Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses

We consider the problem of coding planar graphs by binary strings. Depending on whether O(1)-time queries for adjacency and degree are supported, we present three sets of coding schemes which all take linear time for encoding and decoding. The encoding lengths are significantly shorter than the previously known results in each case. 1 I n t r o d u c t i o n This paper investigates the problem of encoding a graph G with n nodes and m edges into a binary string S. This problem has been extensively studied with three objectives: (1) minimizing the length of S, (2) minimizing the time needed to compute and decode S, and (3) supporting queries efficiently. A number of coding schemes with different trade-offs have been proposed. The adjacency-list encoding of a graph is widely useful but requires 2m[logn] bits. (All logarithms are of base 2.) A folklore scheme uses 2n bits to encode a rooted n-node tree into a string of n pairs of balanced parentheses. Since the total number of such trees is at least ~ . (n-1)!(n-1)!' the minimum number of bits needed to differentiate these trees is the log of this quantity, which is 2no(n). Thus, two bits per edge up to an additive o(1) term is an informationtheoretic tight bound for encoding rooted trees. Works on encodings of certain other graph families can be found in [7, 12, 4, 17, 5, 16]. Let G be a plane graph with n nodes, m edges, f faces, and no self-loop. G need not be connected or simple. We give coding schemes for G which all take O(m + n) time for encoding and decoding. The bit counts of our schemes depend on the level of required query support and the structure of the encoded graphs. For applications that require support of certain queries, Jacobson [6] gave an G(n)-bit encoding for a simple planar graph G that supports traversal in G(log n) time per node visited. Munro and Raman [15] recently gave schemes to encode a planar graph using 2m+Sn+o(m+n) bits while supporting adjacency and degree queries in O(1) time. We reduce this bit count to 2m + 5~n + o(m + n) for any * Research supported in part by NSF Grant CCR-9205982. ** Research supported in part by NSF Grant CCR-9531028.