Peter Eades

Algorithms for drawing graphs: an annotated bibliography

Several data presentation problems involve drawing graphs so that they are easy to read and understand. Examples include circuit schematics and diagrams for information systems analysis and design. In this paper we present a bibliographic survey on algorithms whose goal is to produce aesthetically pleasing drawings of graphs. Research on this topic is spread over the broad spectrum of computer science. This bibliography constitutes a first attempt to encompass both theoretical and application-oriented papers from disparate areas.

Multilevel Visualization of Clustered Graphs

Clustered graphs are graphs with recursive clustering structures over the vertices. This type of structure appears in many systems. Examples include CASE tools, management information systems, VLSI design tools, and reverse engineering systems. Existing layout algorithms represent the clustering structure as recursively nested regions in the plane. However, as the structure becomes more and more complex, two dimensional plane representations tend to be insufficient. In this paper, firstly, we describe some two dimensional plane drawing algorithms for clustered graphs; then we show how to extend two dimensional plane drawings to three dimensional multilevel drawings. We consider two conventions: straight-line convex drawings and orthogonal rectangular drawings; and we show some examples.

Edge crossings in drawings of bipartite graphs

Systems engineers have recently shown interest in algorithms for drawing directed graphs so that they are easy to understand and remember. Each of the commonly used methods has a step which aims to adjust the drawing to decrease the number of arc crossings. We show that the most popular strategy involves an NP-complete problem regarding the minimization of the number of arcs in crossings in a bipartite graph. The performance of the commonly employed “barycenter” heuristic for this problem is analyzed. An alternative method, the “median” heuristic, is proposed and analyzed. The new method is shown to compare favorably with the old in terms of performance guarantees. As a bonus, we show that the median heuristic performs well with regard to the total length of the arcs in the drawing.

FADE: Graph Drawing, Clustering, and Visual Abstraction

A fast algorithm(FADE) for the 2D drawing, geometric clustering and multilevel viewing of large undirected graphs is presented. The algorithm is an extension of the Barnes-Hut hierarchical space decomposition method, which includes edges and multilevel visual abstraction. Compared to the original force directed algorithm, the time overhead is O(e + n log n) where n and e are the numbers of nodes and edges. The improvement is possible since the decomposition tree provides a systematic way to determine the degree of closeness between nodes without explicitly calculating the distance between each node. Different types of regular decomposition trees are introduced. The decomposition tree also represents a hierarchical clustering of the nodes, which improves in a graph theoretic sense as the graph drawing approaches a lower energy state. Finally, the decomposition tree provides a mechanism to view the hierarchical clustering on various levels of abstraction. Larger graphs can be represented more concisely, on a higher level of abstraction, with fewer graphics on screen.

A fast and effective heuristic for the feedback arc set problem

Let G=(V, A) denote a simple connected directed graph, and let n=|V|, m=|A|, where nt-1≤m≤(n2) A feedbackarc set (FAS) of G, denoted R(G), is a (possibly empty)set of arcs whose reversal makes G acyclic. A minimum feedbackarc set of G, denoted R∗(G), is a FAS of minimum cardinality r∗(G); the computation of R∗(G) is called the FASproblem. Berger and Shor have recently published an algorithm which, for a given digraph G, computes a FAS whose cardinality is at most m/2t-c1m/Δ1/2 where Δ is the maximum degree of G and c1 is a constant. Further, they exhibited an infinite class of graphs with the property that for every Gϵ and some constant c2, r∗(G)≥m /2t-c2m/Δ1/2. Thus the Berger-Shor algorithm provides, in a certain asymptotic sense, an optimal solution to the FAS problem. Unfortunately, the Berger-Shor algorithm is complicated and requires runni ng time O(mn). In this paper we present a simple FAS algorithm which guarantees a good (though not optimal) performance bound and executes in time O(m). Further, for the sparse graphs which arise frequently in graph drawing and other applications, our algorithm achieves the same asymptotic performance bound that Berger-Shor does.

Planarity for Clustered Graphs

In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study C-planarity of clustered graphs. Given a clustered graph, the C-planarity testing problem is to determine whether the clustered graph can be drawn without edge crossings, or edge-region crossings. In this paper, we present efficient algorithms for testing C-planarity and finding C-planar embeddings of clustered graphs.

Navigating Clustered Graphs using Force-Directed Methods

Graphs which arise in Information Visualization applications are typically very large: thousands, or perhaps millions of nodes. Current graph drawing methods successfully deal with (at best) a few hundred nodes. This paper describes a strategy for the visualization and navigation of graphs. The strategy has three elements:

Drawing Graphs with Right Angle Crossings

Cognitive experiments show that humans can read graph drawings in which all edge crossings are at right angles equally well as they can read planar drawings; they also show that the readability of a drawing is heavily affected by the number of bends along the edges. A graph visualization whose edges can only cross perpendicularly is called a RAC (Right Angle Crossing) drawing . This paper initiates the study of combinatorial and algorithmic questions related with the problem of computing RAC drawings with few bends per edge. Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings. We establish upper and lower bounds on these quantities by considering two classical graph drawing scenarios: The one where the algorithm can choose the combinatorial embedding of the input graph and the one where this embedding is fixed.

On-line Animated Visualization of Huge Graphs using a Modified Spring Algorithm

On-line graph drawing deals with huge graphs which are partially unknown. At any time, a tiny part of the graph is displayed on the screen. Examples include web graphs and graphs of links in distributed file systems. This paper discusses issues arising in the presentation of such graphs. The paper describes a system for dealing with web graphs using on-line graph drawing.

Effects of Crossing Angles

In visualizing graphs as node-link diagrams, it is commonly accepted and employed as a general rule that the number of link crossings should be minimized whenever possible. However, little attention has been paid to how to handle the remaining crossings in the visualization. The study presented in this paper examines the effects of crossing angles on performance of path tracing tasks. It was found that the effect varied with the size of crossing angles. In particular, task response time decreased as the crossing angle increased. However, the rate of the decrease tended to level off when the angle was close to 90 degrees. One of the implications of this study in graph visualization is that just minimizing the crossing number is not sufficient to reduce the negative impact to the minimum. The angles of remaining crossings should be maximized as well.