Results 1  10
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27
Compact Representations of Separable Graphs
 In Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms
, 2003
"... We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queri ..."
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Cited by 36 (11 self)
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We consider the problem of representing graphs compactly while supporting queries e#ciently. In particular we describe a data structure for representing nvertex unlabeled graphs that satisfy an O(n )separator theorem, c < 1. The structure uses O(n) bits, and supports adjacency and degree queries in constant time, and neighbor listing in constant time per neighbor. This generalizes previous results for graphs with constant genus, such as planar graphs.
An InformationTheoretic Upper Bound of Planar Graphs Using Triangulation
, 2003
"... We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n ..."
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Cited by 24 (5 self)
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We propose a new linear time algorithm to represent a planar graph. Based on a specific triangulation of the graph, our coding takes on average 5.03 bits per node, and 3.37 bits per node if the graph is maximal. We derive from this representation that the number of unlabeled planar graphs with n nodes is at most 2 n+O(log n) where 5.007. The current lower bound is 2 n+(log n) for 4.71. We also show that almost all unlabeled and almost all labeled nnode planar graphs have at least 1.70n edges and at most 2.54n edges.
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 20 (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
Succinct Representations of Planar Maps
, 2008
"... This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. W ..."
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Cited by 19 (3 self)
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This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time.
Succinct representation of triangulations with a boundary. WADS
, 2005
"... Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangul ..."
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Cited by 16 (5 self)
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Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g − 1) lg m bits (which is negligible as long as g ≪ m / lg m). 1
LinearTime Compression of BoundedGenus Graphs into InformationTheoretically Optimal Number of Bits
, 2002
"... ..."
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 15 (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.
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) × ⌊ ⌋
Canonical Decomposition of Outerplanar Maps and Application to Enumeration, Coding and Generation
, 2003
"... In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n node ..."
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Cited by 12 (1 self)
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In this article we define a canonical decomposition of rooted outerplanar maps into a spanning tree and a list of edges. This decomposition, constructible in linear time, implies the existence of bijection between rooted outerplanar maps with n nodes and bicolored rooted ordered trees with n nodes where all the nodes of the last branch are colored white. As a consequence, for rooted outerplanar maps of n nodes, we derive: an enumeration formula, and an asymptotic of 2 3n (log n) ; an optimal data structure of asymptotically 3n bits, built in O(n) time, supporting adjacency and degree queries in worstcase constant time and neighbors query of a ddegree node in worstcase O(d) time...
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 12 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.