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11
A Fast MultiDimensional Algorithm for Drawing Large Graphs
 In Graph Drawing’00 Conference Proceedings
, 2000
"... We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensi ..."
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Cited by 28 (4 self)
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We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a midrange PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...
A Framework for Drawing Planar Graphs with Curves and Polylines
 J. Algorithms
, 1998
"... We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well ..."
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Cited by 15 (3 self)
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We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any nvertex planar graph in an O(n) O(n) grid using polylines that have at most two bends per edge and asymptoticallyoptimal worstcase angular resolution. More significantly, we show how to adapt this algorithm to draw any nvertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. Al...
Constraints in graph drawing algorithms
 Constraints
, 1998
"... Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and g ..."
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Cited by 15 (0 self)
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Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and graphical user interfaces. In this paper, we survey algorithmic techniques for graph drawing that support the expression and satisfaction of userdefined constraints. 1.
Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
Threading splines through 3D channels
"... Given a polygonal channel between obstacles in the plane or in space, we present an algorithm for generating a parametric spline curve with few pieces that traverses the channel and stays inside. While the problem without emphasis on few pieces has trivial solutions, the problem for a limited budget ..."
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Cited by 9 (0 self)
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Given a polygonal channel between obstacles in the plane or in space, we present an algorithm for generating a parametric spline curve with few pieces that traverses the channel and stays inside. While the problem without emphasis on few pieces has trivial solutions, the problem for a limited budget of pieces represents a nonlinear and continuous (`infinite ') feasibility problem. Using tight, twosided, piecewise linear bounds on the potential solution curves, we reformulate the problem as a finite, linear feasibility problem whose solution, by standard linear programming techniques, is a solution of the channel fitting problem. The algorithm allows the user to specify the degree and smoothness of the solution curve and to minimize an objective function, for example, to approximately minimize the curvature of the spline. We describe in detail how to formulate and solve the problem, as well as the problem of fitting parallel curves, for a spline in BernsteinBezier form.
Parameterized Complexity of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 8 (1 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
A NearLinear Area Bound for Drawing Binary Trees
 In Proc. 10th Annu. ACMSIAM Sympos. on Discrete Algorithms
, 2001
"... We present several simple methods to construct planar, strictly upward, strongly orderpreserving, straightline drawings of any nnode binary tree. In particular, it is shown that O(n 1+" ) area is always sucient for an arbitrary constant " > 0. Key Words. Graph drawing, Trees. 1 ..."
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Cited by 5 (0 self)
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We present several simple methods to construct planar, strictly upward, strongly orderpreserving, straightline drawings of any nnode binary tree. In particular, it is shown that O(n 1+" ) area is always sucient for an arbitrary constant " > 0. Key Words. Graph drawing, Trees. 1
Graph Layout Techniques and Multidimensional Data Analysis
, 2000
"... In this paper we explore the relationship between multivariate data analysis and techniques for graph drawing or graph layout. Although both classes of techniques were created for quite different purposes, we find many common principles and implementations. We start with a discussion of the data an ..."
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Cited by 4 (0 self)
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In this paper we explore the relationship between multivariate data analysis and techniques for graph drawing or graph layout. Although both classes of techniques were created for quite different purposes, we find many common principles and implementations. We start with a discussion of the data analysis techniques, in particular multiple correspondence analysis, multidimensional scaling, parallel coordinate plotting, and seriation. We then discuss parallels in the graph layout literature.
The Symmetry Number Problem for Trees
 Information Processing Letters
, 1998
"... For trees, we dene the notion of the socalled symmetry number to measure the size of the maximum subtree that exhibits an axial symmetry in graph drawing. For unrooted unordered trees, we are able to demonstrate a polynomial time algorithm for computing the symmetry number. Keywords: Design of Alg ..."
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Cited by 3 (1 self)
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For trees, we dene the notion of the socalled symmetry number to measure the size of the maximum subtree that exhibits an axial symmetry in graph drawing. For unrooted unordered trees, we are able to demonstrate a polynomial time algorithm for computing the symmetry number. Keywords: Design of Algorithms, Graph Drawing. 1. Introduction Graphs are known to be useful for modeling various scientic/engineering problems in the real world. Because of the popularity of graphs, graph drawing has emerged as a research topic of great importance in graph theory. In many cases, a `pretty drawing' often oers more insights into the nature of a graph. A natural question arises: How to dene `pretty drawings?' Aesthetic guidelines suggested in the literature (see, e.g., [1, 8]) for drawing pretty graphs include minimizing the number of edge crossings, minimizing the variance of edge lengths, minimizing the number of bends, as well as drawing edges orthogonally or using straightline segments. Su...
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...