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Lineartime succinct encodings of planar graphs via canonical orderings
 SIAM Journal on Discrete Mathematics
, 1999
"... Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, rough ..."
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Cited by 20 (6 self)
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Abstract. Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no selfloop or multiple edge. If G is triangulated, we can encode it using 4 m − 1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time 3 is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5 + 2 log 3) min{n, f} −7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.
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
"... 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 th ..."
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Cited by 18 (4 self)
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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.
A Fast and Compact Web Graph Representation
"... Compressed graphs representation has become an attractive research topic because of its applications in the manipulation of huge Web graphs in main memory. By far the best current result is the technique by Boldi and Vigna, which takes advantage of several particular properties of Web graphs. In t ..."
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Cited by 17 (12 self)
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Compressed graphs representation has become an attractive research topic because of its applications in the manipulation of huge Web graphs in main memory. By far the best current result is the technique by Boldi and Vigna, which takes advantage of several particular properties of Web graphs. In this paper we show that the same properties can be exploited with a different and elegant technique, built on RePair compression, which achieves about the same space but much faster navigation of the graph. Moreover, the technique has the potential of adapting well to secondary memory. In addition, we introduce an approximate RePair version that works efficiently with limited main memory.
An Experimental Analysis of a Compact Graph Representation
 In ALENEX04
, 2004
"... In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deleti ..."
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Cited by 16 (6 self)
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In previous work we described a method for compactly representing graphs with small separators, which makes use of small separators, and presented preliminary experimental results. In this paper we extend the experimental results in several ways, including extensions for dynamic insertion and deletion of edges, a comparison of a variety of coding schemes, and an implementation of two applications using the representation.
Optimal Bit Allocation in Compressed 3D Models
 Computational Geometry: Theory and Applications
, 1999
"... To use 3D models on the Internet or in other bandwidthlimited applications, it is often necessary to compress their triangle mesh representations. We consider the problem of balancing two forms of lossy mesh compression: reduction of the number of vertices by simplification, and reduction of the nu ..."
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Cited by 13 (3 self)
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To use 3D models on the Internet or in other bandwidthlimited applications, it is often necessary to compress their triangle mesh representations. We consider the problem of balancing two forms of lossy mesh compression: reduction of the number of vertices by simplification, and reduction of the number of bits per vertex coordinate. Let A(V , B) be a triangle mesh approximation for an original model O. Suppose that A(V , B) has V vertices, each represented using B bits per coordinate. Given a limit F on the file size for A(V , B), what are the optimal values of B and V that minimize the approximation error? Given a desired error bound E, what are optimal B and V ,and how many total bits are needed? We develop answers to these questions by using a shape complexity measure K , which, for any given object approximates the product EV. We give formulae linking B, V, F, E and K,andwe explore a simple algorithm for estimating K and the optimal B and V for piecewise spherical approximations...
3D Geometry Compression and Progressive Transmission
"... Polygonal meshes remain the primary representation for visualization of 3D data in a wide range of industries, including manufacturing, architecture, geographic information systems, medical imaging, robotics, entertainment, and military applications. Because of its widespread use, it is desirable ..."
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Cited by 12 (0 self)
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Polygonal meshes remain the primary representation for visualization of 3D data in a wide range of industries, including manufacturing, architecture, geographic information systems, medical imaging, robotics, entertainment, and military applications. Because of its widespread use, it is desirable to compress polygonal meshes stored in file servers and exchanged over computer networks to reduce storage and transmission time requirements. In this report we describe several schemes that have been recently introduced to represent single and multiresolution polygonal meshes in compressed form, and to progressively transmit polygonal mesh data. The progressive transmission of polygonal meshes allows the decoder process to make part of a singleresolution mesh, or the low resolution levels of detail of a multiresolution mesh, available to the rendering system before the whole bitstream is fully received and decoded. It is desirable to combine compression and progressive transmission, but not all the existing methods exhibit both features. These progressive transmission schemes are closely related to surface simplification or decimation methods, which change the surface topology while approximating the geometry, and can be regarded as lossy compression schemes as well. Finally, we describe in more detail the Topological Surgery and Progressive Forest Split schemes that are currently part of the MPEG4 multimedia standard. 1.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 11 (1 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
On the Optimality of Valencebased Connectivity Coding
, 2003
"... We show that the average entropy of the distribution of valences in valence sequences for the class of manifold 3D triangle meshes and the class of manifold 3D polygon meshes is strictly less than the entropy of these classes themselves. This implies that, apart from a valence sequence, another es ..."
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Cited by 9 (1 self)
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We show that the average entropy of the distribution of valences in valence sequences for the class of manifold 3D triangle meshes and the class of manifold 3D polygon meshes is strictly less than the entropy of these classes themselves. This implies that, apart from a valence sequence, another essential piece of information is needed for valencebased connectivity coding of manifold 3D meshes. Since there is no upper bound on the size of this extra piece of information, the result implies that the question of optimality of valencebased connectivity coding is still open.
Triangle Fixer: Edgebased Connectivity Compression
, 2000
"... Encoding the connectivity of triangle meshes has recently been the subject of intense study and many representations have been proposed [9, 10, 4, 8, 2, 5]. The sudden interest in this area is fueled by the emerging demand for interactive visualization of 3D data sets in a networked environment ( ..."
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Cited by 6 (3 self)
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Encoding the connectivity of triangle meshes has recently been the subject of intense study and many representations have been proposed [9, 10, 4, 8, 2, 5]. The sudden interest in this area is fueled by the emerging demand for interactive visualization of 3D data sets in a networked environment (e.g. VRML over the Internet). Since transmission bandwidth across widearea networks is a scarce resource, compact encodings for 3D models are of great advantage. Common representations for triangle meshes use two lists: a list of vertices and a list of triangles. The list of vertices contains coordinates that specify a physical location for each mesh vertex. This is referred to as the geometry of the triangle mesh. The list of triangles contains triplets of indices into the vertex list that specify the three vertices of each triangle. This is referred to as the connectivity of the triangle mesh. For triangle meshes with # vertices, the triangle list uses at lea
Optimal Bit Allocation in 3D Compression
, 1999
"... To use 3D models on the Internet or in other bandwidthlimited applications, it is often necessary to compress their triangle mesh representations. We consider the problem of balancing two forms of lossy mesh compression: reduction of the number of vertices by simplification, and reduction of the n ..."
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Cited by 6 (0 self)
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To use 3D models on the Internet or in other bandwidthlimited applications, it is often necessary to compress their triangle mesh representations. We consider the problem of balancing two forms of lossy mesh compression: reduction of the number of vertices by simplification, and reduction of the number of bits of resolution used per vertex coordinate via quantization. Let A be a triangle mesh approximation for an original model O. Suppose that A has V vertices, each represented using B bits per coordinate. Given a file size F for A, what are the optimal values of B and V? Given a desired error level E, what are estimates of B and V, and how many total bits are needed? We develop answers to these questions by using a shape complexity measure K that allows us to express the optimal value of B for a general model in terms of V and K alone. We give formulas linking B,V,F,E, and K, and we provide a simple algorithm for estimating the optimal B and V for an existing triangle mesh with a g...