Peleg D. (1999) Proximity-Preserving Labeling Schemes and Their Applications. In: Widmayer P., Neyer G., Eidenbenz S. (eds) Graph-Theoretic Concepts in Computer Science. WG 1999. Lecture Notes in Computer Science, vol 1665. Springer, Berlin, Heidelberg
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 + log2n) bit labels. For the family of all n-vertex unweighted graphs, a labeling scheme is proposed that using O(log2n · k · n1/k) bit labels can provide approximate estimates to the distance, which are accurate up to a factor of √κ. In particular, using O(log3n) 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.