Compact Labeling Scheme for Ancestor Queries

We consider the following problem. Given a rooted tree $T$, label the nodes of $T$ in the most compact way such that, given the labels of two nodes $u$ and $v$, one can determine in constant time, by looking only at the labels, whether $u$ is ancestor of $v$. The best known labeling scheme is rather straightforward and uses labels of length at most $2\log_2 n$ bits each, where $n$ is the number of nodes in the tree. Our main result in this paper is a labeling scheme with maximum label length $\log_2 n + \Oh(\sqrt{\log n})$. Our motivation for studying this problem is enhancing the performance of web search engines. In the context of this application each indexed document is a tree, and the labels of all trees are maintained in main memory. Therefore even small improvements in the maximum label length are important.