Orthogonal Graph Drawing Research Articles

Three-Dimensional Orthogonal Graph Drawing with Optimal Volume

An orthogonal drawing of a graph is an embedding of the graph in the rectangular grid, with vertices represented by axis-aligned boxes, and edges represented by paths in the grid that only possibly intersect at common endpoints. In this paper we study three-dimensional orthogonal drawings and provide lower bounds for three scenarios: (1) drawings where the vertices have a bounded aspect ratio, (2) drawings where the surfaces of vertices are proportional to their degrees, and (3) drawings without any such restrictions. Then we show that these lower bounds are asymptotically optimal, by providing constructions that in all scenarios match the lower bounds within a constant factor.

A note on 3D orthogonal graph drawing

Some recent papers for three-dimensional orthogonal graph drawing use a technique introduced by Kolmogorov and Barzdin. This technique consists of assigning x -coordinates and y -coordinates, and then computing compatible z -coordinates, which was done via vertex-colouring a conflict graph. In this note, we show that compatible z -coordinates can also be found by edge-colouring a (different) conflict graph. As a consequence, the time complexity for computing the drawings decreases, and the volumes of the drawings are reduced as well.

Minimising the Number of Bends and Volume in 3-Dimensional Orthogonal Graph Drawings with a Diagonal Vertex Layout

A 3-dimensional orthogonal drawing of a graph with maximum degree at most 6, positions the vertices at grid-points in the 3-dimensional orthogonal grid, and routes edges along grid-lines such that edge routes only intersect at common end-vertices. Minimising the number of bends and the volume of 3-dimensional orthogonal drawings are established criteria for measuring the aesthetic quality of a given drawing. In this paper we present two algorithms for producing 3-dimensional orthogonal graph drawings with the vertices positioned along the main diagonal of a cube, so-called diagonal drawings. This vertex-layout strategy was introduced in the 3-BENDS algorithm of Eades et al. [Discrete Applied Math. 103:55–87, 2000]. We show that minimising the number of bends in a diagonal drawing of a given graph is NP-hard. Our first algorithm minimises the total number of bends for a fixed ordering of the vertices along the diagonal in linear time. Using two heuristics for determining this vertex-ordering we obtain upper bounds on the number of bends. Our second algorithm, which is a variation of the above-mentioned 3-BENDS algorithm, produces 3-bend drawings with n3+o(n3) volume, which is the best known upper bound for the volume of 3-dimensional orthogonal graph drawings with at most three bends per edge.

A UNIFIED APPROACH TO AUTOMATIC LABEL PLACEMENT

The automatic placement of text or symbol labels corresponding to graphical features is critical in several application areas such as cartography, geographical information systems, and graph drawing. In this paper we present a general framework for solving the problem of assigning text or symbol labels to a set of graphical features in two dimensional drawings or maps. Our approach does not favor the labeling of one type of graphical feature (such as a node, edge, or area) over another. Additionally, the labels are allowed to have arbitrary size and orientation. We also present a fast and simple technique, based on the general framework, for assigning labels to edges of graph drawings. We have implemented our techniques and have performed extensive experimentation on hierarchical and orthogonal drawings of graphs. The resulting label assignments are very practical and indicate the effectiveness of our approach.

Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings

In this paper we present the first non-trivial lower bounds for the total number of bends in 3-D orthogonal drawings of maximum degree six graphs. In particular, we prove lower bounds for the number of bends in 3-D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of c-connected simple graphs and multigraphs (2 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 6) with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3-D orthogonal graph drawings. These results have significant ramifications for the '2-bends' problem, which is one of the most important open problems in the field.

Orthogonal Drawings of Plane Graphs Without Bends

Orthogonal Drawings of Plane Graphs Without Bends

A Software System for Computing Labeled Orthogonal Drawings of Graphs

The paper presents a software system for computing orthogonal drawings of graphs with labels on vertices and edges, while minimizing the area or the total edge length of the drawing. The system is thought mainly to support CASE tools in the automatic visualization of diagrams like UML diagrams and ER-diagrams.We thank Giuseppe Liotta and Maddalena Nonato for their useful discussions.

