P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....binary trees. Let T be a binary tree. 5] presents an algorithm for constructing an upward polyline drawing of T in O(n) area. 6] and[11] present algorithms for constructing an orthogonal drawing of T in O(n) area. 7] gives an algorithm for constructing upward layered drawing in O(n ) area. [2] gives an algorithm for constructing strictly upward drawing drawing of T in O(n log n) area. If T is a Fibonacci tree, AVL tree, balanced tree, respectively) then [2, 10] 3] 2] respectively) give algorithms for constructing an strictly upward drawing of T in O(n) area. 4 Preliminaries In ....

....an orthogonal drawing of T in O(n) area. 7] gives an algorithm for constructing upward layered drawing in O(n ) area. 2] gives an algorithm for constructing strictly upward drawing drawing of T in O(n log n) area. If T is a Fibonacci tree, AVL tree, balanced tree, respectively) then [2, 10] ( 3] 2] respectively) give algorithms for constructing an strictly upward drawing of T in O(n) area. 4 Preliminaries In this section we give de nitions and some known results that will be used throughout the paper. Throughout this paper, by the term drawing, we will mean a planar ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187-200, 1992.

....trees, hierarchical structures and search trees. For a survey of graph drawing algorithms, including algorithms for drawing trees, see [5] In this paper we study h v drawings of binary trees. h v drawings were previously examined by Eades, Lin and Lin [7] and Crescenzi, Di Battista and Piperno [4]. Our results extend to inclusion drawings [6] and to slicing floorplanning [10, 3] The drawing of a rooted binary tree using the h v drawing convention is a planar grid drawing in which tree nodes are represented as points (of integer coordinates) in the plane and tree edges as non overlapping ....

....can be easily converted to upward drawings, our method can be used for deriving (non optimal) upward drawings of binary trees in parallel. In the case that we want to minimise the area of the drawing, by using the fact that for a tree with n nodes there exist upwards layouts of area n(logn 1) [4], the required number of processors reduces to O(n 4 log n) The rest of the paper is organised as follows: In Section 2, we give the necessary terminology. In Section 3, we present our parallel algorithm for deriving optimal drawings for binary trees using the h v convention. Minimum area h v ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista and A. Piperno, "A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees", Computational Geometry: Theory and Applications, Vol. 2, (1992), pp. 187-200.

....points has a long history. From a graph drawing perspective (see [6] for a survey of graph drawing) the traditional questions ask whether a (rooted or free) tree T = V; E) can be embedded in the plane such that some criterion is satisfied: e.g. that the area of the resulting embedding is small [4, 5, 11], that the symmetry Partially supported by an NSERC Postdoctoral Fellowship Partially supported by an NSERC Postgraduate Fellowship Partially supported by an NSERC Research Grant and a B.C. Advanced Systems Institute Fellowship. present in the tree is revealed in the embedding [13] or ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....Order preserving. The curve from the parent to the left child is to the left of the curve from the parent to the right child. Table 1 summarizes the known results for various combinations of the four criteria. The two O(n log n) area bounds in the table, due to Crescenzi, Di Battista, and Piperno [2] and Garg, Goodrich, and Tamassia [6] were obtained by simple recursive algorithms and proved to be tight in the worst case for their corresponding types of drawings. The type of drawings considered by Crescenzi et al. is not order preserving, and thus, one cannot reconstruct the binary tree ....

.... trees) ideal drawings satisfying all criteria 1 4 can be constructed using only O(n) area; see the references [3, 4, 7, 11, 13] Also not in the table are results regarding orthogonal drawings, i.e. drawings in which all line segments are either horizontal or vertical; see the references [1, 2, 6, 8, 11, 14]. 2 planar upward order preserv. straight line area references yes yes no no O(n) 6] yes yes no yes O(n log log n) 11] yes yes (strictly) no yes O(n log n) 2, 10] yes yes (strictly) yes no O(n log n) 6] yes yes (strictly) yes (strongly) yes O(n 1 ) this paper Table 1: Area bounds for ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187-200, 1992. 11

....greater than or equal to that of the child. A drawing satisfying the condition is said to be upward. In addition, a strictly upward drawing means that the parent has y coordinate strictly greater than that of its child. Upward straight line drawing algorithms for rooted trees were presented in [1, 12, 4, 2]. Crescenzi et al. 1] and Shiloach [12] presented O(n log n) area upward straight line drawing algorithms for any rooted trees. Crescenzi et al. 1] proved that Omega (n log n) area is required to draw a class of rooted trees under strictly upward straight line standard and presented an ....

