An Optimal Labeling Scheme for Ancestry Queries

An ancestry labeling scheme assigns labels (bit strings) to the nodes of rooted trees such that ancestry queries between any two nodes in a tree can be answered merely by looking at their corresponding labels. The quality of an ancestry labeling scheme is measured by its label size, that is the maximal number of bits in a label of a tree node. In addition to its theoretical appeal, the design of efficient ancestry labeling schemes is motivated by applications in web search engines. For this purpose, 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 a simple interval-based ancestry scheme with label size $2\log_2 n$ bits (Kannan et al., STOC '88), a considerable amount of work was devoted to improve the bound on the size of a label. The current state of the art upper bound is $\log_2 n + O(\sqrt{\log n})$ bits (Abiteboul et al., SODA '02) which is still far from the known $\log_2 n + \Omega(\log\log n)$ bits lower bound (Alstrup et al., SODA '03). In this paper we close the gap between the known lower and upper bounds, by constructing an ancestry labeling scheme with label size $\log_2 n + O(\log\log n)$ bits. In addition to the optimal label size, our scheme assigns the labels in linear time and can support any ancestry query in constant time.