  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.
