# Proximity-Preserving Labeling Schemes and Their Applications

## Abstract

This paper considers informative labeling schemes for graphs. Specifically, the question introduced is whether it is possible to label the vertices of a graph with short labels in such a way that the *distance* between any two vertices can be inferred from inspecting their labels. A labeling scheme enjoying this property is termed a *proximity-preserving* labeling scheme. It is shown that for the class of *n*-vertex weighted trees with M-bit edge weights, there exists such a proximity-preserving labeling scheme using *O*(*M* log *n* + log^{2} *n*) bit labels. For the family of all *n*-vertex unweighted graphs, a labeling scheme is proposed that using *O*(log^{2} *n* · *k* · *n* ^{1/k }) bit labels can provide approximate estimates to the distance, which are accurate up to a factor of √κ. In particular, using *O*(log^{3} *n*) bit labels the scheme can provide estimates accurate up to a factor of √2 log *n*. (For weighted graphs, one of the log n factors in the label size is replaced by a factor logarithmic in the network’s diameter.) In addition to their theoretical interest, proximity-preserving labeling systems seem to have some relevance in the context of communication networks. We illustrate this by proposing a potential application of our labeling schemes to efficient distributed connection setup in circuit- switched networks.

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