On Compact Encoding of Pagenumber $k$ Graphs

Cyril Gavoille (1, 2) , Nicolas Hanusse (1, 2)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique
2 CEPAGE - Algorithmics for computationally intensive applications over wide scale distributed platforms

In this paper we show an information-theoretic lower bound of kn - o(kn) on the minimum number of bits to represent an unlabeled simple connected n-node graph of pagenumber k. This has to be compared with the efficient encoding scheme of Munro and Raman of 2kn + 2m + o(kn+m) bits (m the number of edges), that is 4kn + 2n + o(kn) bits in the worst-case. For m-edge graphs of pagenumber k (with multi-edges and loops), we propose a 2mlog2k + O(m) bits encoding improving the best previous upper bound of Munro and Raman whenever m ≤ 1 / 2kn/log2 k. Actually our scheme applies to k-page embedding containing multi-edge and loops. Moreover, with an auxiliary table of o(m log k) bits, our coding supports (1) the computation of the degree of a node in constant time, (2) adjacency queries with O(logk) queries of type rank, select and match, that is in O(logk *minlogk / loglogm, loglogk) time and (3) the access to δ neighbors in O(δ) runs of select, rank or match;.

Domaines: Mathématique discrète [cs.DM]

Cyril Gavoille, Nicolas Hanusse. On Compact Encoding of Pagenumber $k$ Graphs. Discrete Mathematics and Theoretical Computer Science, 2008, Vol. 10 no. 3 (3), pp.23-34. ⟨10.46298/dmtcs.436⟩. ⟨hal-00402218⟩