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 @math -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 @math 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 @math bits which is still far from the known @math lower bound. Moreover, the hidden constant factor in the additive @math 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 @math , for the family of trees with at most @math nodes and depth at most @math . 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.
@cite_9: We show that there exists a graph G with n spl middot 2 sup O(log* n) nodes, where any forest with n nodes is a node-induced subgraph of G. Furthermore, the result implies the existence of a graph with n sup k 2 sup O(log* n) nodes that contains all n-node graphs of fixed arboricity k as node-induced subgraphs. We provide a lower bound of spl Omega (n sup k ) for the size of such a graph. The upper bound is obtained through a simple labeling scheme for parent queries in rooted trees.
@cite_4: 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 ofWeb 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.
Adjacency labeling schemes on trees were further investigated in an attempt to reduce the constant factor in the label size. In @cite_4 an adjacency labeling scheme using label size of @math is presented; and in @cite_9 the label size was further reduced to @math . This current state of the art bound implies the existence of a universal graph for the family of @math -node trees with @math nodes.