....of the child. A drawing satisfying the condition is said to be upward. In addition, a strictly upward drawing means that the parent has y coordinate strictly greater than that of its child. Upward straight line drawing algorithms for rooted trees were presented in [1, 12, 4, 2] Crescenzi et al. [1] and Shiloach [12] presented O(n log n) area upward straight line drawing algorithms for any rooted trees. Crescenzi et al. 1] proved that Omega (n log n) area is required to draw a class of rooted trees under strictly upward straight line standard and presented an algorithm to construct an ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, pages 187--200, 1992.

....and existential lower bounds on the area of drawings of graphs. We denote with a an arbitrary constant such that 0 a 1. We denote with b and c fixed constants such that 1 b c. Class of Graphs Drawing Type Area Source rooted tree upward planar straight line grid O(n log n) Omega Gamma n) [17, 98] rooted tree strictly upward planar straight line grid O(n log n) Omega Gamma n logn) 17] degree O(n a ) rooted tree upward planar polyline grid O(n) Omega Gamma n) 56] binary tree upward planar orthogonal grid O(n log log n) Omega Gamma n loglog n) 56] tree planar straight line grid ....

....such that 0 a 1. We denote with b and c fixed constants such that 1 b c. Class of Graphs Drawing Type Area Source rooted tree upward planar straight line grid O(n log n) Omega Gamma n) 17, 98] rooted tree strictly upward planar straight line grid O(n log n) Omega Gamma n logn) [17] degree O(n a ) rooted tree upward planar polyline grid O(n) Omega Gamma n) 56] binary tree upward planar orthogonal grid O(n log log n) Omega Gamma n loglog n) 56] tree planar straight line grid O(n log n) Omega Gamma n) 17, 98] degree O(n a ) tree planar polyline grid O(n) ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....and existential lower bounds on the area of drawings of graphs. We denote with a an arbitrary constant such that 0 a 1. We denote with b and c fixed constants such that 1 b c. Class of Graphs Drawing Type Area Ref. rooted tree upward planar straight line grid Omega Gamma n) O(n log n) [12, 79] rooted tree strictly upward planar straight line grid Omega Gamma n log n) O(n log n) 12] degree O(n a ) rooted tree upward planar polyline grid Omega Gamma n) O(n) 38] binary tree upward planar orthogonal grid Omega Gamma n log log n) O(n log log n) 38] tree planar ....

....such that 0 a 1. We denote with b and c fixed constants such that 1 b c. Class of Graphs Drawing Type Area Ref. rooted tree upward planar straight line grid Omega Gamma n) O(n log n) 12, 79] rooted tree strictly upward planar straight line grid Omega Gamma n log n) O(n log n) [12] degree O(n a ) rooted tree upward planar polyline grid Omega Gamma n) O(n) 38] binary tree upward planar orthogonal grid Omega Gamma n log log n) O(n log log n) 38] tree planar straight line grid Omega Gamma n) O(n log n) 12, 79] degree O(n a ) tree planar polyline grid ....

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P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....resolution rule is verified, and that the minimum angle between two edges incident at the same vertex is at least ff(d) where ff(d) is a predefined function of the maximum degree of the graph. Once the resolution rule is given, the problem of evaluating the area of a drawing naturally arises [4, 9, 16, 17, 15, 6, 19, 20, 28]. In fact, any resolution rule implies a finite minimum area for a drawing of a graph. All known algorithms that compute (weak) proximity drawings produce representations whose area increases exponentially with the number of vertices. As a consequence, the problem of constructing proximity ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....trees, hierarchical structures and search trees. For a survey of graph drawing algorithms, including algorithms for drawing trees, see [5] In this paper we study h v drawings of binary trees. h v drawings were previously examined by Eades, Lin and Lin [7] and Crescenzi, Di Battista and Piperno [4]. Our results extend to inclusion drawings [6] and to slicing floorplanning [10, 3] The drawing of a rooted binary tree using the h v drawing convention is a planar grid drawing in which tree nodes are represented as points (of integer coordinates) in the plane and tree edges as non overlapping ....

....can be easily converted to upward drawings, our method can be used for deriving (non optimal) upward drawings of binary trees in parallel. In the case that we want to minimise the area of the drawing, by using the fact that for a tree with n nodes there exist upwards layouts of area n(log n 1) [4], the 2 required number of processors reduces to O(n 4 log n) The rest of the paper is organised as follows: In Section 2, we give the necessary terminology. In Section 3, we present our parallel algorithm for deriving optimal drawings for binary trees using the h v convention. Minimum area ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista and A. Piperno, "A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees", Computational Geometry: Theory and Applications, Vol. 2, (1992), pp. 187--200.

