Results 1  10
of
32
Succinct Representation of Balanced Parentheses, Static Trees and Planar Graphs
, 1999
"... We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set ..."
Abstract

Cited by 142 (9 self)
 Add to MetaCart
We consider the implementation of abstract data types for the static objects: binary tree, rooted ordered tree and balanced parenthesis expression. Our representations use an amount of space within a lower order term of the information theoretic minimum and support, in constant time, a richer set of navigational operations than has previously been considered in similar work. In the case of binary trees, for instance, we can move from a node to its left or right child or to the parent in constant time while retaining knowledge of the size of the subtree at which we are positioned. The approach is applied to produce succinct representation of planar graphs in which one can test adjacency in constant time. Keywords: abstract data type, succinct representation, binary trees, balanced parenthesis, rooted ordered trees, planar graphs. AMS subject classifications: 68P05, 68Q65 1 Introduction The binary tree is among the most fundamental of data structures. While it is often the c...
Face Fixer: Compressing polygon meshes with properties
 In SIGGRAPHâ€™00 Conference Proceedings
, 2000
"... Most schemes to compress the topology of a surface mesh have been developed for the lowest common denominator: triangulated meshes. We propose a scheme that handles the topology of arbitrary polygon meshes. It encodes meshes directly in their polygonal representation and extends to capture face grou ..."
Abstract

Cited by 88 (18 self)
 Add to MetaCart
Most schemes to compress the topology of a surface mesh have been developed for the lowest common denominator: triangulated meshes. We propose a scheme that handles the topology of arbitrary polygon meshes. It encodes meshes directly in their polygonal representation and extends to capture face groupings in a natural way. Avoiding the triangulation step we reduce the storage costs for typical polygon models that have group structures and property data.
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
, 1998
"... . We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than th ..."
Abstract

Cited by 50 (11 self)
 Add to MetaCart
. We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 Introduction This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different tradeoffs have been proposed. The adjacencylist encoding of a graph is widely useful but requires 2mdlog ne bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted nnode tree into a string of n pairs of balanced parentheses. Since the total number of such trees is...
Compressing the graph structure of the web
 In IEEE Data Compression Conference (DCC
, 2001
"... A large amount of research has recently focused on the graph structure (or link structure) of the World Wide Web. This structure has proven to be extremely useful for improving the performance of search engines and other tools for navigating the web. However, since the graphs in these scenarios invo ..."
Abstract

Cited by 47 (2 self)
 Add to MetaCart
A large amount of research has recently focused on the graph structure (or link structure) of the World Wide Web. This structure has proven to be extremely useful for improving the performance of search engines and other tools for navigating the web. However, since the graphs in these scenarios involve hundreds of millions of nodes and even more edges, highly spaceefficient data structures are needed to fit the data in memory. A first step in this direction was done by the DEC Connectivity Server, which stores the graph in compressed form. In this paper, we describe techniques for compressing the graph structure of the web, and give experimental results of a prototype implementation. We attempt to exploit a variety of different sources of compressibility of these graphs and of the associated set of URLs in order to obtain good compression performance on a large web graph. 1
Wrap&Zip decompression of the connectivity of triangle meshes compressed with Edgebreaker
 Journal of Computational Geometry, Theory and Applications
, 1999
"... The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string a sequence of t symbols from the set {C, L,E,R,S}, each represented by a 1, 2 or ..."
Abstract

Cited by 41 (13 self)
 Add to MetaCart
The Edgebreaker compression (Rossignac, 1999; King and Rossignac, 1999) is guaranteed to encode any unlabeled triangulated planar graph of t triangles with at most 1.84t bits. It stores the graph as a CLERS string a sequence of t symbols from the set {C, L,E,R,S}, each represented by a 1, 2 or 3 bit code. We show here that, in practice, the string can be further compressed to between 0.91t and 1.26t bits using an entropy code. These results improve over the 2.3t bits code proposed by Keeler and Westbrook (1995) and over the various 3D triangle mesh compression techniques published recently (Gumhold and Strasser, 1998; Itai and Rodeh, 1982; Naor, 1990; Touma and Gotsman, 1988; Turan, 1984), which exhibit either larger constants or cannot guarantee a linear worst case storage complexity. The decompression proposed by Rossignac (1999) is complicated and exhibits a nonlinear time complexity. The main contribution reported here is a simpler and efficient decompression algorithm, calle...
Compressing Polygon Mesh Connectivity with Degree Duality Prediction
, 2002
"... In this paper we present a coder for polygon mesh connectivity that delivers the best connectivity compression rates meshes reported so far. Our coder is an extension of the vertexbased coder for triangle mesh connectivity by Touma and Gotsman [26]. We code polygonal connectivity as a sequence of f ..."
Abstract

Cited by 36 (13 self)
 Add to MetaCart
In this paper we present a coder for polygon mesh connectivity that delivers the best connectivity compression rates meshes reported so far. Our coder is an extension of the vertexbased coder for triangle mesh connectivity by Touma and Gotsman [26]. We code polygonal connectivity as a sequence of face and vertex degrees and exploit the correlation between them for mutual predictive compression. Because lowdegree vertices are likely to be surrounded by highdegree faces and vice versa, we predict vertex degrees based on neighboring face degrees and face degrees based on neighboring vertex degrees.
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 ..."
Abstract

Cited by 36 (11 self)
 Add to MetaCart
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.
Compact Routing Tables for Graphs of Bounded Genus
, 2000
"... This paper deals with compact shortest path routing tables on weighted graphs with n nodes. For planar graphs we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log 2+ n) bitoperations per node to extract the route, for any constant > 0. ..."
Abstract

Cited by 31 (13 self)
 Add to MetaCart
This paper deals with compact shortest path routing tables on weighted graphs with n nodes. For planar graphs we show how to construct in linear time shortest path routing tables that require 8n + o(n) bits per node, and O(log 2+ n) bitoperations per node to extract the route, for any constant > 0. We obtain the same bounds for graphs of crossingedge number bounded by o(n= log n), and we generalize for graphs of genus bounded by > 0 yielding a size of n log +O(n) bits per node. Actually we prove a sharp upper bound of 2n log k +O(n) for graphs of pagenumber k, and a lower bound of n log k o(n log k) bits. These results are obtained by the use of dominating sets, compact coding of noncrossing partitions, and kpage representation of 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 ..."
Abstract

Cited by 24 (5 self)
 Add to MetaCart
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.
A Fast General Methodology For InformationTheoretically Optimal Encodings Of Graphs
, 1999
"... . We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtai ..."
Abstract

Cited by 24 (3 self)
 Add to MetaCart
. We propose a fast methodology for encoding graphs with informationtheoretically minimum numbers of bits. Specifically, a graph with property is called a graph. If satisfies certain properties, then an nnode medge graph G can be encoded by a binary string X such that (1) G and X can be obtained from each other in O(n log n) time, and (2) X has at most fi(n)+o(fi(n)) bits for any continuous superadditive function fi(n) so that there are at most 2 fi(n)+o(fi(n)) distinct nnode graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such include all conjunctions over the following groups of properties: (1) G is a planar graph or a plane graph; (2) G is directed or undirected; (3) G is triangulated, triconnected, biconnected, merely connected, or not required to be connected; (4) the nodes of G are labeled with labels from f1; : : : ; ` 1 g for ` 1 n; (5) the edges of G are labeled with labels from f1; : : : ; ` 2 ...