| Summary: | 碩士 === 國立中正大學 === 資訊工程研究所 === 88 === Let G be an n-node m-edge planar graph that has no self-loops. Under the O(logn)-bit word model of computation, we give the best known encoding S of G such that (a) S and G can be obtained from each other in O(m+n) time, (b) the degree of a node and the adjacency of two nodes in G can be determined from S in O(1) time, and (c) the neighbors of a degree-d node in G can be computed from S in O(d) time. When G has (respectively, has no) multiple edges, our S has 2m+3n+o(m+n) (respectively,
2m+2n+o(n)) bits, reducing the bit counts of the encodings of
Chuang, Garg, He, Kao, and Lu by at least 2n. The bit counts of our encodings are close to Tutte''s information-theoretical lower
bound of roughly 3.58m bits for encoding a connected plane graph
without any query support.
Our results are based on two new tools. The first tool, which appears to be fundamental in algorithmic graph theory, is a linear-time algorithm for computing a planar embedding H of G such that each connected component of H admits an orderly spanning tree. The second tool is an improved auxiliary string X for a folklore encoding F of a rooted tree T. The concatenation of F and X is the first known information-theoretically optimal encoding of T, from which the degree, depth, parent, leftmost child, rightmost child, and number of descendents of any node can be determined in O(1) time.
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