....a detailed bibliography. As for the area requirement of upward planar grid drawings of trees, the well known algorithms of Reingold and Tilford [89] and Wetherell and Shannon [112] give drawings with area O(n 2 ) where n is the number of nodes in the tree. Crescenzi, Di Battista, and Piperno [21, 22] have given algorithms for constructing a straight line upward planar grid drawing with area O(n log n) of a given general tree, and with area O(n) of a given complete binary or Fibonacci or AVL tree. In Chapter 3, we study the area requirements of upward planar grid drawings of trees, within two ....

....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 whether O(N) area is still achievable for planar upward drawings. Crescenzi, Di Battista, and Piperno [21] 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 greater than the ones of its children. Namely, they exhibit a family of binary trees that require ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....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 whether O(N) area is still achievable for planar upward drawings. Crescenzi, Di Battista, and Piperno [7] 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 greater than the ones of its children. Namely, they exhibit a family of binary trees that ....

....y coordinate strictly greater than the ones of its 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, and O(log N) width. Their result doesn t settle the question for the standard notion of upward drawing, however, which allows a child node to ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....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 h v 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 h v drawing convention is a planar grid drawing in which tree nodes are represented as points (of integer coordinates) in the plane ....

Crescenzi, P., Di Battista, G. and Piperno, A. A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees. Computational Geometry: Theory and Applications, Vol. 2, pp. 187--200, 1992.

....the algorithm presented in [66] is dynamized; in the same paper it is also shown how to apply the same techniques to tackle the dynamic graph drawing problem for several families of graphs. In [67] a polynomial time algorithm is given for testing if an embedded graph has an upward drawing. In [157] upper and lower bounds are presented on the area requirements of upward drawings of binary trees. Constrained visibility problems are tackled in [203] In [204] a general framework is proposed for drawing acyclic digraphs. The framework basically exploits the vertex orderings induced by a ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, 2, 1992.

....points has a long history. From a graph drawing perspective (see [6] for a survey of graph drawing) the traditional questions ask whether a (rooted or free) tree T = V; E) can be embedded in the plane such that some criterion is satisfied: e.g. that the area of the resulting embedding is small [4, 5, 11], that the symmetry present in the tree is revealed in the embedding [13] or that T is isomorphic to the minimum weight spanning tree [7, 15] or proximity graph [1, 2] of the points in which the vertices are embedded. In essence, the tree is given as input; one needs to construct a set of points ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....various aestheticity criteria. Rectilinear drawings are often considered, when edges are restricted to be horizontal or vertical line segments, with or without bends (see, for example, DLT84] A lot of attention has been given to tree drawing. For example, Crescenzi, Di Battista and Piperno [CDP92], and Garg, Goodrich and Tamassia [GGT93] investigate the problem for upward drawings of trees. Planar upward drawings are studied in [DTT92] See the survey in [DETT94] for more references. In Section 2 we introduce our notation and terminology. In Section 3 we present a generic shift method for ....

P. Crescenzi, G. Di Battista, A. Piperno, A note on optimal area algorithms for upward drawings of binary trees, Computational Geometry: Theory and Applications 2 (1992) 187-200.

....resolution rule is verified, and that the minimum angle between two edges incident at the same vertex is at least ff(d) where ff(d) is a predefined function of the maximum degree of the graph. Once the resolution rule is given, the problem of evaluating the area of a drawing naturally arises [3, 9, 15, 14, 5, 17, 24]. In fact, any resolution rule implies a finite minimum area for a drawing of a graph. All known algorithms that compute (weak) proximity drawings produce representations whose area increases exponentially with the number of vertices. As a consequence, the problem of constructing proximity ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

.... Di Battista and Piperno showed that every binary tree admits an upward drawing into a grid of area O(n log n) where n is the number of nodes, and that this bound is optimal within a constant factor because an infinite family of binary trees exists that require area Omega Gamma n log n) [2]. Moreover, they also show that both complete and Fibonacci trees can be drawn in linear area. In order to achieve this latter result, they introduce the notion of h v drawing. Preprint submitted to Elsevier Science 8 June They develop an algorithm that finds linear area h v drawings for ....

....the first linear area upward drawing algorithm that extends beyond the class of binary trees. Moreover, a straightforward application of the same idea yields an algorithm that draw a Fibonacci tree with n nodes into a grid of area 1:171n O(log n p n) thus improving over the area requirement of [2] (that is about 6n) 1.1 Preliminaries We denote by e the empty tree and by 1 the tree consisting of a single node. Given two binary trees T 1 and T 2 , we denote by T 1 Phi T 2 the binary tree such that T 1 (respectively, T 2 ) is its immediate left (respectively, right) subtree. For any h 0, ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, 2:187--200, 1992.

