10 research outputs found
Crossing minimisation Heuristics for 2-page drawings
The minimisation of edge crossings in a book drawing of a graph G is one of important
goals for a linear VLSI design, and the two-page crossing number of a graph G
provides an upper bound for the standard planar crossing number. We propose several
new heuristics for the 2-page drawing problem and test them on benchmark test
sets like Rome graphs, Random Connected Graphs and some typical graphs. We
get exact results of some structural graphs, and compare some of the experimental
results with the one in paper
Fixed Linear Crossing Minimization by Reduction to the Maximum Cut Problem
Many real-life scheduling, routing and locating problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this NP-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; practically efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we prove that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms
An improved neural network model for the two-page crossing number problem
The simplest graph drawing method is that of putting the vertices of a graph on a line and
drawing the edges as half-circles either above or below the line. Such drawings are called 2-page book drawings. The smallest number of crossings over all 2-page drawings of a graph G is called the 2-page crossing number of G. Cimikowski and Shope have solved the 2-page crossing number problem for an n-vertex and
m-edge graph by using a Hopfield network with 2m
neurons. We present here an improved Hopfield modelwith m neurons. The new model achieves much better performance in the quality of solutions and is more efficient than the model of Cimikowski and Shope for all graphs tested. The parallel time complexity of the algorithm, without considering the crossing number
calculations, is O(m), for the new Hopfield model with m processors clearly outperforming the previous algorithm
Genetic algorithms for the 2-page book drawing problem of graphs
The minimisation of edge crossings in a book drawing of a graph
is one of the important goals for a linear VLSI design, and the 2-page
crossing number of a graph provides an upper bound for the standard
planar crossing number. We design genetic algorithms for the 2-page
drawings, and test them on the benchmark test suits, Rome graphs and
Random Connected Graphs. We also test some circulant graphs, and
get better results than previously presented in the literature. Moreover,
we formalise three conjectures for certain kinds of circulant graphs,
supported by our experimental results
Experimental Evaluation of Book Drawing Algorithms
A -page book drawing of a graph consists of a linear ordering of
its vertices along a spine and an assignment of each edge to one of the
pages, which are half-planes bounded by the spine. In a book drawing, two edges
cross if and only if they are assigned to the same page and their vertices
alternate along the spine. Crossing minimization in a -page book drawing is
NP-hard, yet book drawings have multiple applications in visualization and
beyond. Therefore several heuristic book drawing algorithms exist, but there is
no broader comparative study on their relative performance. In this paper, we
propose a comprehensive benchmark set of challenging graph classes for book
drawing algorithms and provide an extensive experimental study of the
performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Parallelisation of genetic algorithms for the 2-page crossing number problem
Genetic algorithms have been applied to solve the 2-page crossing number problem
successfully, but since they work with one global population, the search time and
space are limited. Parallelisation provides an attractive prospect to improve the efficiency
and solution quality of genetic algorithms. This paper investigates the complexity
of parallel genetic algorithms (PGAs) based on two evaluation measures:
Computation-time to Communication-time and Population-size to Chromosomesize.
Moreover, the paper unifies the framework of PGA models with the function
PGA (subpopulation size; cluster size, migration period; topology), and explores
the performance of PGAs for the 2-page crossing number problem
Heuristic crossing minimisation algorithms for the two-page drawing problem
The minimisation of edge crossings in a book drawing of a graph G is one of the
important goals for a linear VLSI design, and the two-page crossing number of a graph G provides
an upper bound for the standard planar crossing number. We propose several new heuristics for
the two-page drawing problem, and test them on benchmark test suites, Rome graphs and Random
Connected Graphs. We also test some typical graphs, and get some exact results. The results for
some circulant graphs are better than the one presented by Cimikowski. We have a conjecture for
cartesian graphs, supported by our experimental results, and provide direct methods to get optimal
solutions for 3- or 4-row meshes and Halin graphs
Algorithms for the Fixed Linear Crossing Number Problem
Several heuristics and an exact branch-and-bound algorithm are described for the xed linear crossing number problem (FLCNP). An experimental study comparing the heuristics on a large set of test graphs is given. FLCNP is similar to the 2page book crossing number problem in which the vertices of a graph are optimally placed on a horizontal \node line" in the plane, each edge is drawn as an arc in one half-plane (page), and the objective is to minimize the number of edge crossings. In this restricted version of the problem, the order of the vertices along the node line is predetermined and xed. FLCNP belongs to the class of NP-hard optimization problems [33]. The heuristics are tested and compared on a variety of graphs including some \real world" instances of interconnection networks proposed as models for parallel computing. The experimental results indicate that a heuristic based on the neural network model yields near-optimal solutions and outperforms the other heuristics. Also, experiments show the exact algorithm to be feasible for graphs with up to 50 edges, in general, although the quality of the initial upper bound is more critical to running time than graph size