Concatenation trees: A framework for efficient universal cycle and de Bruijn sequence constructions

TL;DR

This paper introduces concatenation trees as a new framework that unifies cycle-joining and concatenation-based methods, enabling efficient universal cycle constructions with constant amortized time per symbol.

Contribution

The paper presents concatenation trees, a novel structure that bridges classic cycle-joining and concatenation methods, allowing for efficient universal cycle generation.

Findings

Universal cycles for permutations and weak orders constructed in O(1) amortized time.

Concatenation trees clarify the relationship between Lyndon word concatenation and cycle-joining.

Potential extension to other universal cycles like de Bruijn sequences.

Abstract

Classic cycle-joining techniques have found widespread application in creating universal cycles for a diverse range of combinatorial objects, such as shorthand permutations, weak orders, orientable sequences, and various subsets of $k$-ary strings, including de Bruijn sequences. In the most favorable scenarios, these algorithms operate with a space complexity of $O(n)$ and require $O(n)$ time to generate each symbol in the sequences. In contrast, concatenation-based methods have been developed for a limited selection of universal cycles. In each of these instances, the universal cycles can be generated far more efficiently, with an amortized time complexity of $O(1)$ per symbol, while still using $O(n)$ space. This paper introduces $\mathit{concatenation~trees}$, which serve as the fundamental structures needed to bridge the gap between cycle-joining constructions based on the pure cycle register and corresponding concatenation-based approaches. They immediately demystify the relationship between the classic Lyndon word concatenation construction of de Bruijn sequences and a corresponding cycle-joining based construction. To underscore their significance, concatenation trees are applied to construct universal cycles for shorthand permutations and weak orders in $O(1)$-amortized time per symbol. Moreover, we provide insights as to how similar results can be obtained for other universal cycles including cut-down de Bruijn sequences and orientable sequences.