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14
Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 34 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
MinimumWidth Grid Drawings of Plane Graphs
 Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science
, 1995
"... Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straightline segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each pl ..."
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Cited by 31 (11 self)
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Given a plane graph G, we wish to draw it in the plane in such a way that the vertices of G are represented as grid points, and the edges are represented as straightline segments between their endpoints. An additional objective is to minimize the size of the resulting grid. It is known that each plane graph can be drawn in such a way in a (n \Gamma 2) \Theta (n \Gamma 2) grid (for n 3), and that no grid smaller than (2n=3 \Gamma 1) \Theta (2n=3 \Gamma 1) can be used for this purpose, if n is a multiple of 3. In fact, for all n 3, each dimension of the resulting grid needs to be at least b2(n \Gamma 1)=3c, even if the other one is allowed to be unbounded. In this paper we show that this bound is tight by presenting a grid drawing algorithm that produces drawings of width b2(n \Gamma 1)=3c. The height of the produced drawings is bounded by 4b2(n \Gamma 1)=3c \Gamma 1. Our algorithm runs in linear time and is easy to implement. 1 Introduction The problem of automatic graph drawing ha...
Planar Drawings and Angular Resolution: Algorithms and Bounds (Extended Abstract)
 IN PROC. 2ND ANNU. EUROPEAN SYMPOS. ALGORITHMS
, 1994
"... We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on th ..."
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Cited by 24 (5 self)
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We investigate the problem of constructing planar straightline drawings of graphs with large angles between the edges. Namely, we study the angular resolution of planar straightline drawings, defined as the smallest angle formed by two incident edges. We prove the first nontrivial upper bound on the angular resolution of planar straightline drawings, and show a continuous tradeoff between the area and the angular resolution. We also give lineartime algorithms for constructing planar straightline drawings with high angular resolution for various classes of graphs, such as seriesparallel graphs, outerplanar graphs, and triangulations generated by nested triangles. Our results are obtained by new techniques that make extensive use of geometric constructions.
Optimizing Area and Aspect Ratio in StraightLine Orthogonal Tree Drawings
 Graph Drawing (Proc. GD '96), volume 1190 of Lecture Notes Comput. Sci
, 1997
"... We investigate the problem of drawing an arbitrary nnode binary tree orthogonally in an integer grid using straightline edges. We show that one can simultaneously achieve good area bounds while also allowing the aspect ratio to be chosen as being O(1) or sometimes even an arbitrary parameter. In a ..."
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Cited by 20 (4 self)
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We investigate the problem of drawing an arbitrary nnode binary tree orthogonally in an integer grid using straightline edges. We show that one can simultaneously achieve good area bounds while also allowing the aspect ratio to be chosen as being O(1) or sometimes even an arbitrary parameter. In addition, we show that one can also achieve an additional desirable aesthetic criterion, which we call "subtree separation." We investigate both upward and nonupward drawings, achieving area bounds of O(n log n) and O(n log log n), respectively, and we show that, at least in the case of upward drawings, our area bound is optimal to within constant factors.
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
A Note on MinimumArea Upward Drawing of Complete and Fibonacci Trees
 Information Processing Letters
, 1996
"... We study the area requirement for upward straightline grid drawing of complete and Fibonacci tree. We prove that a complete tree with n nodes can be drawn in n + O(log n p n) area, and a Fibonacci tree with n nodes can be drawn in 1:17n + O(log n p n) area. Keywords: computational geometry, ..."
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Cited by 11 (0 self)
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We study the area requirement for upward straightline grid drawing of complete and Fibonacci tree. We prove that a complete tree with n nodes can be drawn in n + O(log n p n) area, and a Fibonacci tree with n nodes can be drawn in 1:17n + O(log n p n) area. Keywords: computational geometry, graph drawing. 1 Introduction In this paper we consider planar straightline upward drawings (in short, upward drawings) of rooted trees, that is, drawings such that no two edges intersect, each edge is drawn as a straightline segment, each node is drawn on a point of an integercoordinate grid, and is placed below its parent. Many references about upward drawing of arbitrary graphs can be found in the annotated bibliography maintained by Di Battista, Eades and Tamassia [1]. Upward drawings have applications in program animation and in data structure visualization, and, more generally, are a convenient representation of hierarchical structures. Since such drawings have to be presented on s...
Area Requirement of Visibility Representations of Trees
, 1996
"... We study the area requirement of barvisibility and rectanglevisibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give lineartime algorithms that construct representations with asymptotically optimal area. ..."
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Cited by 11 (7 self)
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We study the area requirement of barvisibility and rectanglevisibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give lineartime algorithms that construct representations with asymptotically optimal area.
Area Requirement of Gabriel Drawings
, 1996
"... In this paper we investigate the area requirement of proximity drawings and we prove an exponential lower bound. Namely, our main contribution is to show the existence of a class of Gabrieldrawable graphs that require exponential area for any Gabriel drawing and any resolution rule. Also, we extend ..."
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Cited by 5 (4 self)
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In this paper we investigate the area requirement of proximity drawings and we prove an exponential lower bound. Namely, our main contribution is to show the existence of a class of Gabrieldrawable graphs that require exponential area for any Gabriel drawing and any resolution rule. Also, we extend the result to an infinite class of proximity drawings.
Advances in the Theory and Practice of Graph Drawing
 Theor. Comp. Sci
, 1996
"... The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph d ..."
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Cited by 4 (0 self)
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The visualization of conceptual structures is a key component of support tools for complex applications in science and engineering. Foremost among the visual representations used are drawings of graphs and ordered sets. In this talk, we survey recent advances in the theory and practice of graph drawing. Specific topics include bounds and tradeoffs for drawing properties, threedimensional representations, methods for constraint satisfaction, and experimental studies. 1 Introduction In this paper, we survey selected research trends in graph drawing, and overview some recent results of the author and his collaborators. Graph drawing addresses the problem of constructing geometric representations of graphs, a key component of support tools for complex applications in science and engineering. Graph drawing is a young research field that has growth very rapidly in the last decade. One of its distinctive characteristics is to have furthered collaborative efforts between computer scien...
AreaEfficient Algorithms for Upward StraightLine Tree Drawings (Extended Abstract)
 in: Proc. COCOON ‘96
, 1996
"... ) ChanSu Shin 1 and Sung Kwon Kim 2 and KyungYong Chwa 1 1 Dept. of Computer Science, Korea Advanced Institute of Science and Technology, Taejon 305701, Korea, fcssin, kychwag@jupiter.kaist.ac.kr. 2 Dept. of Computer Science and Engineering, ChungAng University, Seoul 156756, Korea, ..."
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Cited by 2 (0 self)
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) ChanSu Shin 1 and Sung Kwon Kim 2 and KyungYong Chwa 1 1 Dept. of Computer Science, Korea Advanced Institute of Science and Technology, Taejon 305701, Korea, fcssin, kychwag@jupiter.kaist.ac.kr. 2 Dept. of Computer Science and Engineering, ChungAng University, Seoul 156756, Korea, ksk@point.cse.cau.ac.kr. Abstract. In this paper, we investigate planar upward straightline grid drawing problems for boundeddegree rooted trees so that a drawing takes up as little area as possible. A planar upward straightline grid tree drawing satisfies the following four constraints: (1) all vertices are placed at distinct grid points (grid ), (2) all edges are drawn as straight lines (straightline ), (3) no two edges in the drawing intersect (planar ), and (4) no parents are placed below their children (upward ). Our results are summarized as follows. First, we show that a boundeddegree tree T with n vertices admits an upward straightline drawing with area O(n log log n...