Number Of Arc Crossings Research Articles

An Improved Bound on the One-Sided Minimum Crossing Number in Two-Layered Drawings

Given a bipartite graph G = (V,W,E), a two-layered drawing consists of placing nodes in the first node set V on a straight line L1 and placing nodes in the second node set W on a parallel line L2. The one-sided crossing minimization problem asks one to find an ordering of nodes in V to be placed on L1 so that the number of arc crossings is minimized. In this paper we use a 1.4664-approximation algorithm for this problem. This improves the previously best bound 3 due to P. Eades and N. C. Wormald [Edge crossing in drawing bipartite graphs, Algorithmica 11 (1994), 379-403].

On the one-sided crossing minimization in a bipartite graph with large degrees

Given a bipartite graph G = ( V , W , E ) , a 2-layered drawing consists of placing nodes in the first node set V on a straight line L 1 and placing nodes in the second node set W on a parallel line L 2 . For a given ordering of nodes in W on L 2 , the one-sided crossing minimization problem asks to find an ordering of nodes in V on L 1 so that the number of arc crossings is minimized. A well-known lower bound LB on the minimum number of crossings is obtained by summing up min { c uv , c vu } over all node pairs u , v ∈ V , where c uv denotes the number of crossings generated by arcs incident to u and v when u precedes v in an ordering. In this paper, we prove that there always exists a solution whose crossing number is at most ( 1.2964 + 12 / ( δ - 4 ) ) LB if the minimum degree δ of a node in V is at least 5.

Arc crossing minimization in graphs with GRASP

Graphs are commonly used to represent information in many fields of science and engineering. Automatic drawing tools generate comprehensible graphs from data, taking into account a variety of properties, enabling users to see important relationships in the data. The goal of limiting the number of arc crossings is a well-admitted criterion for a good drawing. In this paper, we present a Greedy Randomized Adaptive Search Procedure (GRASP) for the problem of minimizing are crossings in graphs. Computational experiments with 200 graphs with up to 350 vertices are presented to assess the merit of the method. We show that simple heuristics are very fast but result in inferior solutions, while high-quality solutions have been found with complex meta-heuristics but demand an impractical amount of computer time. The proposed GRASP algorithm is better in speed than complex meta-heuristics and in quality than simple heuristics. Thus, it is a clear candidate for practical implementations in graph drawing software systems.

An experimental study of the basis for graph drawing algorithms

Designers of graph drawing algorithms and systems claim to illuminate application data by producing layouts that optimise measurable aesthetic qualities. Examples of these aesthetics include symmetry (where possible, a symmetrical view of the graph should be displayed), minimise arc crossing (the number of arc crossings in the display should be minimised), and minimise bends (the total number of bends in polyline arcs should be minimised). The aim of this paper is to describe our work to validate these claims by performing empirical studies of human understanding of graphs drawn using various layout aesthetics. This work is important since it helps indicate to algorithm and system designers what are the aesthetic qualities most important to aid understanding, and consequently to build more effective systems.