Optimal three-dimensional orthogonal graph drawing in the general position model

Let G be a graph with maximum degree at most six. A three-dimensional orthogonal drawing of G positions the vertices at grid-points in the three-dimensional orthogonal grid, and routes edges along grid lines such that edge routes only intersect at common end-vertices. In this paper, we consider three-dimensional orthogonal drawings in the general position model; here no two vertices are in a common grid-plane. Minimising the number of bends in an orthogonal drawing is an important aesthetic criterion, and is NP-hard for general position drawings. We present an algorithm for producing general position drawings with an average of at most 227 bends per edge. This result is the best known upper bound on the number of bends in three-dimensional orthogonal drawings, and is optimal for general position drawings of K7. The same algorithm produces drawings with two bends per edge for graphs with maximum degree at most five; this is the only known non-trivial class of graphs admitting two-bend drawings.

The DFS-heuristic for orthogonal graph drawing

In this paper, we present a new heuristic for orthogonal graph drawings, which creates drawings by performing a depth-first search and placing the nodes in the order they are encountered. This DFS-heuristic works for graphs with arbitrarily high degrees, and particularly well for graphs with maximum degree 3. It yields drawings with at most one bend per edge, and a total number of m−n+1 bends for a graph with n nodes and m edges; this improves significantly on the best previous bound of m−2 bends.

Fully Dynamic 3-Dimensional Orthogonal Graph Drawing

In a 3-dimensional orthogonal drawing of a graph, vertices are mapped to grid points on an integer lattice and edges are routed along integer grid lines. In this paper, we present a layout scheme that draws any graph with n vertices of maximum degree 6, using at most 6 bends per edge and in a volume of O(n2). The advantage of our strategy over other drawing methods is that our method is fully dynamic, allowing both insertion and deletion of vertices and edges, while maintaining the volume and bend bounds. The drawing can be obtained in O(n) time and insertions/deletions can be performed in O(1) time. Multiple edges and self loops are permitted. A more elaborate construction that uses only 5 bends per edge, and a simpler, more balanced layout that requires at most 7 bends per edge are also described.

THE THREE-PHASE METHOD: A UNIFIED APPROACH TO ORTHOGONAL GRAPH DRAWING

In this paper, we study orthogonal graph drawings from a practical point of view. Most previously existing algorithms restricted the attention to graphs of maximum degree four. Here we study orthogonal drawing algorithms that work for any input graph, and discuss different models for such drawings. Then we introduce the three-phase method, a generic technique to create high-degree orthogonal drawings. This approach simplifies the description and implementation of orthogonal graph drawing, and can be applied to global as well as interactive and incremental settings.

Three-dimensional orthogonal graph drawing algorithms

We use basic results from graph theory to design algorithms for constructing three-dimensional, intersection-free orthogonal grid drawings of n vertex graphs of maximum degree 6. The best previous result generated a drawing bounded by an O( n )× O( n )× O( n ) box, with each edge route containing up to 16 bends. Our algorithms initiate the study of bend/bounding box trade-off issues for three-dimensional grid drawings and produce drawings with the following characteristics: 1. at most 7 bends per edge route, bounded by an O( n )× O( n )× O( n ) box; 2. at most 6 bends per edge route, bounded by an O( n )× O( n )× O(n) box; 3. at most 5 bends per edge route, bounded by an O( n )× O(n)× O(n) box; 4. at most 3 bends per edge route, bounded by an O( n)×O( n)×O( n) box; and 5. for maximum degree 4 graphs, at most 3 bends per edge route, bounded by and O( n)×O( n)×O(1) box.

