## A Fast General Methodology For Information-Theoretically Optimal Encodings Of Graphs (1999)

Citations: | 23 - 3 self |

### BibTeX

@MISC{He99afast,

author = {Xin He and Ming-yang Kao and Hsueh-I Lu},

title = {A Fast General Methodology For Information-Theoretically Optimal Encodings Of Graphs},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property is called a -graph. If satisfies certain properties, then an n-node m-edge -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 super-additive function fi(n) so that there are at most 2 fi(n)+o(fi(n)) distinct n-node -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 ...

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Citation Context ...plane triangulation. For instance, π can be the property of being a directed unlabeled biconnected simple plane graph. These examples are novel applications of small cycle separators of planar graphs =-=[11,12]-=-. Note that the rooted trees are the only other nontrivial class of graphs with a known polynomial-time information-theoretically optimal coding scheme, which encodes a tree as nested parentheses usin... |

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Citation Context ...ng methodology. Let |X| be the number of bits in a binary string X. Let |G| be the number of nodes in a graph G. Let |S| be the number of elements, counting multiplicity, in a multiset S. Fact 1 (see =-=[1,3]-=-). Let X1, X2, . . ., Xk be O(1) binary strings. Let n = |X1| + |X2| + · · · + |Xk|. Then there exists an O(log n)-bit string χ, obtainable in O(n) time, such that given the concatenation of χ, X1, X2... |

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Citation Context ...d Rodeh [6] gave an encoding of 3 n log n + O(n) bits. For unlabeled general graphs, Naor [14] gave an 2 encoding of 1 2n2 − n log n + O(n) bits. For applications that require query support, Jacobson =-=[7]-=- gave a Θ(n)-bit encoding for a connected and simple planar graph G that supports traversal in Θ(log n) time per node visited. Munro and Raman [13] improved this result and gave schemes to encode bina... |

140 | Succinct Representation of Balanced Parentheses and Static Trees
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Citation Context ...For applications that require query support, Jacobson [7] gave a Θ(n)-bit encoding for a connected and simple planar graph G that supports traversal in Θ(log n) time per node visited. Munro and Raman =-=[13]-=- improved this result and gave schemes to encode binary trees, rooted ordered trees, and planar graphs. For a general planar G, they used 2m + 8n + o(m + n) bits while supporting adjacency and degree ... |

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Citation Context ...plane triangulation. For instance, π can be the property of being a directed unlabeled biconnected simple plane graph. These examples are novel applications of small cycle separators of planar graphs =-=[11,12]-=-. Note that the rooted trees are the only other nontrivial class of graphs with a known polynomial-time information-theoretically optimal coding scheme, which encodes a tree as nested parentheses usin... |

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Citation Context ...ct m-edge plane triangulations where β(m) = ( 8 3 − log2 3)m + o(m) ≈ 1.08m + o(m) [17] and that there are 22m+o(n) distinct m-edge n-node triconnected plane graphs that may be non-simple [18]. Turán =-=[16]-=- used 4m bits to encode a plane graph G that may have self-loops. Keeler and Westbrook [10] improved this bit count to 3.58m. They also gave coding schemes for several families of plane graphs. In par... |

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Citation Context ...upport. The bit count can be further reduced if only O(1)time adjacency queries are supported, or if G is simple, triconnected or triangulated [2]. For certain graph families, Kannan, Naor and Rudich =-=[8]-=- gave schemes that encode each node with O(log n) bits and support O(log n)-time testing of adjacency between two nodes. For dense graphs and complement graphs, Kao, Occhiogrosso, and Teng [9] devised... |

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Citation Context .... For dense graphs and complement graphs, Kao, Occhiogrosso, and Teng [9] devised two compressed representations from adjacency lists to speed up basic graph search techniques. Galperin and Wigderson =-=[4]-=- and Papadimitriou and Yannakakis [15] 2sinvestigated complexity issues arising from encoding a graph by a small circuit that computes its adjacency matrix. Section 2 discusses the general encoding me... |

