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

We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property $\pi$ is called a {\em $\pi$-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 superadditive function $\beta(n)$ so that there are at most $2^{\beta(n)+o(\beta(n))}$ 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 $\{1,\ldots, \ell_1\}$ for $\ell_1\leq n$; (5) the edges of G are labeled with labels from $\{1,\ldots, \ell_2\}$ for $\ell_2\leq m$; and (6) each node (respectively, edge) of G has at most $\ell_3=O(1)$ self-loops (respectively, $\ell_4=O(1)$ multiple edges). Moreover, $\ell_3$ and $\ell_4$ 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.