## Planar Upward Tree Drawings with Optimal Area (1996)

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Venue: | Internat. J. Comput. Geom. Appl |

Citations: | 18 - 4 self |

### BibTeX

@ARTICLE{Garg96planarupward,

author = {Ashim Garg and Michael T. Goodrich and Roberto Tamassia},

title = {Planar Upward Tree Drawings with Optimal Area},

journal = {Internat. J. Comput. Geom. Appl},

year = {1996},

volume = {6},

pages = {333--356}

}

### Years of Citing Articles

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### Abstract

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 linear-time algorithms for constructing optimal area drawings. Let T be a bounded-degree 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 left-to-right 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 ...

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Citation Context ...ay all the symmetries in G, or, if G contains a Hamiltonian cycle, we may wish to draw G as a regular polygon with chords. The interest in this area has been growing significantly of late (see, e.g., =-=[7, 11, 16, 17, 20]-=-). For example, the annotated bibliography maintained by Di Battista, Eades, and Tamassia [10] mentions more than 250 papers in graph drawing. Important domains of application for graph drawing algori... |

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Citation Context ...isual languages, and VLSI layout. Perhaps the most studied graph drawing problem is that of producing a planar drawing of a planar graph (e.g., see the classic work of Tutte on planar convex drawings =-=[27]-=- and the recent results on planar straight-line drawings [11, 12, 17, 20, 25]). But there are a variety of other interesting graph drawing problems that are also being investigated of late, such as re... |

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Citation Context ... with chords. The interest in this area has been growing significantly of late (see, e.g., [7, 11, 16, 17, 20]). For example, the annotated bibliography maintained by Di Battista, Eades, and Tamassia =-=[10]-=- mentions more than 250 papers in graph drawing. Important domains of application for graph drawing algorithms include software engineering, project management, visual languages, and VLSI layout. Perh... |

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Citation Context ...aph drawing problem is that of producing a planar drawing of a planar graph (e.g., see the classic work of Tutte on planar convex drawings [27] and the recent results on planar straight-line drawings =-=[11, 12, 17, 20, 25]-=-). But there are a variety of other interesting graph drawing problems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e... |

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Citation Context ..., see any undergraduate text in data structures). The difficulty is that most of the known techniques for constructing planar upward drawings of trees require\Omega\Gamma N 2 ) area in the worst case =-=[22, 23]-=-. 1.2 Previous Work If we relax the upward requirement, however, then, as independently shown by Leiserson [19] and Valiant [28], one can construct an O(N)-area planar orthogonal grid drawing of an N-... |

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Citation Context ...ake up as little area as possible. This is motivated by the finite resolution of all of our current technologies for rendering a drawing, and also by circuit-area optimization criteria in VLSI layout =-=[2, 19, 28]-=-. In the following, we assume the existence of a resolution rule that implies a finite minimum area for the drawing of any graph. A typical resolution rule is to require grid drawings, where the verti... |

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Citation Context ...ere are a variety of other interesting graph drawing problems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e.g., see =-=[24, 26]-=- and O'Rourke's computational geometry column [21]), or dynamically maintaining a drawing under a sequence of insertions and deletions of vertices and edges, as studied by Cohen et al. [7]. 1.1 The Pr... |

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Citation Context ... tree of T �� rooted at u, and the right child associated with the rest of T �� . Tree S has 2N \Gamma 1 nodes, height at most log 3=2 N , and can be constructed in O(N) time (e.g., see Guibas=-= et al. [15]-=-). (a) (b) (c) (d) Figure 7: Illustration of the Algorithm of Theorem 6: (a) Example of a drawing produced by the algorithm of Lemma 2. (b) A tree T and the separators that join blocks. (c) Truncated ... |

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Citation Context ...tween partial tree and subtree.) A separator of a binary tree T is an edge of T whose removal divides T into two partial trees, each with at least N=3 nodes and at most 2N=3 nodes (e.g., see Chazelle =-=[6]). A-=- recursive decomposition of T by separators defines a binary tree S, called separator tree, where each leaf of S corresponds to a node of T , and each internal node �� of S corresponds to a partia... |

