Informative labeling schemes for graphs

This paper introduces and studies the notion of informative labeling schemes for arbitrary graphs. Let f(W) be a function on subsets of vertices W. An f labeling scheme labels the vertices of a weighted graph G in such a way that f(W) can be inferred efficiently for any vertex subset W of G by merely inspecting the labels of the vertices of W, without having to use any additional information sources. The paper develops f labeling schemes for some functions f over the class of n-vertex trees, including Sep Level, the separation level of any two vertices in the tree, LCA, the least common ancestor of any two vertices, and Center, the center of any three given vertices in the tree. These schemes use O(log² n)-bit labels, which is asymptotically optimal. We then turn to weighted graphs and consider the function Steiner(W), denoting the weight of the Steiner tree spanning the vertices of W in the graph. For n-vertex weighted trees with M-bit edge weights, it is shown that there exists a Steiner labeling scheme using O((M+log n) log n) bit labels, which is asymptotically optimal. In the full paper it is shown that for the class of arbitrary n-vertex graphs with M-bit edge weights, there exists an approximate-Steiner labeling scheme, providing an estimate (up to a logarithmic factor) for Steiner(W) using O((M +log n) log² n) bit labels.