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Citation Context ...nd thus free of degree-one nodes, they [5] improved the bit count to at most 2.835m bits. This bit count was later reduced to at most 3log2 3 m + O(1) ≈ 2.378m + O(1) by Chuang, Garg, He, Kao, and Lu =-=[2]-=-. These coding 2 schemes all take linear time for encoding and decoding, but their bit counts are not information-theoretically optimal. For labeled planar graphs, Itai and Rodeh [6] gave an encoding ... |

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Citation Context ... that there are 22m+o(n) distinct m-edge n-node triconnected plane graphs that may be non-simple [18]. Turán [16] used 4m bits to encode a plane graph G that may have self-loops. Keeler and Westbrook =-=[10]-=- improved this bit count to 3.58m. They also gave coding schemes for several families of plane graphs. In particular, they used 1.53m bits for a triangulated simple G, and 3m bits for a connected G fr... |

41 |
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Citation Context ...phs, Kao, Occhiogrosso, and Teng [9] devised two compressed representations from adjacency lists to speed up basic graph search techniques. Galperin and Wigderson [4] and Papadimitriou and Yannakakis =-=[15]-=- 2sinvestigated complexity issues arising from encoding a graph by a small circuit that computes its adjacency matrix. Section 2 discusses the general encoding methodology. Sections 3 and 4 use the me... |

29 | Succinct representation of general unlabeled graphs
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Citation Context ...d decoding, but their bit counts are not information-theoretically optimal. For labeled planar graphs, Itai and Rodeh [6] gave an encoding of 3 n log n + O(n) bits. For unlabeled general graphs, Naor =-=[14]-=- gave an 2 encoding of 1 2n2 − n log n + O(n) bits. For applications that require query support, Jacobson [7] gave a Θ(n)-bit encoding for a connected and simple planar graph G that supports traversal... |

22 |
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Citation Context ..., He, Kao, and Lu [2]. These coding 2 schemes all take linear time for encoding and decoding, but their bit counts are not information-theoretically optimal. For labeled planar graphs, Itai and Rodeh =-=[6]-=- gave an encoding of 3 n log n + O(n) bits. For unlabeled general graphs, Naor [14] gave an 2 encoding of 1 2n2 − n log n + O(n) bits. For applications that require query support, Jacobson [7] gave a ... |

21 | Linear-time succinct encodings of planar graphs via canonical orderings
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- 1999
(Show Context)
Citation Context ...s of plane graphs. In particular, they used 1.53m bits for a triangulated simple G, and 3m bits for a connected G free of self-loops and degree-one nodes. For a simple triangulated G, He, Kao, and Lu =-=[5]-=- improved the bit count to 4m + O(1). For 3 a simple G that is triconnected and thus free of degree-one nodes, they [5] improved the bit count to at most 2.835m bits. This bit count was later reduced ... |

21 |
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(Show Context)
Citation Context ... as nested parentheses using 2(n − 1) bits in O(n) time. Previously, Tutte proved that there are 2β(m)+o(β(m)) distinct m-edge plane triangulations where β(m) = ( 8 3 − log2 3)m + o(m) ≈ 1.08m + o(m) =-=[17]-=- and that there are 22m+o(n) distinct m-edge n-node triconnected plane graphs that may be non-simple [18]. Turán [16] used 4m bits to encode a plane graph G that may have self-loops. Keeler and Westbr... |

7 |
Simple and efficient compression schemes for dense and complement graphs
- Kao, Occhiogrosso, et al.
- 1999
(Show Context)
Citation Context ... Rudich [8] gave schemes that encode each node with O(log n) bits and support O(log n)-time testing of adjacency between two nodes. For dense graphs and complement graphs, Kao, Occhiogrosso, and Teng =-=[9]-=- devised two compressed representations from adjacency lists to speed up basic graph search techniques. Galperin and Wigderson [4] and Papadimitriou and Yannakakis [15] 2sinvestigated complexity issue... |

5 | Succinct representations of graphs - an - 1984 |