Ioannis G. Tollis

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.

Planar grid embedding in linear time

The authors consider the problem of constructing a planar grid embedding for G, where G is a planar graph with n vertices, which maps vertices to distinct grid points and edges to nonintersecting grid paths. A new algorithm is presented that runs in O(n) time and produces grid embeddings with the following properties: (1) the total number of bends is at most 2.4n+2; (2) the number of bends along each edge is at most 4; (3) the length of every edge is O(n); and (4) the area of the embedding is O(n/sup 2/). >

A vision and strategy for the virtual physiological human in 2010 and beyond

European funding under framework 7 (FP7) for the virtual physiological human (VPH) project has been in place now for nearly 2 years. The VPH network of excellence (NoE) is helping in the development of common standards, open-source software, freely accessible data and model repositories, and various training and dissemination activities for the project. It is also helping to coordinate the many clinically targeted projects that have been funded under the FP7 calls. An initial vision for the VPH was defined by framework 6 strategy for a European physiome (STEP) project in 2006. It is now time to assess the accomplishments of the last 2 years and update the STEP vision for the VPH. We consider the biomedical science, healthcare and information and communications technology challenges facing the project and we propose the VPH Institute as a means of sustaining the vision of VPH beyond the time frame of the NoE.

Area requirement and symmetry display of planar upward drawings

In this paper we investigate the problem of constructing planar straight-line drawings of acyclic digraphs such that all the edges flow in the same direction, e.g., from bottom to top. Our contribution is twofold. First we show the existence of a family of planar acyclic digraphs that require exponential area for any such drawing. Second, motivated by the preceding lower bound, we relax the straight-line constraint and allow bends along the edges. We present a linear-time algorithm that produces drawings of planarst-graphs with a small number of bends, asymptotically optimal area, and such that symmetries and isomorphisms of the digraph are displayed. If the digraph has no transitive edges, then the drawing obtained has no bends. Also, a variation of the algorithm produces drawings with exact minimum area.

Algorithms for area-efficient orthogonal drawings

Abstract An orthogonal drawing of a graph is a drawing such that vertices are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n vertices. If the maximum degree is four, then the drawing produced by our first algorithm needs area at most (roughly) 0.76n2, and introduces at most 2n + 2 bends. Also, each edge of such a drawing has at most two bends. Our algorithm is based on forming and placing pairs of vertices of the graph. If the maximum degree is three, then the drawing produced by our second algorithm needs at most (roughly) ( 1 4 )n 2 area and, if the graph is biconnected, at most ⌊ n 2 ⌋ + 3 bends. These upper bounds match the upper bounds known for planar graphs of maximum degree 3. This algorithm produces optimal drawings (within a constant of 2) with respect to the number of bends, since there is a lower bound of n 2 + 1 in the number of bends for orthogonal drawings of maximum degree 3 graphs.

A vision and strategy for the virtual physiological human: 2012 update.

European funding under Framework 7 (FP7) for the virtual physiological human (VPH) project has been in place now for 5 years. The VPH Network of Excellence (NoE) has been set up to help develop common standards, open source software, freely accessible data and model repositories, and various training and dissemination activities for the project. It is also working to coordinate the many clinically targeted projects that have been funded under the FP7 calls. An initial vision for the VPH was defined by the FP6 STEP project in 2006. In 2010, we wrote an assessment of the accomplishments of the first two years of the VPH in which we considered the biomedical science, healthcare and information and communications technology challenges facing the project (Hunter et al. 2010 Phil. Trans. R. Soc. A 368, 2595–2614 (doi:10.1098/rsta.2010.0048)). We proposed that a not-for-profit professional umbrella organization, the VPH Institute, should be established as a means of sustaining the VPH vision beyond the time-frame of the NoE. Here, we update and extend this assessment and in particular address the following issues raised in response to Hunter et al.: (i) a vision for the VPH updated in the light of progress made so far, (ii) biomedical science and healthcare challenges that the VPH initiative can address while also providing innovation opportunities for the European industry, and (iii) external changes needed in regulatory policy and business models to realize the full potential that the VPH has to offer to industry, clinics and society generally.

A framework for dynamic graph drawing

In this paper we give a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing. We present dynamic algorithms for drawing planar graphs that use a variety of drawing standards (such as polyline, straight-line, orthogonal, grid, upward, and visibility drawings), and address aesthetic criteria that are important for readability, such as the display of planarity, symmetry, and reachability. Also, we provide techniques that are especially tailored for important subclasses of planar graphs such as trees and series-parallel digraphs. Our dynamic drawing algorithms have the important property of performing “smooth updates” of the drawing. Of special geometric interest is the possibility of performing point-location and window queries on the implicit representation of the drawing.

Dynamic Graph Drawings: Trees, Series-Parallel Digraphs, and Planar ST -Digraphs

Drawing graphs is an important problem that combines elements of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. We present efficient dynamic drawing algorithms for trees and series-parallel digraphs. As further applications of the model, we give dynamic drawing algorithms for planar st-digraphs and planar graphs. Our algorithms adopt a variety of representations (e.g., straight line, polyline, visibility) and update the drawing in a smooth way.

A unified approach to labeling graphical features

The automatic placement of text or symbol labels corresponding to graphical objects 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 labcls arc allowed to have arbitrary size and orientation. We have applied our framework to drawings of graphs. 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.

Improved Algorithms and Bounds for Orthogonal Drawings

An orthogonal drawing of a graph is a drawing such that nodes are placed on grid points and edges are drawn as sequences of vertical and horizontal segments. In this paper we present linear time algorithms that produce orthogonal drawings of graphs with n nodes. If the maximum degree is four, then the drawing produced by our algorithm needs area no greater than 0.8n2 and no more than 1.9n bends. Notice that our upper bound on the bends is below the lower bound for planar orthogonal drawings of planar graphs. To the best of our knowledge, this is the first algorithm for orthogonal drawings that needs area less than n2. If the maximum degree is three, then the drawing produced by our algorithm needs (n/2+1)×n/2 area and at most n/2+3 bends. These upper bounds match the upper bounds known for triconnected planar graphs of degree 3.