Short and simple labels for small distances and other functions

Abstract

We present a labeling scheme for rooted trees which allows to compute, from the label of v alone, unique identifiers for the ancestors of v that are at distance at most d from v. For any constant d our labeling scheme produce labels of length log n + O(√log n), and for d ∈ O(√log n) the labels are still of length O(log n). In particular, given the labels of two nodes u and v we can determine from the labels alone whether u is the parent of v or vice versa, whether u and v are siblings, and whether u and v are at distance at most d from each other. The need for such labeling scheme arises in several application areas, including in particular communication networks and search engines for large collections of Web XML files. In the latter application XML files are viewed as trees, and typical queries ask for XML files containing a particular set of nodes with specific ancestor, parent, or sibling relationships among them.