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Citation Context ...ake up as little area as possible. This is motivated by the finite resolution of all of our current technologies for rendering a drawing, and also by circuit-area optimization criteria in VLSI layout =-=[2, 19, 28]-=-. In the following, we assume the existence of a resolution rule that implies a finite minimum area for the drawing of any graph. A typical resolution rule is to require grid drawings, where the verti... |

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Citation Context ...here is a variety of other interesting graph drawing problems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e.g., see =-=[24, 28]-=- and O'Rourke's computational geometry column [22]), or dynamically maintaining a drawing under a sequence of insertions and deletions of vertices and edges, as studied by Cohen et al. [6]. 1.1 The Pr... |

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Citation Context ...ake up as little area as possible. This is motivated by the finite resolution of all of our current technologies for rendering a drawing, and also by circuit-area optimization criteria in VLSI layout =-=[2, 19, 28]-=-. In the following, we assume the existence of a resolution rule that implies a finite minimum area for the drawing of any graph. A typical resolution rule is to require grid drawings, where the verti... |

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Citation Context ...aph drawing problem is that of producing a planar drawing of a planar graph (e.g., see the classic work of Tutte on planar convex drawings [27] and the recent results on planar straight-line drawings =-=[11, 12, 17, 20, 25]-=-). But there are a variety of other interesting graph drawing problems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e... |

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Citation Context ... the edges have integer coordinates. Indeed, this consideration recently motivated the re-examination of straightline drawings of planar directed graphs, because they require exponentially-large area =-=[9]-=-, whereas several researchers have recently shown that planar graph drawings require only quadratic area, and that such drawings can be produced in linear time [12, 17, 25]. Moreover, some very nice r... |

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Citation Context ...ay all the symmetries in G, or, if G contains a Hamiltonian cycle, we may wish to draw G as a regular polygon with chords. The interest in this area has been growing significantly of late (see, e.g., =-=[7, 11, 16, 17, 20]-=-). For example, the annotated bibliography maintained by Di Battista, Eades, and Tamassia [10] mentions more than 250 papers in graph drawing. Important domains of application for graph drawing algori... |

32 |
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Citation Context ...ull of the drawing, then the drawing needs\Omega\Gamma N log N) area. Thus, a natural question is whether O(N) area is still achievable for planar upward drawings. Crescenzi, Di Battista, and Piperno =-=[8]-=- have recently provided a negative answer to this question for the case of strictly upward grid drawings, where the nodes have integer coordinates, and the parent of a node has y-coordinate strictly g... |

29 | Drawing graphs in the plane with high resolution - Formann, Hagerup, et al. - 1993 |

28 |
Drawing planar graphs using the lmc-ordering
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Citation Context ...ay all the symmetries in G, or, if G contains a Hamiltonian cycle, we may wish to draw G as a regular polygon with chords. The interest in this area has been growing significantly of late (see, e.g., =-=[7, 11, 16, 17, 20]-=-). For example, the annotated bibliography maintained by Di Battista, Eades, and Tamassia [10] mentions more than 250 papers in graph drawing. Important domains of application for graph drawing algori... |

28 |
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(Show Context)
Citation Context ..., see any undergraduate text in data structures). The difficulty is that most of the known techniques for constructing planar upward drawings of trees require\Omega\Gamma N 2 ) area in the worst case =-=[22, 23]-=-. 1.2 Previous Work If we relax the upward requirement, however, then, as independently shown by Leiserson [19] and Valiant [28], one can construct an O(N)-area planar orthogonal grid drawing of an N-... |

26 |
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Citation Context |

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18 |
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Citation Context ...is not above its parent. In addition, the issue is clouded somewhat by the fact that producing the exact minimization of the area of the drawing of a tree is NP-hard under several drawing conventions =-=[1, 4, 13]-=-. Nevertheless, Crescenzi et al. give O(N) area planar straight-line upward grid drawings of complete binary trees and Fibonacci trees. They do not, however, give a general construction for other type... |

