430 research outputs found
Linear-Time Algorithms for Geometric Graphs with Sublinearly Many Edge Crossings
We provide linear-time algorithms for geometric graphs with sublinearly many
crossings. That is, we provide algorithms running in O(n) time on connected
geometric graphs having n vertices and k crossings, where k is smaller than n
by an iterated logarithmic factor. Specific problems we study include Voronoi
diagrams and single-source shortest paths. Our algorithms all run in linear
time in the standard comparison-based computational model; hence, we make no
assumptions about the distribution or bit complexities of edge weights, nor do
we utilize unusual bit-level operations on memory words. Instead, our
algorithms are based on a planarization method that "zeroes in" on edge
crossings, together with methods for extending planar separator decompositions
to geometric graphs with sublinearly many crossings. Incidentally, our
planarization algorithm also solves an open computational geometry problem of
Chazelle for triangulating a self-intersecting polygonal chain having n
segments and k crossings in linear time, for the case when k is sublinear in n
by an iterated logarithmic factor.Comment: Expanded version of a paper appearing at the 20th ACM-SIAM Symposium
on Discrete Algorithms (SODA09
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
Crossing Patterns in Nonplanar Road Networks
We define the crossing graph of a given embedded graph (such as a road
network) to be a graph with a vertex for each edge of the embedding, with two
crossing graph vertices adjacent when the corresponding two edges of the
embedding cross each other. In this paper, we study the sparsity properties of
crossing graphs of real-world road networks. We show that, in large road
networks (the Urban Road Network Dataset), the crossing graphs have connected
components that are primarily trees, and that the remaining non-tree components
are typically sparse (technically, that they have bounded degeneracy). We prove
theoretically that when an embedded graph has a sparse crossing graph, it has
other desirable properties that lead to fast algorithms for shortest paths and
other algorithms important in geographic information systems. Notably, these
graphs have polynomial expansion, meaning that they and all their subgraphs
have small separators.Comment: 9 pages, 4 figures. To appear at the 25th ACM SIGSPATIAL
International Conference on Advances in Geographic Information Systems(ACM
SIGSPATIAL 2017
On Visibility Representations of Non-planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an
assignment of axis-aligned rectangles to vertices such that for every edge
there exists a horizontal or vertical line of sight between the rectangles
assigned to its endpoints. Testing whether a graph has an RVR is known to be
NP-hard. In this paper, we study the problem of finding an RVR under the
assumption that an embedding in the plane of the input graph is fixed and we
are looking for an RVR that reflects this embedding. We show that in this case
the problem can be solved in polynomial time for general embedded graphs and in
linear time for 1-plane graphs (i.e., embedded graphs having at most one
crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of
1-plane graphs. These forbidden configurations can be tested for in linear
time, and so in linear time we can test whether a 1-plane graph has an RVR and
either compute such a representation or report a negative witness. Finally, we
discuss some extensions of our study to the case when the embedding is not
fixed but the RVR can have at most one crossing per edge
Crossing Minimization for 1-page and 2-page Drawings of Graphs with Bounded Treewidth
We investigate crossing minimization for 1-page and 2-page book drawings. We
show that computing the 1-page crossing number is fixed-parameter tractable
with respect to the number of crossings, that testing 2-page planarity is
fixed-parameter tractable with respect to treewidth, and that computing the
2-page crossing number is fixed-parameter tractable with respect to the sum of
the number of crossings and the treewidth of the input graph. We prove these
results via Courcelle's theorem on the fixed-parameter tractability of
properties expressible in monadic second order logic for graphs of bounded
treewidth.Comment: Graph Drawing 201
On Graph Crossing Number and Edge Planarization
Given an n-vertex graph G, a drawing of G in the plane is a mapping of its
vertices into points of the plane, and its edges into continuous curves,
connecting the images of their endpoints. A crossing in such a drawing is a
point where two such curves intersect. In the Minimum Crossing Number problem,
the goal is to find a drawing of G with minimum number of crossings. The value
of the optimal solution, denoted by OPT, is called the graph's crossing number.
This is a very basic problem in topological graph theory, that has received a
significant amount of attention, but is still poorly understood
algorithmically. The best currently known efficient algorithm produces drawings
with crossings on bounded-degree graphs, while only a
constant factor hardness of approximation is known. A closely related problem
is Minimum Edge Planarization, in which the goal is to remove a
minimum-cardinality subset of edges from G, such that the remaining graph is
planar. Our main technical result establishes the following connection between
the two problems: if we are given a solution of cost k to the Minimum Edge
Planarization problem on graph G, then we can efficiently find a drawing of G
with at most \poly(d)\cdot k\cdot (k+OPT) crossings, where is the maximum
degree in G. This result implies an O(n\cdot \poly(d)\cdot
\log^{3/2}n)-approximation for Minimum Crossing Number, as well as improved
algorithms for special cases of the problem, such as, for example, k-apex and
bounded-genus graphs
A Note on the Practicality of Maximal Planar Subgraph Algorithms
Given a graph , the NP-hard Maximum Planar Subgraph problem (MPS) asks for
a planar subgraph of with the maximum number of edges. There are several
heuristic, approximative, and exact algorithms to tackle the problem, but---to
the best of our knowledge---they have never been compared competitively in
practice. We report on an exploratory study on the relative merits of the
diverse approaches, focusing on practical runtime, solution quality, and
implementation complexity. Surprisingly, a seemingly only theoretically strong
approximation forms the building block of the strongest choice.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time
Thomassen characterized some 1-plane embedding as the forbidden configuration
such that a given 1-plane embedding of a graph is drawable in straight-lines if
and only if it does not contain the configuration [C. Thomassen, Rectilinear
drawings of graphs, J. Graph Theory, 10(3), 335-341, 1988].
In this paper, we characterize some 1-plane embedding as the forbidden
configuration such that a given 1-plane embedding of a graph can be re-embedded
into a straight-line drawable 1-plane embedding of the same graph if and only
if it does not contain the configuration. Re-embedding of a 1-plane embedding
preserves the same set of pairs of crossing edges.
We give a linear-time algorithm for finding a straight-line drawable 1-plane
re-embedding or the forbidden configuration.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016). This is an extended
abstract. For a full version of this paper, see Hong S-H, Nagamochi H.:
Re-embedding a 1-Plane Graph into a Straight-line Drawing in Linear Time,
Technical Report TR 2016-002, Department of Applied Mathematics and Physics,
Kyoto University (2016
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