Average Case Analysis for Tree Labelling Schemes

Abstract

We study how to label the vertices of a tree in such a way that we can decide the distance of two vertices in the tree given only their labels. For trees, Gavoille et al. [7] proved that for any such distance labelling scheme, the maximum label length is at least ${1 \over 8} {\rm log}^{2} n - O({\rm log} n)$ bits. They also gave a separator-based labelling scheme that has the optimal label length ${\it \Theta}({\rm log} {n} \cdot {\rm log}(H_{n}(T)))$, where Hn(T) is the height of the tree. In this paper, we present two new distance labelling schemes that not only achieve the optimal label length ${\it \Theta}({\rm log} n \cdot {\rm log} (H_{n}(T)))$, but also have a much smaller expected label length under certain tree distributions. With these new schemes, we also can efficiently find the least common ancestor of any two vertices based on their labels only.