Labeling schemes for tree representation

This paper deals with compact label-based representations for trees. Consider an n-node undirected connected graph G with a predefined numbering on the ports of each node. The all-ports tree labeling ℒ_all gives each node v of G a label containing the port numbers of all the tree edges incident to v. The upward tree labeling ℒ_up labels each node v by the number of the port leading from v to its parent in the tree. Our measure of interest is the worst case and total length of the labels used by the scheme, denoted M_up(T) and S_up(T) for ℒ_up and M_all(T) and S_all(T) for ℒ_all. The problem studied in this paper is the following: Given a graph G and a predefined port labeling for it, with the ports of each node v numbered by 0, . . . , deg(v) - 1, select a rooted spanning tree for G minimizing (one of) these measures. We show that the problem is polynomial for M_up(T), S_up(T) and S_all(T) but NP-hard for M_all(T) (even for 3-regular planar graphs). We show that for every graph G and port numbering there exists a spanning tree T for which S_up(T) = O(n log log n). We give a tight bound of O(n) in the cases of complete graphs with arbitrary labeling and arbitrary graphs with symmetric port assignments. We conclude by discussing some applications for our tree representation schemes.