On Minimizing Crossings in Storyline Visualizations

TL;DR

This paper investigates the complexity of minimizing crossings in storyline visualizations, providing tight bounds for special cases and an FPT algorithm for the general case.

Contribution

It establishes tight bounds for crossing minimization in tree-structured meetings and introduces an FPT algorithm for the general case.

Findings

For tree-structured meetings, the number of crossings is tightly bounded by O(n log n).

There exist storylines requiring Omega(n log n) crossings, matching the bounds.

The problem is fixed-parameter tractable with respect to the number of characters.

Abstract

In a storyline visualization, we visualize a collection of interacting characters (e.g., in a movie, play, etc.) by $x$-monotone curves that converge for each interaction, and diverge otherwise. Given a storyline with $n$ characters, we show tight lower and upper bounds on the number of crossings required in any storyline visualization for a restricted case. In particular, we show that if (1) each meeting consists of exactly two characters and (2) the meetings can be modeled as a tree, then we can always find a storyline visualization with $O(n\log n)$ crossings. Furthermore, we show that there exist storylines in this restricted case that require $\Omega(n\log n)$ crossings. Lastly, we show that, in the general case, minimizing the number of crossings in a storyline visualization is fixed-parameter tractable, when parameterized on the number of characters $k$. Our algorithm runs in time $O(k!^2k\log k + k!^2m)$, where $m$ is the number of meetings.