A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs

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We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property pi is called a pi-graph. If pi satisfies certain properties, then an n-node m-edge pi-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 beta(n)+o(beta(n)) bits for any continuous super-additive function beta(n) so that there are at most 2beta(n)+o(beta(n)) distinct n-node pi-graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such pi 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 1,..., ell₁ for ell₁ <= n; (5) the edges of G are labeled with labels from 1,..., ell₂ for ell₂ <= m; and (6) each node (respectively, edge) of G has at most ell₃ = O(1) self-loops (respectively, ell₄ = O(1) multiple edges). Moreover, ell₃ and ell₄ are not required to be O(1) for the cases of pi being a plane triangulation. These examples are novel applications of small cycle separators of planar graphs and are the only nontrivial classes of graphs, other than rooted trees, with known polynomial-time information-theoretically optimal coding schemes.