Compact Ancestry Labeling Schemes for Trees of Small Depth

An ancestry labeling scheme labels the nodes of any tree in such a way that ancestry queries between any two nodes in a tree can be answered just by looking at their corresponding labels. The common measure to evaluate the quality of an ancestry labeling scheme is by its label size, that is the maximal number of bits stored in a label, taken over all n-node trees. The design of ancestry labeling schemes finds applications in XML search engines. In the context of these applications, even small improvements in the label size are important. In fact, the literature about this topic is interested in the exact label size rather than just its order of magnitude. As a result, following the proposal of an original scheme of size 2 logn bits, a considerable amount of work was devoted to improve the bound on the label size. The current state of the art upper bound is logn + O( √ logn) bits which is still far from the known logn + Ω(log logn) lower bound. Moreover, the hidden constant factor in the additive O( √ log n) term is large, which makes this term dominate the label size for typical current XML trees. In attempt to provide good performances for real XML data, we rely on the observation that the depth of a typical XML tree is bounded from above by a small constant. Having this in mind, we present an ancestry labeling scheme of size logn + 2 log d + O(1), for the family of trees with at most n nodes and depth at most d. In addition to our main result, we prove a result that may be of independent interest concerning the existence of a linear universal graph for the family of forests with trees of bounded depth. ∗This research is supported in part by the ANR project ALADDIN, by the INRIA project GANG, and by COST Action 295 DYNAMO.