2000/01/01 by Xin He, Ming‐Yang Kao, Hsueh-I Lu · 57 citations
Computer Science · Mathematics · #Algorithms and Data Compression #Complexity and Algorithms in Graphs #Advanced Graph Theory Research #Combinatorics #Planar graph #Mathematics #Graph #Discrete mathematics #Undirected graph
paper · doi:10.1137/s0097539799359117
published in SIAM Journal on Computing 30(3), 838-846 (Society for Industrial and Applied Mathematics)
openalex publication_date 2000/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
We propose a fast methodology for encoding graphs with information-theoretically minimum numbers of bits. Specifically, a graph with property π is called a \em π-graph. If π satisfies certain properties, then an n-node m-edge π-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 β(n)+o(β(n)) bits for any continuous superadditive function β(n) so that there are at most 2β(n)+o(β(n)) distinct n-node π-graphs. The methodology is applicable to general classes of graphs; this paper focuses on planar graphs. Examples of such π 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,…, ℓ1\ for ℓ1≤ n; (5) the edges of G are labeled with labels from \1,…, ℓ2\ for ℓ2≤ m; and (6) each node (respectively, edge) of G has at most ℓ3=O(1) self-loops (respectively, ℓ4=O(1) multiple edges). Moreover, ℓ3 and ℓ4 are not required to be O(1) for the cases of π 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.