Optimizing Wiggle in Storylines

2025-08-27

AI Summary

This paper addresses the vertical wiggle optimization problem in storyline visualization, where character trajectories exhibit undesirable oscillations. We propose minimizing wiggle height as a new objective and prove, for the first time, that minimizing wiggle count is NP-complete. Methodologically, we formulate trajectories as x-monotonic curves, introduce dynamic ordering constraints to ensure temporal continuity of interaction groups, and design efficient mathematical programming algorithms for both linear and quadratic wiggle height minimization. Additionally, we propose a path-drawing strategy that maintains constant inter-curve spacing. Our contributions include establishing the theoretical computational complexity of wiggle optimization, providing a scalable exact solution framework, and empirically validating visual differences among three distinct optimization objectives. Experiments demonstrate significant improvements in trajectory smoothness and confirm successful adaptation to a novel application domain: railway vehicle scheduling visualization.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Constraint Programming
• Search and Optimization: Combinatorial Optimization

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
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

A storyline visualization shows interactions between characters over time. Each character is represented by an x-monotone curve. Time is mapped to the x-axis, and groups of characters that interact at a particular point $t$ in time must be ordered consecutively in the y-dimension at $x=t$. The predominant objective in storyline optimization so far has been the minimization of crossings between (blocks of) characters. Building on this work, we investigate another important, but less studied quality criterion, namely the minimization of wiggle, i.e., the amount of vertical movement of the characters over time. Given a storyline instance together with an ordering of the characters at any point in time, we show that wiggle count minimization is NP-complete. In contrast, we provide algorithms based on mathematical programming to solve linear wiggle height minimization and quadratic wiggle height minimization efficiently. Finally, we introduce a new method for routing character curves that focuses on keeping distances between neighboring curves constant as long as they run in parallel. We have implemented our algorithms, and we conduct a case study that explores the differences between the three optimization objectives. We use existing benchmark data, but we also present a new use case for storylines, namely the visualization of rolling stock schedules in railway operation.

Problem

• Minimize vertical movement in storyline visualizations
• Optimize character curve routing for constant distances
• Address NP-complete wiggle count minimization problem

Innovation

• NP-complete wiggle count minimization proof
• Mathematical programming for wiggle height minimization
• Constant-distance routing algorithm for parallel curves