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

TL;DR

StorySets introduces a novel visualization for uncertain set systems by representing elements as vertical glyphs and sets as x-monotone curves that pass through uncertainty bins. It integrates storyline and parallel-coordinate concepts to minimize crossings and curve wiggle, using a two-step optimization: ordering elements via TSP-inspired methods and then ordering the set curves with an exact algorithm that guarantees minimal crossings for a fixed element order. The method supports both storyline and star variants, providing clear containment cues and handling multi-dimensional discrete data without overplotting. Quantitative and qualitative evaluations on synthetic and real data demonstrate near-realtime feasibility and the ability to reveal containment and uncertainty patterns, with potential for interactive exploration and broader applicability to uncertainty visualization tasks.

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.

Figures (10)

• Figure 1: (a) The input: the (un)certainty of an element being a member of a set; (b--f) different visual representations of an uncertain set system with three sets $\{A, B, C\}$ and six elements $\{1, 2, 3, 4, 5, 6\}$.
• Figure 2: Different design variants of StorySets.
• Figure 3: (a) and (b) show the same order of a subset of elements. Even though we do not know the actual order of set curves during Step I, we can estimate the upper bound by looking at the assigned bins of two sets. Depending on the order of remaining elements such crossings are either necessary or can be removed with the curve ordering algorithm.
• Figure 4: The case distinctions of the curve ordering algorithm. Either $s_1$ or $s_2$ needs to be kept as representative after kernelization. During lookahead, it is determined that $s_3$ must be above $s_2$ as $s_3$ switches to a different bin that is above the bin in which $s_2$ remains. The lookahead determines that initially $s_3$ must be below $s_4$. The order of $s_3$ and $s_4$ can be kept until the element in which $s_4$ switches to a bin below.
• Figure 5: Our experiments revealed that a small number of iterations is sufficient. According to (a), most instances needed at most ten iterations before no improvement could be measured. Note that the last bin represents all instances that needed ten or more iterations. According to (b), the first two to three iterations improved the metrics by far the most.

Theorems & Definitions (2)

• Theorem 1
• proof