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25
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.
Optimal Algorithms to Embed Trees in a Point Set
, 1995
"... We present optimal \Theta(n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted tree embeddings and degreeconstrained embeddings. In the rooted tree embedding problem we are given a rooted tree T with n nodes and a set of n po ..."
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Cited by 20 (1 self)
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We present optimal \Theta(n log n) time algorithms to solve two tree embedding problems whose solution previously took quadratic time or more: rooted tree embeddings and degreeconstrained embeddings. In the rooted tree embedding problem we are given a rooted tree T with n nodes and a set of n points P with one designated point p and are asked to find a straightline embedding of T into P with the root at point p. In the degreeconstrained embedding problem we are given a set of n points P where each point is assigned a positive degree and the degrees sum to 2n \Gamma 2 and are asked to embed a tree in P that respects the degrees assigned to each point of P .
Planar Upward Tree Drawings with Optimal Area
 Internat. J. Comput. Geom. Appl
, 1996
"... Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and pro ..."
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Cited by 19 (3 self)
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Rooted trees are usually drawn planar and upward, i.e., without crossings and without any parent placed below its child. In this paper we investigate the area requirement of planar upward drawings of rooted trees. We give tight upper and lower bounds on the area of various types of drawings, and provide lineartime algorithms for constructing optimal area drawings. Let T be a boundeddegree rooted tree with N nodes. Our results are summarized as follows: ffl We show that T admits a planar polyline upward grid drawing with area O(N ), and with width O(N ff ) for any prespecified constant ff such that 0 ! ff ! 1. ffl If T is a binary tree, we show that T admits a planar orthogonal upward grid drawing with area O(N log log N ). ffl We show that if T is ordered, it admits an O(N log N)area planar upward grid drawing that preserves the lefttoright ordering of the children of each node. ffl We show that all of the above area bounds are asymptotically optimal in the worst case. ffl ...
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
Straightline Drawings of Binary Trees with Linear Area and Good Aspect Ratio
 Proceedings 10th International Symposium on Graph Drawing
, 2002
"... Trees are usually drawn planar, i.e. without any crossings. In this paper we investigate the area requirement of (nonupward) planar straightline drawings of binary trees. Let T be a binary tree with n vertices. We show that T admits a planar straightline grid drawing with area O(n) and with any p ..."
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Cited by 12 (2 self)
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Trees are usually drawn planar, i.e. without any crossings. In this paper we investigate the area requirement of (nonupward) planar straightline drawings of binary trees. Let T be a binary tree with n vertices. We show that T admits a planar straightline grid drawing with area O(n) and with any prespecified aspect ratio in the range [1; n ], where is a constant such that 0 < 1. We also show that such a drawing can be constructed in O(n log n) time.
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...
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 6 (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