Dualities in Tree Representations

TL;DR

This paper explores the relationships between different tree representations, specifically BP and DFUDS, providing characterizations that unify these structures and potentially simplify queries related to range minimum and interval problems.

Contribution

It offers a rigorous characterization of trees that unify BP and DFUDS representations, advancing understanding of their dualities and potential for query optimization.

Findings

Unified framework for BP and DFUDS representations

Characterization of trees where BP equals reversed DFUDS

Improved methods for Range Minimum and Interval Query problems

Abstract

A characterization of the tree $T^*$ such that $\mathrm{BP}(T^*)=\overleftrightarrow{\mathrm{DFUDS}(T)}$, the reversal of $\mathrm{DFUDS}(T)$ is given. An immediate consequence is a rigorous characterization of the tree $\hat{T}$ such that $\mathrm{BP}(\hat{T})=\mathrm{DFUDS}(T)$. In summary, $\mathrm{BP}$ and $\mathrm{DFUDS}$ are unified within an encompassing framework, which might have the potential to imply future simplifications with regard to queries in $\mathrm{BP}$ and/or $\mathrm{DFUDS}$. Immediate benefits displayed here are to identify so far unnoted commonalities in most recent work on the Range Minimum Query problem, and to provide improvements for the Minimum Length Interval Query problem.