A simple and optimal ancestry labeling scheme for trees

TL;DR

This paper introduces a simple and asymptotically optimal ancestry labeling scheme for trees, improving label size over previous solutions and generalizing interval-based methods with minimal modifications.

Contribution

It presents a new, simple, and asymptotically optimal ancestry labeling scheme that generalizes previous interval-based approaches with minimal changes.

Findings

Achieves label size of lg n + 2 lg lg n + 3

Provides a simple framework for optimal ancestry labeling

Builds on and simplifies previous complex solutions

Abstract

We present a $\lg n + 2 \lg \lg n+3$ ancestry labeling scheme for trees. The problem was first presented by Kannan et al. [STOC 88'] along with a simple $2 \lg n$ solution. Motivated by applications to XML files, the label size was improved incrementally over the course of more than 20 years by a series of papers. The last, due to Fraigniaud and Korman [STOC 10'], presented an asymptotically optimal $\lg n + 4 \lg \lg n+O(1)$ labeling scheme using non-trivial tree-decomposition techniques. By providing a framework generalizing interval based labeling schemes, we obtain a simple, yet asymptotically optimal solution to the problem. Furthermore, our labeling scheme is attained by a small modification of the original $2 \lg n$ solution.