Abstract:The force-directed paradigm is one of the few generic approaches to drawing graphs. Since force-directed algorithms can be extended easily, they are used frequently. Most of these algorithms are, however, quite slow on large graphs, as they compute a quadratic number of forces in each iteration. We give a new algorithm that takes only O(m + n log n) time per iteration when laying out a graph with n vertices and m edges. Our algorithm approximates the true forces using the so-called well-separated pair decomposition. We perform experiments on a large number of graphs and show that we can strongly reduce the runtime, even on graphs with less than a hundred vertices, without a significant influence on the quality of the drawings (in terms of the number of crossings and deviation in edge lengths).
Stick graphs are intersection graphs of horizontal and vertical line
segments that all touch a line of slope $-1$ and lie above this
line. De Luca et al. [De Luca et al. GD'18] considered the
recognition problem of stick graphs when no order is given ($\textsf{STICK}$),
when the order of either one of the two sets is given ($\textsf{STICK}_{\textsf A}$), and
when the order of both sets is given ($\textsf{STICK}_{\textsf{AB}}$). They showed
how to solve $\textsf{STICK}_{\textsf{AB}}$ efficiently.
In this paper, we improve
the running time of their algorithm, and we solve $\textsf{STICK}_{\textsf A}$ efficiently.
Further, we consider variants of these problems where the lengths of
the sticks are given as input. We show that these variants of
$\textsf{STICK}$, $\textsf{STICK}_{\textsf A}$, and $\textsf{STICK}_{\textsf{AB}}$ are all NP-complete.
On the positive side, we give an efficient solution for $\textsf{STICK}_{\textsf{AB}}$
with fixed stick lengths if there are no isolated vertices.
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