Labeling Schemes for Small Distances in Trees

Abstract

We consider labeling schemes for trees, supporting various relationships between nodes at small distance. For instance, we show that given a tree T and an integer k we can assign labels to each node of T such that given the label of two nodes we can decide, from these two labels alone, if the distance between v and w is at most k and, if so, compute it. For trees with n nodes and $k\geq 2$, we give a lower bound on the maximum label length of $\log n + \Omega(\log \log n)$ bits, and for constant k, we give an upper bound of log n + O(log log n). Bounds for ancestor, sibling, connectivity, and bi- and triconnectivity labeling schemes are also presented.