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Lineartime compression of boundedgenus graphs into informationtheoretically optimal number of bits
 In: 13th Symposium on Discrete Algorithms (SODA
, 2002
"... 1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer ..."
Abstract

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1 I n t roduct ion This extended abstract summarizes a new result for the graph compression problem, addressing how to compress a graph G into a binary string Z with the requirement that Z can be decoded to recover G. Graph compression finds important applications in 3D model compression of Computer Graphics [12, 1720] and compact routing table of Computer Networks [7}. For brevity, let a ~rgraph stand for a graph with property n. The informationtheoretically optimal number of bits required to represent an nnode ngraph is [log 2 N~(n)], where N,~(n) is the number of distinct nnode *rgraphs. Although determining or approximating the close forms of N ~ (n) for nontrivial classes of n is challenging, we provide a lineartime methodology for graph compression schemes that are informationtheoretically optimal with respect to continuous uperadditive functions (abbreviated as optimal for the rest of the extended abstract). 1 Specifically, if 7r satisfies certain properties, then we can compress any nnode medge 1rgraph G into a binary string Z such that G and Z can be computed from each other in O(m + n) time, and that the bit count of Z is at most fl(n) + o(fl(n)) for any continuous uperadditive function fl(n) with log 2 N~(n) < fl(n) + o(fl(n)). Our methodology is applicable to general classes of graphs; this extended abstract focuses on graphs with sublinear genus. 2 For example, if the input nnode,rgraph G is equipped with an embedding on its genus surface, which is a reasonable assumption for graphs arising from 3D model compression, then our methodology is applicable to any 7r satisfying the following statements: