# Informative Labeling Schemes for Graphs

- First Online:

DOI: 10.1007/3-540-44612-5_53

- Cite this paper as:
- Peleg D. (2000) Informative Labeling Schemes for Graphs. In: Nielsen M., Rovan B. (eds) Mathematical Foundations of Computer Science 2000. MFCS 2000. Lecture Notes in Computer Science, vol 1893. Springer, Berlin, Heidelberg

## Abstract

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 SepLevel, 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(log^{2} 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 +* logn) log^{2}*n*) bit labels.

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