Proximity-preserving labeling schemes

This article 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 + log² n) bit labels. For the family of all n-vertex unweighted graphs, a labeling scheme is proposed that using O(log² n · K · n^1/k) bit labels can provide approximate estimates to the distance, which are accurate up to a factor of √8k. In particular, using O(log³ 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.)