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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.
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 ...
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...
Parallel hv Drawings of Binary Trees
, 1994
"... . In this paper we present a method to obtain optimal hv and inclusion drawings in parallel. Based on parallel tree contraction, our method computes optimal (with respect to a class of cost functions of the enclosing rectangle) drawings in O(log 2 n) parallel time by using a polynomial number of ..."
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Cited by 3 (1 self)
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. In this paper we present a method to obtain optimal hv and inclusion drawings in parallel. Based on parallel tree contraction, our method computes optimal (with respect to a class of cost functions of the enclosing rectangle) drawings in O(log 2 n) parallel time by using a polynomial number of EREW processors. The method can be extended to compute optimal inclusion layouts in the case where each leaf l of the tree is represented by rectangle l x \Theta l y . Our method also yields an NC algorithm for the slicing floorplanning problem. Whether this problem was in NC was an open question [2]. 1 Introduction In this paper we examine drawings of rooted binary trees. We study the hv drawing convention studied by Crescenzi, Di Battista and Piperno [3] and Eades, Lin and Lin [7]. Our results extend to the inclusion convention [6], and to slicing floorplanning [10, 2]. The drawing of a rooted binary tree using the hv drawing convention is a planar grid drawing in which tree nodes are ...
On MultiStack Boundary Labeling Problems
"... Boundary labeling is a relatively new labeling method. It targets the areas of technical drawings and medical maps, where it is often common to explain certain parts of the drawing with large text labels arranged on its boundary, so that other parts of the drawing are not obscured. According to thi ..."
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Cited by 3 (2 self)
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Boundary labeling is a relatively new labeling method. It targets the areas of technical drawings and medical maps, where it is often common to explain certain parts of the drawing with large text labels arranged on its boundary, so that other parts of the drawing are not obscured. According to this labeling method, we are given a rectangle R, which encloses a set of n sites. Each site si is associated with an axisparallel rectangular label li. The labels must be placed in distinct positions on the boundary of R and to be connected to their corresponding sites with polygonal lines, called leaders, so that a) labels are pairwise disjoint and b) leaders do not intersect each other. In this paper, we examine labelings with more than one stacks of uniform labels on each side of R and we aim to maximize the (uniform) label size.
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...
A Linear Algorithm for Compact BoxDrawings of Trees
 University of Lethbridge
, 2002
"... In a boxdrawing of a tree each node is drawn by a rectangular box of prescribed size, no two boxes overlap each other, all boxes corresponding to siblings of the tree have the same xcoordinate at their left sides, and a parent node is drawn at a given distance apart from its rst child. A box draw ..."
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In a boxdrawing of a tree each node is drawn by a rectangular box of prescribed size, no two boxes overlap each other, all boxes corresponding to siblings of the tree have the same xcoordinate at their left sides, and a parent node is drawn at a given distance apart from its rst child. A box drawing of a tree is compact if it attains the minimum possible rectangular area enclosing the drawing. We give a lineartime algorithm for nding a compact boxdrawing of a tree. A known algorithm does not always nd a compact boxdrawing and takes time O(n ) if a tree has n nodes.
Computational Geometry Theory and Applications Linear area upward drawings of AVL trees"
"... We prove that any AVL tree admits a lineararea straightline strictlyupward planar grid drawing, that is, a drawing in which (a) each edge is mapped into a single straightline segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point w ..."
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We prove that any AVL tree admits a lineararea straightline strictlyupward planar grid drawing, that is, a drawing in which (a) each edge is mapped into a single straightline segment, (b) each node is placed below its parent, (c) no two edges intersect, and (d) each node is mapped into a point with integer coordinates. © 1998 Elsevier Science B.V.