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20
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 14 (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.
Drawing Trees with Perfect Angular Resolution and Polynomial Area
"... Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require e ..."
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Cited by 14 (9 self)
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Abstract. We study methods for drawing trees with perfect angular resolution, i.e., with angles at each vertex, v, equal to 2π/d(v). We show: 1. Any unordered tree has a crossingfree straightline drawing with perfect angular resolution and polynomial area. 2. There are ordered trees that require exponential area for any crossingfree straightline drawing having perfect angular resolution. 3. Any ordered tree has a crossingfree Lombardistyle drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area. Thus, our results explore what is achievable with straightline drawings and what more is achievable with Lombardistyle drawings, with respect to drawings of trees with perfect angular resolution. 1
Quantifying the SpaceEfficiency of 2D Graphical Representations of Trees
"... Abstract — A mathematical evaluation and comparison of the spaceefficiency of various 2D graphical representations of tree structures is presented. As part of the evaluation, a novel metric called the mean area exponent is introduced that quantifies the distribution of area across nodes in a tree r ..."
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Cited by 9 (1 self)
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Abstract — A mathematical evaluation and comparison of the spaceefficiency of various 2D graphical representations of tree structures is presented. As part of the evaluation, a novel metric called the mean area exponent is introduced that quantifies the distribution of area across nodes in a tree representation, and that can be applied to a broad range of different representations of trees. Several representations are analyzed and compared by calculating their mean area exponent as well as the area they allocate to nodes and labels. Our analysis inspires a set of design guidelines as well as a few novel tree representations that are also presented. Index Terms—Tree visualization, graph drawing, efficiency metrics. 1
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 7 (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
How to Draw a Clustered Tree
, 2007
"... The visualization of clustered graphs is a classical algorithmic topic that has several practical applications and is attracting increasing research interest. In this paper we deal with the visualization of clustered trees, a problem that is somehow foundational with respect to the one of visualizin ..."
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Cited by 5 (2 self)
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The visualization of clustered graphs is a classical algorithmic topic that has several practical applications and is attracting increasing research interest. In this paper we deal with the visualization of clustered trees, a problem that is somehow foundational with respect to the one of visualizing a general clustered graph. We show many, in our opinion, surprising results that put in evidence how drawing clustered trees has many sharp differences with respect to drawing “plain” trees. We study a wide class of drawing standards, giving both negative and positive results. Namely, we show that there are clustered trees that do not have any drawing in certain standards and others that require exponential area. On the contrary, for many drawing conventions there are efficient algorithms that allow to draw clustered trees with polynomial asymptotic optimal area.
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...
Track Planarity Testing and Embedding
 PROC. SOFTWARE SEMINAR: THEORY AND PRACTICE OF INFORMATICS, SOFSEM 2004
, 2004
"... A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that ..."
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Cited by 4 (3 self)
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A track graph is a graph with its vertex set partitioned into horizontal levels. It is track planar if there are permutations of the vertices on each level such that all edges can be drawn as weak monotone curves without crossings. The novelty and generalisation over level planar graphs is that horizontal edges connecting consecutive vertices on the same level are allowed. We show that track planarity can be reduced to level planarity in linear time. Hence, there are time algorithms for the track planarity test and for the computation of a track planar embedding.
Tree Drawings on the Hexagonal Grid
"... We consider straightline drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ ..."
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Cited by 4 (1 self)
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We consider straightline drawings of trees on a hexagonal grid. The hexagonal grid is an extension of the common grid with inner nodes of degree six. We restrict the number of directions used for the edges fromeachnodetoitschildrenfromonetofive, andtofivepatterns: straight, Y, ψ, X, and full. The ψ–drawings generalize hv or strictly upward drawings to ternary trees. Weshowthatcompleteternarytreeshavea ψ–drawingonasquareofsize O(n 1.262) and general ternary trees can be drawn within O(n 1.631) area. Bothboundsareoptimal.Sub–quadraticboundsarealsoobtainedfor X– pattern drawings of complete tetra trees, and for full–pattern drawings of complete penta trees, which are 4–ary and 5–ary trees. These results parallel and complement the ones of Frati [8] for straight–line orthogonal drawings of ternary trees. Moreover, we provide an algorithm for compacted straight–line drawings of penta trees on the hexagonal grid, such that the direction of the edges from a node to its children is given by our patterns and these edges have the same length. However, drawing trees on a hexagonal grid within a prescribed area or with unit length edges is NP–hard.