23,545 research outputs found
An Heuristic for the Construction of Intersection Graphs
International audienceMost methods for generating Euler diagrams describe the detection of the general structure of the final drawing as the first step. This information is generally encoded using a graph, where nodes are the regions to be represented and edges represent adjacency. A planar drawing of this graph will then indicate how to draw the sets in order to depict all the set intersections. In this paper we present an heuristic to construct this structure, the intersection graph. The final Euler diagram can be constructed by drawing the sets boundaries around the nodes of the intersection graph, either manually or automatically
Progressive Simplification of Polygonal Curves
Simplifying polygonal curves at different levels of detail is an important
problem with many applications. Existing geometric optimization algorithms are
only capable of minimizing the complexity of a simplified curve for a single
level of detail. We present an -time algorithm that takes a polygonal
curve of n vertices and produces a set of consistent simplifications for m
scales while minimizing the cumulative simplification complexity. This
algorithm is compatible with distance measures such as the Hausdorff, the
Fr\'echet and area-based distances, and enables simplification for continuous
scaling in time. To speed up this algorithm in practice, we present
new techniques for constructing and representing so-called shortcut graphs.
Experimental evaluation of these techniques on trajectory data reveals a
significant improvement of using shortcut graphs for progressive and
non-progressive curve simplification, both in terms of running time and memory
usage.Comment: 20 pages, 20 figure
Density Functional Theory for Hard Particles in N Dimensions
Recently it has been shown that the heuristic Rosenfeld functional derives
from the virial expansion for particles which overlap in one center. Here, we
generalize this approach to any number of intersections. Starting from the
virial expansion in Ree-Hoover diagrams, it is shown in the first part that
each intersection pattern defines exactly one infinite class of diagrams.
Determining their automorphism groups, we sum over all its elements and derive
a generic functional. The second part proves that this functional factorizes
into a convolute of integral kernels for each intersection center. We derive
this kernel for N dimensional particles in the N dimensional, flat Euclidean
space. The third part focuses on three dimensions and determines the
functionals for up to four intersection centers, comparing the leading order to
Rosenfeld's result. We close by proving a generalized form of the Blaschke,
Santalo, Chern equation of integral geometry.Comment: 2 figure
Drawing Arrangement Graphs In Small Grids, Or How To Play Planarity
We describe a linear-time algorithm that finds a planar drawing of every
graph of a simple line or pseudoline arrangement within a grid of area
O(n^{7/6}). No known input causes our algorithm to use area
\Omega(n^{1+\epsilon}) for any \epsilon>0; finding such an input would
represent significant progress on the famous k-set problem from discrete
geometry. Drawing line arrangement graphs is the main task in the Planarity
puzzle.Comment: 12 pages, 8 figures. To appear at 21st Int. Symp. Graph Drawing,
Bordeaux, 201
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