Linear-time compression of bounded-genus graphs into information-theoretically optimal number of bits

Abstract

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, 17-20] and compact routing table of Computer Networks [7]. For brevity, let a ¦Ð-graph stand for a graph with property ¦Ð. The information-theoretically optimal number of bits required to represent an n-node ¦Ð-graph is ⌈log2N¦Ð(n)⌉, where N¦Ð(n) is the number of distinct n-node ¦Ð-graphs. Although determining or approximating the close forms of N¦Ð(n) for nontrivial classes of ¦Ð is challenging, we provide a linear-time methodology for graph compression schemes that are information-theoretically optimal with respect to continuous super-additive functions (abbreviated as optimal for the rest of the extended abstract). Specifically, if ¦Ð satisfies certain properties, then we can compress any n-node m-edge ¦Ð-graph 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 ¦Â(n) + o(¦Â(n)) for any continuous super-additive function ¦Â(n) with log2N¦Ð(n) iU ¦Â(n) + o(¦Â(n)). Our methodology is applicable to general classes of graphs; this extended abstract focuses on graphs with sublinear genus. For example, if the input n-node ¦Ð-graph 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 ¦Ð satisfying the following statements:F1. The genus of any n-node ¦Ð-graph is o(n/log2n);F2. Any subgraph of a ¦Ð-graph remains a ¦Ð-graph;F3. log N¦Ð(n) = ¦¸(n); andF4. There is an integer k = O(1) such that it takes O(n) time to determine whether an O(log(k)n)-node graph satisfies property ¦Ð.For instance, ¦Ð can be the property of being a directed 3-colorable simple graph with genus no more than ten. The result is a novel application of planarization algorithm for bounded-genus graphs [5] and separator decomposition tree of planar graphs [9]. Rooted trees were the only known nontrivial class of graphs with linear-time optimal coding schemes. He, Kao, and Lu [11] provided O(n log n)-time compression schemes for planar and plane graphs that are optimal. Our results significantly enlarge the classes of graphs that admit efficient optimal compression schemes. More results on various versions of graph compression problems or succinct graph representations can be found in [1-4, 6, 8, 10, 14, 15] and the references therein.