## Gavoille, Cyril and Hanusse, Nicolas - On Compact Encoding of Pagenumber $k$ Graphs

dmtcs:436 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, Vol. 10 no. 3
On Compact Encoding of Pagenumber $k$ Graphs

Authors: Gavoille, Cyril and Hanusse, Nicolas

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;.

Source : oai:HAL:hal-00402218v1
Volume: Vol. 10 no. 3
Published on: January 1, 2008
Submitted on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]