....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 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 greater than the ones of its children. Namely, they exhibit a family of binary trees that ....

P. Crescenzi, G. Di Battista, and A. Piperno, "A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees," to appear in Computational Geometry: Theory and Applications.

....an O(n log n) area planar polyline upward grid drawing that preserves the left to right ordering of the children of each vertex. The drawing can be constructed in O(n) time. Also, the above area bound is asymptotically optimal in the worst case. Previously, Crescenzi, Di Battista, and Piperno [10] considered strictly upward grid drawings, where the vertices have integer coordinates, and the parent of a vertex has y coordinate strictly greater than the ones of its children. They show that a rooted tree with n vertices, admits a strictly upward planar straight line grid drawing with O(n log ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Computational Geometry: Theory and Applications, 2:187--200, 1992.

....Order preserving. The curve from the parent to the left child is to the left of the curve from the parent to the right child. Table 1 summarizes the known results for various combinations of the four criteria. The two O(n log n) area bounds in the table, due to Crescenzi, Di Battista, and Piperno [2] and Garg, Goodrich, and Tamassia [6] were obtained by simple recursive algorithms and proved to be tight in the worst case for their corresponding types of drawings. The type of drawings considered by Crescenzi et al. is not order preserving, and thus, one cannot reconstruct the binary tree ....

.... trees) ideal drawings satisfying all criteria 1 4 can be constructed using only O(n) area; see the references [3, 4, 7, 10, 12] Also not in the table are results regarding orthogonal drawings, i.e. drawings in which all line segments are either horizontal or vertical; see the references [1, 2, 6, 8, 10, 13]. Note that despite its naturalness, our strong definition of order preserving drawings (criterion 3) seems to be unstudied before. One may insist on an even stronger condition where the curves are not only monotone increasing decreasing in the x direction, but strictly increasing decreasing. ....

[Article contains additional citation context not shown here]

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

....area Omega (c aen ) In particular, this implies that there exist bounded degree planar graphs for which asymptotically optimal resolution can be achieved only at the expense of an exponential increase in the area. Previously, area trade off results were proved for planar drawings (see, e.g. [1, 2, 3, 6, 4, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19]) depending on various restrictions on the representation (e.g. upward, straight line, orthogonal, polyline) Our result is the first continuous area trade off result for planar drawings. In Section 5, we give linear time algorithms for constructing planar straight line drawings with high ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comp. Geom. Theory Appl., 2:187--200, 1992.

....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 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 greater than the ones of its children. Namely, they exhibit a family of binary trees that ....

....is a straight line drawing (see Fig. 1.a) if each edge is a straight line segment. Gamma is a polyline drawing (see Fig. 1. b) if each edge is a polygonal chain, and we call bends the intermediate vertices of the chain that are Tree Type Previous Bounds Our Bounds Upward Polyline O(N log N) [8] Theta(N ) Strictly Upward Straight Line Theta(N log N) 8] Upward Ordered Polyline Theta(N log N) Upward Straight Line (complete binary trees and Fibonacci trees) O(N) 8] Non Upward Orthogonal Theta(N ) 19, 28] Upward Orthogonal Theta(N log log N) Leaves on Hull Orthogonal ....

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P. Crescenzi, G. Di Battista, and A. Piperno, "A Note on Optimal Area Algorithms for Upward Drawings of Binary Trees," to appear in Computational Geometry: Theory and Applications.

....12] give algorithms for constructing upward planar drawings of st digraphs and investigate area bounds and symmetry display. Tamassia and Vitter [32] show that the above drawing algorithms can be e#ciently parallelized. Upward planar drawings of trees and series parallel digraphs are studied in [29, 31, 6, 13, 15] and [1, 2] respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial time algorithm is given for testing the upward planarity of digraphs with a prescribed embedding. Thomassen [33] characterizes the ....

P. Crescenzi, G. Di Battista, and A. Piperno, A note on optimal area algorithms for upward drawings of binary trees, Comput. Geom. Theory Appl., 2 (1992), pp. 187--200.

....12] give algorithms for constructing upward planar drawings of st digraphs, and investigate area bounds and symmetry display. Tamassia and Vitter [32] show that the above drawing algorithms can be efficiently parallelized. Upward planar drawings of trees and series parallel digraphs are studied in [29, 31, 6, 13, 15] and [1, 2] respectively. In [8] it is shown that for the special case of bipartite digraphs, upward planarity is equivalent to planarity. In [3, 4] a polynomial time algorithm is given for testing the upward planarity of digraphs with a prescribed embedding. Thomassen [33] characterizes the ....

P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

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P. Crescenzi, G. Di Battista, and A. Piperno. A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. Theory Appl., 2:187--200, 1992.

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