15 | On the area of binary tree layouts
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Citation Context ..., one can construct an O(N)-area planar orthogonal grid drawing of an N-node tree T , where the nodes are placed at integer grid points and the edges follow paths of the grid. However, Brent and Kung =-=[5]-=- show that if the leaves of an N-node complete binary tree are constrained to be on the convex hull of the drawing, then the drawing needs\Omega\Gamma N log N) area. Thus, a natural question is whethe... |

15 | Linear-area upward drawings of AVL trees - Crescenzi, Penna, et al. - 1998 |

15 |
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Citation Context ...children. Namely, they exhibit a family of binary trees that require\Omega\Gamma N log N) area in any strictly upward planar grid drawing. This lower bound is tight within a constant factor: Shiloach =-=[26]-=- and Crescenzi, Di Battista, and Piperno [7] give linear-time algorithms that construct a strictly upward planar straight-line grid drawing of an N-node rooted tree with O(N log N) area, O(N) height, ... |

14 |
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Citation Context ...is not above its parent. In addition, the issue is clouded somewhat by the fact that producing the exact minimization of the area of the drawing of a tree is NP-hard under several drawing conventions =-=[1, 4, 13]-=-. Nevertheless, Crescenzi et al. give O(N) area planar straight-line upward grid drawings of complete binary trees and Fibonacci trees. They do not, however, give a general construction for other type... |

14 |
Minimum size h-v drawings
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(Show Context)
Citation Context ... enclosing rectangles of the drawings of the subtrees rooted at the children of a node are disjoint. Hence, hv-drawings are a special case of upward planar straight-line drawings. Eades, Lin, and Lin =-=[14]-=- show how to construct in O(N p N log N) time a minimum-area hv-drawing of an N-node binary tree. However, they do not provide specific bounds on the area requirement of hv-drawings. Related results o... |

11 | Area requirement of visibility representations of trees
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(Show Context)
Citation Context ...nary trees and Fibonacci trees. They do not, however, give a general construction for other types of trees. Related results on the area requirement of visibility representations of trees are given in =-=[18]-=-. 1.3 Our Results In this paper we show that, for any rooted bounded-degree tree T with N nodes, one can construct a planar upward grid drawing of T with O(N) area in O(N) time, and that such drawing ... |

10 |
Computational geometry column 18
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Citation Context ...roblems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e.g., see [24, 26] and O'Rourke's computational geometry column =-=[21]-=-), or dynamically maintaining a drawing under a sequence of insertions and deletions of vertices and edges, as studied by Cohen et al. [7]. 1.1 The Problem An important criterion for a drawing of a gr... |

10 |
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- 1986
(Show Context)
Citation Context ...ere are a variety of other interesting graph drawing problems that are also being investigated of late, such as representing G by means of visibility between geometric figures in the plane (e.g., see =-=[24, 26]-=- and O'Rourke's computational geometry column [21]), or dynamically maintaining a drawing under a sequence of insertions and deletions of vertices and edges, as studied by Cohen et al. [7]. 1.1 The Pr... |

9 | Nice drawings of graphs and trees are computationally hard - Brandenburg |

4 | Area-E cient Graph Layouts (for VLSI - Leiserson - 1980 |

4 | A Uni ed Approach and Visibility Representation of Planar Graphs - Tamassia, Tollis - 1986 |

3 |
O(n log n)-work parallel algorithm for straight-line grid embedding of planar graphs
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(Show Context)
Citation Context |

2 |
Nice Drawings of Graphs and Trees are Computationally Hard
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(Show Context)
Citation Context ...is not above its parent. In addition, the issue is clouded somewhat by the fact that producing the exact minimization of the area of the drawing of a tree is NP-hard under several drawing conventions =-=[1, 4, 13]-=-. Nevertheless, Crescenzi et al. give O(N) area planar straight-line upward grid drawings of complete binary trees and Fibonacci trees. They do not, however, give a general construction for other type... |

1 | Area requirement and symmetry display ofplanarupward drawings - Battista, Tamassia, et al. - 1992 |

1 | The complexity ofdrawing trees nicely - Supowit, Reingold - 1983 |