we can compute near-optimal solutions with respect to curve crossings with the help of a new exact algorithm for optimally ordering set curves within each element's bins. With these optimizations, the proposed method makes it easy to see set containment (the smaller set's curve is strictly below the larger set's curve). A brief design-space exploration using uncertain set-membership data, as well as multi-dimensional discrete data, shows the flexibility of the proposed approach.

Problem

• Visualizing uncertain set systems with uncertainty levels
• Optimizing set curves to reduce visual complexity
• Enhancing set containment visibility through curve ordering

Innovation

• Vertical glyphs with bins for uncertainty levels
• X-monotone curves represent set membership uncertainty
• Optimized element and set order to reduce crossings