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135
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 (4 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
Comparing and Evaluating Layout Algorithms within GraphEd
 J. Visual Languages and Computing
, 1995
"... This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd ..."
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Cited by 15 (2 self)
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This paper is organized as follows. In section 2, we present an overview on the GraphEd system and the implemented graph drawing algorithms. Section 3 explains our evaluation experiments, and Section 4 shows our results. In Section 5 we give a subjective ranking of layout criteria. 2 GraphEd
A Framework for Drawing Planar Graphs with Curves and Polylines
 J. Algorithms
, 1998
"... We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well ..."
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Cited by 15 (3 self)
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We describe a unified framework of aesthetic criteria and complexity measures for drawing planar graphs with polylines and curves. This framework includes several visual properties of such drawings, including aspect ratio, vertex resolution, edge length, edge separation, and edge curvature, as well as complexity measures such as vertex and edge representational complexity and the area of the drawing. In addition to this general framework, we present algorithms that operate within this framework. Specifically, we describe an algorithm for drawing any nvertex planar graph in an O(n) O(n) grid using polylines that have at most two bends per edge and asymptoticallyoptimal worstcase angular resolution. More significantly, we show how to adapt this algorithm to draw any nvertex planar graph using cubic Bézier curves, with all vertices and control points placed within an O(n) O(n) integer grid so that the curved edges achieve a curvilinear analogue of good angular resolution. Al...
Graph Drawing
 Lecture Notes in Computer Science
, 1997
"... INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducte ..."
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Cited by 14 (3 self)
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INTRODUCTION Graph drawing addresses the problem of constructing geometric representations of graphs, and has important applications to key computer technologies such as software engineering, database systems, visual interfaces, and computeraideddesign. Research on graph drawing has been conducted within several diverse areas, including discrete mathematics (topological graph theory, geometric graph theory, order theory), algorithmics (graph algorithms, data structures, computational geometry, vlsi), and humancomputer interaction (visual languages, graphical user interfaces, software visualization). This chapter overviews aspects of graph drawing that are especially relevant to computational geometry. Basic definitions on drawings and their properties are given in Section 1.1. Bounds on geometric and topological properties of drawings (e.g., area and crossings) are presented in Section 1.2. Section 1.3 deals with the time complexity of fundamental graph drawin
Compact floorplanning via orderly spanning trees
 Journal of Algorithms
"... Floorplanning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)time algorithm to construct a floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is no ..."
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Cited by 14 (1 self)
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Floorplanning is a fundamental step in VLSI chip design. Based upon the concept of orderly spanning trees, we present a simple O(n)time algorithm to construct a floorplan for any nnode plane triangulation. In comparison with previous floorplanning algorithms in the literature, our solution is not only simpler in the algorithm itself, but also produces floorplans which require fewer module types. An equally important aspect of our new algorithm lies in its ability to fit the floorplan area in a rectangle of size (n − 1) × ⌊ ⌋
Planar Drawings of Plane Graphs
, 2000
"... this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results. ..."
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Cited by 13 (3 self)
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this paper first we review known two methods to find such drawings, then explain a hidden relation between them, and finally survey related results.
AGDLibrary: A Library of Algorithms for Graph Drawing
, 1997
"... A graph drawing algorithm produces a layout of a graph in two or threedimensional space that should be readable and easy to understand. Since the aesthetic criteria differ from one application area to another, it is unlikely that a definition of the "optimal drawing" of a graph in a strict math ..."
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Cited by 13 (5 self)
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A graph drawing algorithm produces a layout of a graph in two or threedimensional space that should be readable and easy to understand. Since the aesthetic criteria differ from one application area to another, it is unlikely that a definition of the "optimal drawing" of a graph in a strict mathematical sense exists. A large number of graph drawing algorithms taking different aesthetic criteria into account have already been proposed. In this paper we describe the design and implementation of the AGDLibrary, a library of Algorithms for Graph Drawing. The library offers a broad range of existing algorithms for twodimensional graph drawing and tools for implementing new algorithms. The library is written in C++ using the LEDA platform for combinatorial and geometric computing ([16, 17]). The algorithms are implemented independently of the underlying visualization or graphics system by using a generic layout interface. Most graph drawing algorithms place a set of restriction...
Simultaneous embedding of a planar graph and its dual on the grid
, 2002
"... Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar ..."
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Cited by 13 (8 self)
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Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straightline segments and the only crossings are between primaldual pairs of edges. We provide a lineartime algorithm that simultaneously embeds a 3connected planar graph and its dual on a (2n −2) ×(2n −2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.
A polynomial bound for untangling geometric planar graphs
, 2007
"... ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this ..."
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Cited by 13 (1 self)
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ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this question in the affirmative with ɛ = 1/4. The previous best known bound was Ω ( � log n / log log n). We also consider untangling geometric trees. It is known that every nvertex geometric tree can be untangled while keeping at least � n/3 vertices fixed, while the best upper bound was O((n log n) 2/3). We answer a question of Spillner and Wolff