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Algorithms for Drawing Clustered Graphs
, 1997
"... In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics ..."
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Cited by 25 (2 self)
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In the mid 1980s, graphics workstations became the main platforms for software and information engineers. Since then, visualization of relational information has become an essential element of software systems. Graphs are commonly used to model relational information. They are depicted on a graphics workstation as graph drawings. The usefulness of the relational model depends on whether the graph drawings effectively convey the relational information to the users. This thesis is concerned with finding good drawings of graphs. As the amount of information that we want to visualize becomes larger and the relations become more complex, the classical graph model tends to be inadequate. Many extended models use a node hierarchy to help cope with the complexity. This thesis introduces a new graph model called the clustered graph. The central theme of the thesis is an investigation of efficient algorithms to produce good drawings for clustered graphs. Although the criteria for judging the qua...
Refinement of ThreeDimensional Orthogonal Graph Drawings
, 2001
"... In this paper we introduce a number of techniques for the re nement of threedimensional orthogonal drawings of maximum degree six graphs. We have implemented several existing algorithms for threedimensional orthogonal graph drawing including a number of heuristics to improve their performance. The ..."
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Cited by 21 (3 self)
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In this paper we introduce a number of techniques for the re nement of threedimensional orthogonal drawings of maximum degree six graphs. We have implemented several existing algorithms for threedimensional orthogonal graph drawing including a number of heuristics to improve their performance. The performance of the re nements on the produced drawings is then evaluated in an extensive experimental study. We measure the aesthetic criteria of the bounding box volume, the average and maximum number of bends per edge, and the average and maximum edge length. On the same set of graphs used in Di Battista et al. [3], our main re nement algorithm improves the above aesthetic criteria by 80%, 38%, 10%, 54% and 49%, respectively.
Two Algorithms for Three Dimensional Orthogonal Graph Drawing
, 1997
"... We use basic results from graph theory to design two algorithms for constructing 3dimensional, intersectionfree orthogonal grid drawings of n vertex graphs of maximum degree 6. Our first algorithm gives drawings bounded by an O( p n) \Theta O( p n) \Theta O( p n) box; each edge route conta ..."
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Cited by 20 (1 self)
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We use basic results from graph theory to design two algorithms for constructing 3dimensional, intersectionfree orthogonal grid drawings of n vertex graphs of maximum degree 6. Our first algorithm gives drawings bounded by an O( p n) \Theta O( p n) \Theta O( p n) box; each edge route contains at most 7 bends. The best previous result generated edge routes containing up to 16 bends per route. Our second algorithm gives drawings having at most 3 bends per edge route. The drawings lie in an O(n) \Theta O(n) \Theta O(n) bounding box. Together, the two algorithms initiate the study of bend/bounding box tradeoff issues for 3dimensional grid drawings. 1 Introduction The 3dimensional orthogonal grid consists of grid points whose (X; Y; Z)coordinates are all integers, together with the axisparallel grid lines determined by these points. A 3dimensional orthogonal grid drawing of a graph G places the vertices of G at grid points and routes the edges of G along sequences of cont...
Turnregularity and optimal area drawings of orthogonal representations
, 2000
"... Given an orthogonal representation H with n vertices and bends, we study the problem of computing a planar orthogonal drawing of H with small area. This problem has direct applications to the development of practical graph drawing techniques for information visualization and VLSI layout. In this pap ..."
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Cited by 17 (5 self)
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Given an orthogonal representation H with n vertices and bends, we study the problem of computing a planar orthogonal drawing of H with small area. This problem has direct applications to the development of practical graph drawing techniques for information visualization and VLSI layout. In this paper, we introduce the concept of turnregularity of an orthogonal representation H, provide combinatorial characterizations of it, and show that if H is turnregular (i.e., all its faces are turnregular), then a planar orthogonal drawing of H with minimum area can be computed in O(n) time, and a planar orthogonal drawing of H with minimum area and minimum total edge length within that area can be computed in O(n 7/4 log n) time. We also apply our theoretical results to the design and implementation of new practical heuristic methods for constructing planar orthogonal drawings. An experimental study conducted on a test suite of orthogonal representations of randomly generated biconnected 4planar graphs shows that the percentage of turnregular faces is quite high and that our heuristic drawing methods perform better than previous ones.
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 16 (4 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...
Dynamic Grid Embedding with Few Bends and Changes
, 1998
"... In orthogonal graph drawing, edges are represented by sequences of horizontal and vertical straight line segments. For graphs of degree at most four, this can be achieved by embedding the graph in a grid. The number of bends displayed is an important criterion for layout quality. A wellknown al ..."
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Cited by 8 (2 self)
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In orthogonal graph drawing, edges are represented by sequences of horizontal and vertical straight line segments. For graphs of degree at most four, this can be achieved by embedding the graph in a grid. The number of bends displayed is an important criterion for layout quality. A wellknown algorithm of Tamassia eciently embeds a planar graph with xed combinatorial embedding and vertex degree at most four in the grid such that the number of bends is minimum [23].
A Pairing Technique for AreaEfficient Orthogonal Drawings (Extended Abstract)
 in Proc. of Symp. for Graph Drawing
"... 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 nodes. If the maximum degree is four, the ..."
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Cited by 4 (1 self)
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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 nodes. If the maximum degree is four, then the drawing produced by our first algorithm needs area no greater than 0.76n&sup2;, and introduces no more than 2n + 2 bends. Also, every 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 1/4 n&sup2; area, and at most &lfloor;n/2 + 2l + 1&rfloor; bends (&lfloor;n/2&rfloor; + 3 bends, if the graph is biconnected), where l is the number of biconnected components that are...
Drawing Clustered Graphs on . . .
 J. GRAPH ALGORITHMS APPL
, 1999
"... 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 3 (0 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 orthogonal grid rectangular cluster drawings. If the input graph has n vertices, then the algorithm produces in O(n) time a drawing with O(n²) area and at most 3 bends in each edge. This result is as good as existing results for classical planar graphs. Further, we show that our algorithm is optimal in terms of the number of bends per edge.
Efficient Algorithms for Drawing Planar Graphs
, 1999
"... x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ................. ..."
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Cited by 1 (0 self)
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x 1 Introduction 1 1.1 Historical Background . . .............................. 4 1.2 Drawing Styles . ................................... 4 1.2.1 Polyline drawings .............................. 5 1.2.2 Planar drawings ............................... 5 1.2.3 Straight line drawings ............................ 6 1.2.4 Orthogonal drawings . . ........................... 7 1.2.5 Grid drawings ................................ 8 1.3 Properties of Drawings ................................ 9 1.4 Scope of this Thesis .................................. 10 1.4.1 Rectangular drawings . . . ......................... 11 1.4.2 Orthogonal drawings . . ........................... 12 1.4.3 Boxrectangular drawings ........................... 14 1.4.4 Convex drawings . . ............................. 16 1.5 Summary ....................................... 16 2 Preliminaries 20 2.1 Basic Terminology .................................. 20 2.1.1 Graphs and Multigraphs ........................... 20 i CO...
Drawing Clustered Graphs on an Orthogonal Grid (Extended Abstract)
 J. GRAPH ALGORITHMS APPL
, 1997
"... 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 produc ..."
Abstract
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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 orthogonal grid rectangular cluster drawings. We present an algorithm which produces such drawings with O(n²) area and with at most 3 bends in each edge. This result is as good as existing results for classical planar graphs. Further, we show that the bend performance of our algorithm is optimal.