9,352 research outputs found
Rectangular Layouts and Contact Graphs
Contact graphs of isothetic rectangles unify many concepts from applications
including VLSI and architectural design, computational geometry, and GIS.
Minimizing the area of their corresponding {\em rectangular layouts} is a key
problem. We study the area-optimization problem and show that it is NP-hard to
find a minimum-area rectangular layout of a given contact graph. We present
O(n)-time algorithms that construct -area rectangular layouts for
general contact graphs and -area rectangular layouts for trees.
(For trees, this is an -approximation algorithm.) We also present an
infinite family of graphs (rsp., trees) that require (rsp.,
) area.
We derive these results by presenting a new characterization of graphs that
admit rectangular layouts using the related concept of {\em rectangular duals}.
A corollary to our results relates the class of graphs that admit rectangular
layouts to {\em rectangle of influence drawings}.Comment: 28 pages, 13 figures, 55 references, 1 appendi
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
Exact and fixed-parameter algorithms for metro-line crossing minimization problems
A metro-line crossing minimization problem is to draw multiple lines on an
underlying graph that models stations and rail tracks so that the number of
crossings of lines becomes minimum. It has several variations by adding
restrictions on how lines are drawn. Among those, there is one with a
restriction that line terminals have to be drawn at a verge of a station, and
it is known to be NP-hard even when underlying graphs are paths. This paper
studies the problem in this setting, and propose new exact algorithms. We first
show that a problem to decide if lines can be drawn without crossings is solved
in polynomial time, and propose a fast exponential algorithm to solve a
crossing minimization problem. We then propose a fixed-parameter algorithm with
respect to the multiplicity of lines, which implies that the problem is FPT.Comment: 19 pages, 15 figure
Geometric versions of the 3-dimensional assignment problem under general norms
We discuss the computational complexity of special cases of the 3-dimensional
(axial) assignment problem where the elements are points in a Cartesian space
and where the cost coefficients are the perimeters of the corresponding
triangles measured according to a certain norm. (All our results also carry
over to the corresponding special cases of the 3-dimensional matching problem.)
The minimization version is NP-hard for every norm, even if the underlying
Cartesian space is 2-dimensional. The maximization version is polynomially
solvable, if the dimension of the Cartesian space is fixed and if the
considered norm has a polyhedral unit ball. If the dimension of the Cartesian
space is part of the input, the maximization version is NP-hard for every
norm; in particular the problem is NP-hard for the Manhattan norm and the
Maximum norm which both have polyhedral unit balls.Comment: 21 pages, 9 figure
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
Lombardi Drawings of Graphs
We introduce the notion of Lombardi graph drawings, named after the American
abstract artist Mark Lombardi. In these drawings, edges are represented as
circular arcs rather than as line segments or polylines, and the vertices have
perfect angular resolution: the edges are equally spaced around each vertex. We
describe algorithms for finding Lombardi drawings of regular graphs, graphs of
bounded degeneracy, and certain families of planar graphs.Comment: Expanded version of paper appearing in the 18th International
Symposium on Graph Drawing (GD 2010). 13 pages, 7 figure
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