14,665 research outputs found
Convergence theorems for some layout measures on random lattice and random geometric graphs
This work deals with convergence theorems and bounds on the
cost of several layout measures for lattice graphs, random
lattice graphs and sparse random geometric graphs. For full
square lattices, we give optimal layouts for the problems
still open. Our convergence theorems can be viewed as an
analogue of the Beardwood, Halton and Hammersley theorem for
the Euclidian TSP on random points in the -dimensional
cube. As the considered layout measures are
non-subadditive, we use percolation theory to obtain our
results on random lattices and random geometric graphs. In
particular, we deal with the subcritical regimes on these
class of graphs.Postprint (published version
Orientation-Constrained Rectangular Layouts
We construct partitions of rectangles into smaller rectangles from an input
consisting of a planar dual graph of the layout together with restrictions on
the orientations of edges and junctions of the layout. Such an
orientation-constrained layout, if it exists, may be constructed in polynomial
time, and all orientation-constrained layouts may be listed in polynomial time
per layout.Comment: To appear at Algorithms and Data Structures Symposium, Banff, Canada,
August 2009. 12 pages, 5 figure
Area-Universal Rectangular Layouts
A rectangular layout is a partition of a rectangle into a finite set of
interior-disjoint rectangles. Rectangular layouts appear in various
applications: as rectangular cartograms in cartography, as floorplans in
building architecture and VLSI design, and as graph drawings. Often areas are
associated with the rectangles of a rectangular layout and it might hence be
desirable if one rectangular layout can represent several area assignments. A
layout is area-universal if any assignment of areas to rectangles can be
realized by a combinatorially equivalent rectangular layout. We identify a
simple necessary and sufficient condition for a rectangular layout to be
area-universal: a rectangular layout is area-universal if and only if it is
one-sided. More generally, given any rectangular layout L and any assignment of
areas to its regions, we show that there can be at most one layout (up to
horizontal and vertical scaling) which is combinatorially equivalent to L and
achieves a given area assignment. We also investigate similar questions for
perimeter assignments. The adjacency requirements for the rectangles of a
rectangular layout can be specified in various ways, most commonly via the dual
graph of the layout. We show how to find an area-universal layout for a given
set of adjacency requirements whenever such a layout exists.Comment: 19 pages, 16 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
Visualizing and Interacting with Concept Hierarchies
Concept Hierarchies and Formal Concept Analysis are theoretically well
grounded and largely experimented methods. They rely on line diagrams called
Galois lattices for visualizing and analysing object-attribute sets. Galois
lattices are visually seducing and conceptually rich for experts. However they
present important drawbacks due to their concept oriented overall structure:
analysing what they show is difficult for non experts, navigation is
cumbersome, interaction is poor, and scalability is a deep bottleneck for
visual interpretation even for experts. In this paper we introduce semantic
probes as a means to overcome many of these problems and extend usability and
application possibilities of traditional FCA visualization methods. Semantic
probes are visual user centred objects which extract and organize reduced
Galois sub-hierarchies. They are simpler, clearer, and they provide a better
navigation support through a rich set of interaction possibilities. Since probe
driven sub-hierarchies are limited to users focus, scalability is under control
and interpretation is facilitated. After some successful experiments, several
applications are being developed with the remaining problem of finding a
compromise between simplicity and conceptual expressivity
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