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A Better Heuristic for Orthogonal Graph Drawings
 COMPUT. GEOM. THEORY APPL
, 1998
"... An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is ben ..."
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Cited by 71 (6 self)
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An orthogonal drawing of a graph is an embedding in the plane such that all edges are drawn as sequences of horizontal and vertical segments. We present a linear time and space algorithm to draw any connected graph orthogonally on a grid of size n \Theta n with at most 2n + 2 bends. Each edge is bent at most twice. In particular for nonplanar and nonbiconnected planar graphs, this is a big improvement. The algorithm is very simple, easy to implement, and it handles both planar and nonplanar graphs at the same time.
Algorithms for AreaEfficient Orthogonal Drawings
 Computational Geometry: Theory and Applications
, 1996
"... 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 t ..."
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Cited by 22 (6 self)
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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 first algorithm needs area at most (roughly) 0:76n 2 , 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 b n 2 c + 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 fo...
A Graph Drawing and Translation Service on the WWW
 In Proceedings of Graph Drawing '96
, 1999
"... . Both practitioners and researchers can take better advantage of the latest developments in graph drawing if implementations of graph drawing algorithms are made available on the WWW. We envision a graph drawing and translation service for the WWW with dual objectives: drawing userspecified gr ..."
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Cited by 12 (2 self)
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. Both practitioners and researchers can take better advantage of the latest developments in graph drawing if implementations of graph drawing algorithms are made available on the WWW. We envision a graph drawing and translation service for the WWW with dual objectives: drawing userspecified graphs, and translating graphdescriptions and graph drawings from one format to another. As a first step toward realizing this vision, we have developed a prototype service which is available at http://loki.cs.brown.edu:8081/graphserver/home.html. 1 Introduction Motivated by numerous applications, new graph drawing algorithms are continually being developed. By making implementations of graph drawing algorithms available on the WWW, we can help both practitioners and researchers to use the latest technological innovations. This, however, also requires tackling the problem of the overabundance of formats for describing graphs and drawings. While there are efforts in this direction [3], t...
Issues in Interactive Orthogonal Graph Drawing
, 1995
"... . Several applications require human interaction during the design process. The user is given the ability to alter the graph as the design progresses. Interactive Graph Drawing gives the user the ability to dynamically interact with the drawing. In this paper we discuss features that are essential f ..."
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Cited by 7 (1 self)
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. Several applications require human interaction during the design process. The user is given the ability to alter the graph as the design progresses. Interactive Graph Drawing gives the user the ability to dynamically interact with the drawing. In this paper we discuss features that are essential for an interactive drawing system. We also describe some possible interactive drawing scenaria and present results on two of them. In these results we assume that the underline drawing is always orthogonal and the maximum degree of any vertex is at most four at the end of any update operation. 1 Introduction Graphs have been extensively used to represent various important concepts or objects. Examples of such objects include parallel computer architectures, networks, state graphs, entityrelationship diagrams, subroutine call graphs, automata, dataflow graphs, Petri nets, VLSI circuits, etc. In all of these cases, we require that the graph be represented (or drawn) in the plane so that we c...
2Visibility Drawings of Planar Graphs
, 1997
"... In a 2visibility drawing the vertices of a given graph are represented by rectangular boxes and the adjacency relations are expressed by horizontal and vertical lines drawn between the boxes. In this paper we want to emphasize this model as a practical alternative to other representations of graphs ..."
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Cited by 6 (1 self)
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In a 2visibility drawing the vertices of a given graph are represented by rectangular boxes and the adjacency relations are expressed by horizontal and vertical lines drawn between the boxes. In this paper we want to emphasize this model as a practical alternative to other representations of graphs, and to demonstrate the quality of the produced drawings. We give several approaches, heuristics as well as provably good algorithms, to represent planar graphs within this model. To this, we present a polynomial time algorithm to compute a bendminimum orthogonal drawing under the restriction that the number of bends at each edge is at most 1.
Computing Orthogonal Drawings in a Variable Embedding Setting
 In [57
, 1998
"... This paper addresses the classical graph drawing problem of designing an algorithm that computes an orthogonal representation with the minimum number of bends. The algorithm receives as input a 4planar graph with a given ordering of the edges around the vertices and is allowed to change such orderi ..."
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Cited by 6 (2 self)
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This paper addresses the classical graph drawing problem of designing an algorithm that computes an orthogonal representation with the minimum number of bends. The algorithm receives as input a 4planar graph with a given ordering of the edges around the vertices and is allowed to change such ordering to reach the optimum. While the general problem has been shown to be NP complete [10], polynomial time algorithms have been devised for graphs whose vertex degree is at most three [5]. We show the first algorithm whose time complexity is exponential only in the number of vertices of degree four of the input graph. This settles a problem left as open in [6]. Our algorithm is further extended to handle graphs with vertices of degree higher than four. The analysis of the algorithm is supported by several experiments on the structure of a large set of input graphs. 1 Introduction and Overview Graph drawing is concerned with the design of methods for the automatic display of graphs so as to ...
Orthogonal Grid Drawing of Clustered Graphs
, 1996
"... Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which pro ..."
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Cited by 4 (2 self)
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Clustered graphs are graphs with recursive clustering structures over the vertices. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which belong to that cluster. In this paper, we present an algorithm which produces planar drawings of clustered graphs in a convention known as orthogonalgrid rectangular cluster drawings. The drawing produced by the algorithm has constant number of bends on each edge and has O(n 2 ) area, which is as good as existing results for classical graph drawings. 1 Introduction Clustered graphs are graphs with recursive clustering structures over the vertices (see Fig. 1). This type of clustering structure appears in many systems. Examples include CASE tools [19], management information systems [10], and VLSI design tools [8]. For graphical representation, the clustering structure is represented by a simple region that contains the drawing of all the vertices which ...
Where to Draw the Line
, 1996
"... Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be rep ..."
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Cited by 2 (0 self)
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Graph Drawing (also known as Graph Visualization) tackles the problem of representing graphs on a visual medium such as computer screen, printer etc. Many applications such as software engineering, data base design, project planning, VLSI design, multimedia etc., have data structures that can be represented as graphs. With the ever increasing complexity of these and new applications, and availability of hardware supporting visualization, the area of graph drawing is increasingly getting more attention from both practitioners and researchers. In a typical drawing of a graph, the vertices are represented as symbols such as circles, dots or boxes, etc., and the edges are drawn as continuous curves joining their end points. Often, the edges are simply drawn as (straight or poly) lines joining their end points (and hence the title of this thesis), followed by an optional transformation into smooth curves. The goal of research in graph drawing is to develop techniques for constructing good...