38 research outputs found

### Ordering Metro Lines by Block Crossings

A problem that arises in drawings of transportation networks is to minimize
the number of crossings between different transportation lines. While this can
be done efficiently under specific constraints, not all solutions are visually
equivalent. We suggest merging crossings into block crossings, that is,
crossings of two neighboring groups of consecutive lines. Unfortunately,
minimizing the total number of block crossings is NP-hard even for very simple
graphs. We give approximation algorithms for special classes of graphs and an
asymptotically worst-case optimal algorithm for block crossings on general
graphs. That is, we bound the number of block crossings that our algorithm
needs and construct worst-case instances on which the number of block crossings
that is necessary in any solution is asymptotically the same as our bound

### Approximating the Minimum Logarithmic Arrangement Problem

We study a graph reordering problem motivated by compressing massive graphs such as social networks and inverted indexes. Given a graph, G = (V, E), the Minimum Logarithmic Arrangement problem is to find a permutation, ?, of the vertices that minimizes ?_{(u, v) ? E} (1 + ? lg |?(u) - ?(v)| ?).
This objective has been shown to be a good measure of how many bits are needed to encode the graph if the adjacency list of each vertex is encoded using relative positions of two consecutive neighbors under the ? order in the list rather than using absolute indices or node identifiers, which requires at least lg n bits per edge.
We show the first non-trivial approximation factor for this problem by giving a polynomial time ?(log k)-approximation algorithm for graphs with treewidth k

### Edge Routing with Ordered Bundles

Edge bundling reduces the visual clutter in a drawing of a graph by uniting
the edges into bundles. We propose a method of edge bundling drawing each edge
of a bundle separately as in metro-maps and call our method ordered bundles. To
produce aesthetically looking edge routes it minimizes a cost function on the
edges. The cost function depends on the ink, required to draw the edges, the
edge lengths, widths and separations. The cost also penalizes for too many
edges passing through narrow channels by using the constrained Delaunay
triangulation. The method avoids unnecessary edge-node and edge-edge crossings.
To draw edges with the minimal number of crossings and separately within the
same bundle we develop an efficient algorithm solving a variant of the
metro-line crossing minimization problem. In general, the method creates clear
and smooth edge routes giving an overview of the global graph structure, while
still drawing each edge separately and thus enabling local analysis

### On Semantic Word Cloud Representation

We study the problem of computing semantic-preserving word clouds in which
semantically related words are close to each other. While several heuristic
approaches have been described in the literature, we formalize the underlying
geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this
model each word is associated with rectangle with fixed dimensions, and the
goal is to represent semantically related words by ensuring that the two
corresponding rectangles touch. We design and analyze efficient polynomial-time
algorithms for some variants of the WRAC problem, show that several general
variants are NP-hard, and describe a number of approximation algorithms.
Finally, we experimentally demonstrate that our theoretically-sound algorithms
outperform the early heuristics

### On Families of Planar DAGs with Constant Stack Number

A $k$-stack layout (or $k$-page book embedding) of a graph consists of a
total order of the vertices, and a partition of the edges into $k$ sets of
non-crossing edges with respect to the vertex order. The stack number of a
graph is the minimum $k$ such that it admits a $k$-stack layout.
In this paper we study a long-standing problem regarding the stack number of
planar directed acyclic graphs (DAGs), for which the vertex order has to
respect the orientation of the edges. We investigate upper and lower bounds on
the stack number of several families of planar graphs: We prove constant upper
bounds on the stack number of single-source and monotone outerplanar DAGs and
of outerpath DAGs, and improve the constant upper bound for upward planar
3-trees. Further, we provide computer-aided lower bounds for upward (outer-)
planar DAGs