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170
Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs
 Journal of Graph Algorithms and Applications
, 1998
"... We describe a new approach for clusterbased drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSPtype decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounde ..."
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Cited by 20 (10 self)
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We describe a new approach for clusterbased drawing of large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSPtype decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph embedded in the plane into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D0(G)), where n and m are the number of vertices and edges of the graph G, andD0(G) is the time it takes to obtain an initial embedding of G in the plane. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n×n grid and the running time reduces to O(n log n).
Planar graphs, via wellorderly maps and trees
 In 30 th International Workshop, Graph  Theoretic Concepts in Computer Science (WG), volume 3353 of Lecture Notes in Computer Science
, 2004
"... Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequ ..."
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Cited by 19 (4 self)
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Abstract. The family of wellorderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a wellorderly map. We show that the number of wellorderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worstcase, a rate of 4.91 bits per node and of 2.82 bits per edge. 1
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 (4 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 ...
Greedy Drawings of Triangulations
, 2007
"... Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the fol ..."
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Cited by 19 (1 self)
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Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the following conjecture: Any 3connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s = v1,v2,...,vk = t in a drawing is said to be distance decreasing if �vi − t � < �vi−1 − t�, 2 ≤ i ≤ k where �... � denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Walter Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the KnasterKuratowskiMazurkiewicz Theorem, that some drawing of G belonging to this class is greedy. 1 1
Strictly Convex Drawings of Planar Graphs
, 2004
"... Every threeconnected planar graph with n vertices has a drawing on an O(n7=3) \Theta O(n7=3) grid in which all faces are strictly convex polygons. ..."
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Cited by 18 (1 self)
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Every threeconnected planar graph with n vertices has a drawing on an O(n7=3) \Theta O(n7=3) grid in which all faces are strictly convex polygons.
Improved Bounds for Intersecting Triangles and Halving Planes
, 1993
"... If a configuration of m triangles in the plane has only n points as vertices, then there must be a set of max # #m/(2n  5)# #(m 3 /(n 6 log 2 n)) triangles having a common intersection. As a consequence the number of halving planes for a threedimensional point set is O(n 8/3 log 2/3 ..."
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Cited by 16 (1 self)
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If a configuration of m triangles in the plane has only n points as vertices, then there must be a set of max # #m/(2n  5)# #(m 3 /(n 6 log 2 n)) triangles having a common intersection. As a consequence the number of halving planes for a threedimensional point set is O(n 8/3 log 2/3 n). For all m and n there exist configurations of triangles in which the largest common intersection involves max # #m/(2n  5)# O(m 2 /n 3 ) triangles; the upper and lower bounds match for m = O(n 2 ). The best previous bounds were #(m 3 /(n 6 log 5 n)) for intersecting triangles, and O(n 8/3 log 5/3 n) for halving planes. 1 Introduction Suppose we have a set of triangles which share many of their corners. If there are n corners, and more than 2n triangles, then the graph of triangle edges must not be planar and so some two triangles must overlap. A recent paper of Aranov et al. [1] generalized this observation. Suppose some m triangles again share n corners. The re...
Succinct greedy graph drawing in the hyperbolic plane
 In Proc. 16th Int. Symp. Graph Drawing
, 2008
"... Abstract. We describe an efficient method for drawing any nvertex simple graph G in the hyperbolic plane. Our algorithm produces greedy drawings, which support greedy geometric routing, so that a message M between any pair of vertices may be routed geometrically, simply by having each vertex that r ..."
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Cited by 16 (3 self)
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Abstract. We describe an efficient method for drawing any nvertex simple graph G in the hyperbolic plane. Our algorithm produces greedy drawings, which support greedy geometric routing, so that a message M between any pair of vertices may be routed geometrically, simply by having each vertex that receives M pass it along to any neighbor that is closer in the hyperbolic metric to the message’s eventual destination. More importantly, for networking applications, our algorithm produces succinct drawings, in that each of the vertex positions in one of our embeddings can be represented using O(log n) bits and the calculation of which neighbor to send a message to may be performed efficiently using these representations. These properties are useful, for example, for routing in sensor networks, where storage and bandwidth are limited. 1
An Algorithm to Construct Greedy Drawings of Triangulations
, 2010
"... We show an algorithm to construct a greedy drawing of every given triangulation. The algorithm relies on two main results. First, we show how to construct greedy drawings of a fairly simple class of graphs, called triangulated binary cactuses. Second, we show that every triangulation can be spanned ..."
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Cited by 16 (2 self)
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We show an algorithm to construct a greedy drawing of every given triangulation. The algorithm relies on two main results. First, we show how to construct greedy drawings of a fairly simple class of graphs, called triangulated binary cactuses. Second, we show that every triangulation can be spanned by a triangulated binary cactus. Further, we discuss how to extend our techniques in order to prove that every triconnected planar graph admits a greedy drawing. Such a result, which proves a conjecture by Papadimitriou and Ratajczak, was independently shown by Leighton and Moitra.