Better Labeling Schemes for Nearest Common Ancestors through Minor-Universal Trees

Abstract

A labeling scheme for nearest common ancestors assigns a distinct binary string, called the label, to every node of a tree, so that given the labels of two nodes (and no further information about the topology of the tree) we can compute the label of their nearest common ancestor. The goal is to make the labels as short as possible. Alstrup, Gavoille, Kaplan, and Rauhe [Theor. Comput. Syst. 37(3):441-456 2004] showed that $O(\log n)$-bit labels are enough. More recently, Alstrup, Halvorsen, and Larsen [SODA 2014] refined this to only $2.772\log n$, and provided a lower bound of $1.008\log n$. We connect designing a labeling scheme for nearest common ancestors to the existence of a tree, called a minor-universal tree, that contains every tree on $n$ nodes as a topological minor. Even though it is not clear if a labeling scheme must be based on such a notion, we argue that the existing schemes can be reformulated as such, and it allows us to obtain clean and good bounds on the length of the labels. As the main upper bound, we show that $2.318\log n$-bit labels are enough. Surprisingly, the notion of a minor-universal tree for binary trees on $n$ nodes has been already used in a different context by Hrubes et al. [CCC 2010], and Young, Chu, and Wong [J. ACM 46(3):416-435, 1999] introduced a closely related notion of a universal tree. On the lower bound side, we show that the size of any minor-universal tree for trees on $n$ nodes is $Ω(n^{2.174})$. This highlights a natural limitation for all approaches based on such trees. Our lower bound technique also implies that the size of any universal tree in the sense of Young et al. is $Ω(n^{2.185})$, thus dramatically improves their lower bound of $Ω(n\log n)$. We complement the existential results with a generic transformation that decreases the query time to constant in any scheme based on a minor-universal tree.