228 research outputs found
Spanners for Geometric Intersection Graphs
Efficient algorithms are presented for constructing spanners in geometric
intersection graphs. For a unit ball graph in R^k, a (1+\epsilon)-spanner is
obtained using efficient partitioning of the space into hypercubes and solving
bichromatic closest pair problems. The spanner construction has almost
equivalent complexity to the construction of Euclidean minimum spanning trees.
The results are extended to arbitrary ball graphs with a sub-quadratic running
time.
For unit ball graphs, the spanners have a small separator decomposition which
can be used to obtain efficient algorithms for approximating proximity problems
like diameter and distance queries. The results on compressed quadtrees,
geometric graph separators, and diameter approximation might be of independent
interest.Comment: 16 pages, 5 figures, Late
Track Layouts of Graphs
A \emph{-track layout} of a graph consists of a (proper) vertex
-colouring of , a total order of each vertex colour class, and a
(non-proper) edge -colouring such that between each pair of colour classes
no two monochromatic edges cross. This structure has recently arisen in the
study of three-dimensional graph drawings. This paper presents the beginnings
of a theory of track layouts. First we determine the maximum number of edges in
a -track layout, and show how to colour the edges given fixed linear
orderings of the vertex colour classes. We then describe methods for the
manipulation of track layouts. For example, we show how to decrease the number
of edge colours in a track layout at the expense of increasing the number of
tracks, and vice versa. We then study the relationship between track layouts
and other models of graph layout, namely stack and queue layouts, and geometric
thickness. One of our principle results is that the queue-number and
track-number of a graph are tied, in the sense that one is bounded by a
function of the other. As corollaries we prove that acyclic chromatic number is
bounded by both queue-number and stack-number. Finally we consider track
layouts of planar graphs. While it is an open problem whether planar graphs
have bounded track-number, we prove bounds on the track-number of outerplanar
graphs, and give the best known lower bound on the track-number of planar
graphs.Comment: The paper is submitted for publication. Preliminary draft appeared as
Technical Report TR-2003-07, School of Computer Science, Carleton University,
Ottawa, Canad
Computing the Greedy Spanner in Linear Space
The greedy spanner is a high-quality spanner: its total weight, edge count
and maximal degree are asymptotically optimal and in practice significantly
better than for any other spanner with reasonable construction time.
Unfortunately, all known algorithms that compute the greedy spanner of n points
use Omega(n^2) space, which is impractical on large instances. To the best of
our knowledge, the largest instance for which the greedy spanner was computed
so far has about 13,000 vertices.
We present a O(n)-space algorithm that computes the same spanner for points
in R^d running in O(n^2 log^2 n) time for any fixed stretch factor and
dimension. We discuss and evaluate a number of optimizations to its running
time, which allowed us to compute the greedy spanner on a graph with a million
vertices. To our knowledge, this is also the first algorithm for the greedy
spanner with a near-quadratic running time guarantee that has actually been
implemented
Height representation of XOR-Ising loops via bipartite dimers
The XOR-Ising model on a graph consists of random spin configurations on
vertices of the graph obtained by taking the product at each vertex of the
spins of two independent Ising models. In this paper, we explicitly relate loop
configurations of the XOR-Ising model and those of a dimer model living on a
decorated, bipartite version of the Ising graph. This result is proved for
graphs embedded in compact surfaces of genus g.
Using this fact, we then prove that XOR-Ising loops have the same law as
level lines of the height function of this bipartite dimer model. At
criticality, the height function is known to converge weakly in distribution to
a Gaussian free field.
As a consequence, results of this paper shed a light on the occurrence of the
Gaussian free field in the XOR-Ising model. In particular, they prove a
discrete analogue of Wilson's conjecture, stating that the scaling limit of
XOR-Ising loops are "contour lines" of the Gaussian free field.Comment: 41 pages, 10 figure
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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