Proximity-preserving labeling schemes and their applications

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 label- ing scheme using O(M log n + log² n) bit labels. For the family of all n-vertex unweighted graphs, a labeling scheme is proposed that using O(log² n κ n^1/κ) bit labels can provide apprimate estimates to the distance, which are accurate up to a factor of √8κ. In particular, using O(log³ n) it 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 communica- tion networks. We illustrate this by proposing a potential application of our labeling schemes to efficient distributed connection setup in circuit- switched networks.