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 + log²n) 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 approximate estimates to the distance, which are accurate up to a factor of <UEQN NOTAT="TEX" LOC="INLINE"> $\sqrt{8\kappa }.$</UEQN> In particular, using O(log³n) bit labels, the scheme can provide estimates accurate up to a factor of <UEQN NOTAT="TEX" LOC="INLINE">$\sqrt{2 \log n}$.</UEQN> (For weighted graphs, one of the log n factors in the label size is replaced by a factor logarithmic in the network's diameter.) © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 167–176, 2000