A Simple and Optimal Ancestry Labeling Scheme for Trees

We consider ancestry labeling schemes: Given a rooted tree T , assign a binary string (label) to each node, such that given the labels of any two nodes, one can determine whether the first is an ancestor of the second in T . Recently, Fraigniaud and Korman [STOC’10] showed that such labels can be assigned using log n+O(log log n) bits per label, solving a long standing open question and matching a lower bound of log n + Ω(log log n) due to Alstrup et al. [SODA’03]. In this paper we present an alternative ancestry labeling scheme using log n+ 2 log log n+O(1) bits. Similar to the previously known schemes, our scheme relies on intervals and can encode any tree in O(n) time. Rather than using complicated decompositions, our scheme uses approximate interval sizes, which gives a clean algorithm that is simple enough to be taught and implemented.