5,029 research outputs found
Block Crossings in Storyline Visualizations
Storyline visualizations help visualize encounters of the characters in a
story over time. Each character is represented by an x-monotone curve that goes
from left to right. A meeting is represented by having the characters that
participate in the meeting run close together for some time. In order to keep
the visual complexity low, rather than just minimizing pairwise crossings of
curves, we propose to count block crossings, that is, pairs of intersecting
bundles of lines.
Our main results are as follows. We show that minimizing the number of block
crossings is NP-hard, and we develop, for meetings of bounded size, a
constant-factor approximation. We also present two fixed-parameter algorithms
and, for meetings of size 2, a greedy heuristic that we evaluate
experimentally.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and
curves, respectively, while clusters are represented by simple closed regions.
A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region,
or region-region crossings. Determining the complexity of testing whether a
clustered graph admits a c-planar drawing is a long-standing open problem in
the Graph Drawing research area. An obvious necessary condition for c-planarity
is the planarity of the graph underlying the clustered graph. However, such a
condition is not sufficient and the consequences on the problem due to the
requirement of not having edge-region and region-region crossings are not yet
fully understood.
In order to shed light on the c-planarity problem, we consider a relaxed
version of it, where some kinds of crossings (either edge-edge, edge-region, or
region-region) are allowed even if the underlying graph is planar. We
investigate the relationships among the minimum number of edge-edge,
edge-region, and region-region crossings for drawings of the same clustered
graph. Also, we consider drawings in which only crossings of one kind are
admitted. In this setting, we prove that drawings with only edge-edge or with
only edge-region crossings always exist, while drawings with only region-region
crossings may not. Further, we provide upper and lower bounds for the number of
such crossings. Finally, we give a polynomial-time algorithm to test whether a
drawing with only region-region crossings exist for biconnected graphs, hence
identifying a first non-trivial necessary condition for c-planarity that can be
tested in polynomial time for a noticeable class of graphs
On the complexity of optimal homotopies
In this article, we provide new structural results and algorithms for the
Homotopy Height problem. In broad terms, this problem quantifies how much a
curve on a surface needs to be stretched to sweep continuously between two
positions. More precisely, given two homotopic curves and
on a combinatorial (say, triangulated) surface, we investigate the problem of
computing a homotopy between and where the length of the
longest intermediate curve is minimized. Such optimal homotopies are relevant
for a wide range of purposes, from very theoretical questions in quantitative
homotopy theory to more practical applications such as similarity measures on
meshes and graph searching problems.
We prove that Homotopy Height is in the complexity class NP, and the
corresponding exponential algorithm is the best one known for this problem.
This result builds on a structural theorem on monotonicity of optimal
homotopies, which is proved in a companion paper. Then we show that this
problem encompasses the Homotopic Fr\'echet distance problem which we therefore
also establish to be in NP, answering a question which has previously been
considered in several different settings. We also provide an O(log
n)-approximation algorithm for Homotopy Height on surfaces by adapting an
earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the
planar setting
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth
We investigate crossing minimization for 1-page and 2-page book drawings. We
show that computing the 1-page crossing number is fixed-parameter tractable
with respect to the number of crossings, that testing 2-page planarity is
fixed-parameter tractable with respect to treewidth, and that computing the
2-page crossing number is fixed-parameter tractable with respect to the sum of
the number of crossings and the treewidth of the input graph. We prove these
results via Courcelle's theorem on the fixed-parameter tractability of
properties expressible in monadic second order logic for graphs of bounded
treewidth.Comment: Graph Drawing 201
Stress-Minimizing Orthogonal Layout of Data Flow Diagrams with Ports
We present a fundamentally different approach to orthogonal layout of data
flow diagrams with ports. This is based on extending constrained stress
majorization to cater for ports and flow layout. Because we are minimizing
stress we are able to better display global structure, as measured by several
criteria such as stress, edge-length variance, and aspect ratio. Compared to
the layered approach, our layouts tend to exhibit symmetries, and eliminate
inter-layer whitespace, making the diagrams more compact
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