Algorithms for ranking and unranking the combinatorial set of RNA secondary structures

TL;DR

This paper introduces efficient algorithms for ranking and unranking RNA secondary structures, represented by Motzkin words, with improved polynomial time complexity and no preprocessing needed, advancing computational methods in RNA structure analysis.

Contribution

The paper develops novel algorithms for ranking and unranking RNA secondary structures using AND/OR trees, offering better efficiency and no preprocessing compared to prior methods.

Findings

Algorithms operate in polynomial time: O(m^2 (n - m)) and O(m (n - m)^2).

New algorithms outperform existing methods by eliminating preprocessing steps.

Provides a bijection between RNA structures and AND/OR tree variants.

Abstract

In this paper, we study the combinatorial set of RNA secondary structures of length $n$ with $m$ base-pairs. For a compact representation, we encode an RNA secondary structure by the corresponding Motzkin word. For this combinatorial set, we construct an AND/OR tree structure, find a bijection between the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree. The developed ranking and unranking algorithms have polynomial time complexity $O(m^2 (n - m))$ for $m < n - 2 m$ and $O(m (n - m)^2)$ for $m > n - 2 m$. In contrast to the existing algorithms, the new algorithms do not require preprocessing steps and have better time complexity.