Results 1  10
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12
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 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
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 6 (6 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 6 (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
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 3 (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.
Lower Bounds on the Area Requirements of SeriesParallel Graphs
, 2009
"... We show that there exist seriesparallel graphs requiring Ω(n2 √ logn) area in any straightline or polyline grid drawing. Such a result is achieved in two steps. First, we show that, in any straightline or polyline drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two ..."
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We show that there exist seriesparallel graphs requiring Ω(n2 √ logn) area in any straightline or polyline grid drawing. Such a result is achieved in two steps. First, we show that, in any straightline or polyline drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. [Information Processing Letters, 2003]. Second, we show a family of seriesparallel graphs requiring Ω(2 √ logn) width and Ω(2 √ logn) height in any straightline or polyline grid drawing. Combining the two results, the Ω(n2 √ logn) area lower bound is achieved. 2 1
MinimumLayer Upward . . .
, 2010
"... An upward drawing of a rooted tree T is a planar straightline drawing of T where the vertices of T are placed on a set of horizontal lines, called layers, such that for each vertex u of T, no child of u is placed on a layer vertically above the layer on which u has been placed. In this paper we giv ..."
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An upward drawing of a rooted tree T is a planar straightline drawing of T where the vertices of T are placed on a set of horizontal lines, called layers, such that for each vertex u of T, no child of u is placed on a layer vertically above the layer on which u has been placed. In this paper we give alineartime algorithm toobtainan upwarddrawing ofagiven rooted tree T on the minimum number of layers. Moreover, if the given tree T is not rooted, we can select a vertex r of T in linear time such that an upward drawing of T rooted at r would require the minimum number of layers among all the upward drawings of T with any of its vertices as the root. We also extend our results on a rooted tree to give an algorithm for an upwarddrawing of arooted ordered tree. To the best of our knowledge, there is no previous algorithm for obtaining an upward drawing of a tree on the minimum number of layers.
Upward Planar Drawings . . .
, 2006
"... In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popula ..."
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In this paper we present a new characterization of switchregular upward embeddings, a concept introduced by Di Battista and Liotta in 1998. This characterization allows us to define a new efficient algorithm for computing upward planar drawings of embedded planar digraphs. If compared with a popular approach described by Bertolazzi, Di Battista, Liotta, and Mannino, our algorithm computes drawings that are significantly better in terms of total edge length and aspect ratio, especially for lowdensity digraphs. Also, we experimentally prove that the running time of the drawing process is reduced in most cases.