Nearest Common Ancestors: Universal Trees and Improved Labeling Schemes

TL;DR

This paper introduces universal trees for the nearest common ancestor problem, providing explicit constructions that lead to improved labeling schemes with smaller label sizes for rooted trees.

Contribution

It presents the first explicit constructions of NCA-universal trees with sizes that improve existing labeling schemes for rooted trees.

Findings

Constructed NCA-universal trees of size n^{2.318} and n^{1.894}.

Developed NCA-labeling schemes with smaller label sizes.

Improved upon previous labeling schemes by Alstrup, Halvorsen, and Larsen.

Abstract

We investigate the nearest common ancestor (NCA) function in rooted trees. As the main conceptual contribution, the paper introduces universal trees for the NCA function: For a given family of rooted trees, an NCA-universal tree $S$ is a rooted tree such that any tree $T$ of the family can be embedded into $S$ such that the embedding of the NCA in $T$ of two nodes of $T$ is equal to the NCA in $S$ of the embeddings of the two nodes. As the main technical result we give explicit constructions of NCA-universal trees of size $n^{2.318}$ for the family of rooted $n$-vertex trees and of size $n^{1.894}$ for the family of rooted binary $n$-vertex trees. A direct consequence is the explicit construction of NCA-labeling schemes with labels of size $2.318\log_2 n$ and $1.894\log_2 n$ for the two families of rooted trees. This improves on the best known such labeling schemes established by Alstrup, Halvorsen and Larsen [SODA 2014].