A Parallel Algorithm for Planar Orthogonal Grid Drawings

In this paper we consider the problem of constructing planar orthogonal grid drawings (or more simply, layouts) of graphs, with the goal of minimizing the number of bends along the edges. We present optimal parallel algorithms that construct graph layouts with O(n) maximum edge length, O(n2) area, and at most 2n+4 bends (for biconnected graphs) and 2.4n+2 bends (for simply connected graphs). All three of these quality measures for the layouts are optimal in the worst case for biconnected graphs. The algorithm runs on a CREW PRAM in O( log n) time with n/ log n processors, thus achieving optimal time and processor utilization. Applications include VLSI layout, graph drawing, and wireless communication.

Techniques for the Refinement of Orthogonal Graph Drawings

Current orthogonal graph drawing algorithms produce drawings which are generally good. However, many times the quality of orthogonal drawings can be signicantly improved with a postprocessing technique, called renement, which improves aesthetic qualities of a drawing such as area, bends, crossings, and total edge length. Renement is separate from layout and works by analyzing and then ne-tuning the existing drawing in an ecient manner. In this paper we dene the problem and goals of orthogonal drawing renement, review measures of a graph drawing’s quality, and introduce a methodology which eciently renes any orthogonal graph drawing. We have implemented our techniques in C++ and conducted experiments over a set of drawings from ve well known orthogonal drawing systems. Experimental analysis shows our techniques to produce an average 37% improvement in area, 23% in bends, 25% in crossings, and 37% in total edge length.

Efficient Orthogonal Drawings of High Degree Graphs

Most of the work that appears in the two-dimensional orthogonal graph drawing literature deals with graphs whose maximum degree is four. In this paper we present an algorithm for orthogonal drawings of simple graphs with degree higher than four. Vertices are represented by rectangular boxes of perimeter less than twice the degree of the vertex. Our algorithm is based on creating groups / pairs of vertices of the graph. The orthogonal drawings produced by our algorithm have area at most (m-1)\(\times\)( m / 2 +2) . Two important properties of our algorithm are that the drawings exhibit a small total number of bends (less than m ), and that there is at most one bend per edge.

A Split&Push Approach to 3D Orthogonal Drawing

We present a method for constructing orthogonal drawings of graphs of maximum degree six in three dimensions. Such a method is based on generating the final drawing through a sequence of steps, starting from a degenerate drawing. At each step the drawing splits into two pieces and finds a structure more similar to its final version. Also, we test the effectiveness of our approach by performing an experimental comparison with several existing algorithms.

Difference Metrics for Interactive Orthogonal Graph Drawing Algorithms

Preserving the map is major goal of interactive graph drawing algorithms. Several models have been proposed for formalizing the notion of mental map. Additional work needs to be done to formulate and validate metrics which can be used in practice. This paper introduces a framework for defining and validating metrics to measure the difference between two drawings of the same graph.

直交グラフ描画法を用いた室配置手法 : タブー探索法を用いた対話型多目的最適化

This paper proposes a new method for finding an optimal floor layout iti a possibly non-rectangular site, based on an orthogonal graph drawing algorithm. With the proposed method, we can obtain a floor layout that allows the existence of non-rectangular rooms, by which we resolve the drawbacks of the existing methods that admit only rectangular rooms. Introducing two types of criteria concerning area and shape for each room, we design an interactive multi-objective optimization algorithm that produces a number of satisfactory solutions. We have implemented the algorithm and tested it for several practical cases.

Orthogonal drawings of graphs for the automation of VLSI circuit design

This article shows the recent developments on orthogonal drawings of graphs which have applications for the automation of VLSI circuit design. Meanwhile, a number of problem are posed for further research.

Algorithms for Incremental Orthogonal Graph Drawing in Three Dimensions

We present two algorithms for orthogonal graph drawing in three dimensional space. For a graph with n vertices of maximum degree six, the 3-D drawing is produced in linear time, has volume at most 4:63n 3 and has at most three bends per edge. If the degree of the graph is arbitrary, the vertices are represented by solid 3-D boxes whose surface is proportional to their degree. The produced drawing has two bends per edge. Both algorithms guarantee no crossings and can be used under an interactive setting (i.e., vertices arrive and enter the drawing on-line), as well.