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

TL;DR

The paper introduces concatenation trees, built on bifurcated ordered trees (BOTs) and a Right-Current-Left (RCL) traversal, to unify cycle-joining constructions with fast concatenation-based universal-cycle generation. By converting PCR-based cycle-joining trees into concatenation trees and applying RCL traversals, the authors show that universal cycles for various combinatorial objects—including de Bruijn sequences, shorthand permutations, weak orders, and orientable sequences—can be generated in $O(1)$-amortized time per symbol using polynomial space. The framework provides explicit algorithmic details (via a local Child function) and proves key structural claims ensuring correctness, while also yielding known sequences (Granddaddy, Grandmama, Granny, Grandpa) as natural consequences. This approach unifies diverse prior constructions under a single theory and suggests broad potential for extending fast concatenation techniques to additional feedback-register families like CCR, PSR/CSR, and PRR, with significant implications for efficient combinatorial generation and cryptographic applications.

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.

Figures (12)

• Figure 1: Initial steps to building a universal cycle for $\mathbf{W}_{3}$.
• Figure 2: Two tree structures for creating a universal cycle for $\mathbf{W}_{4}$.
• Figure 3: Cycle-joining trees for $n=6$ and $k=2$ derived from the four simple parent rules. The node $001101$ is joined to a different parent cycle in each tree. In particular, the edge $0011\textcolor{blue}{0}1 \text{--} 001111$ in $\mathbb{T}_1$ is obtained by flipping its last $0$.
• Figure 4: All eight bifurcated ordered trees (BOTs) with $n{=}3$ nodes. Each left-child descends from a blue $\bullet$, while each right-child descends from a red $\bullet$.
• Figure 5: A BOT with its $n{=}12$ nodes labeled as they appear in RCL order.

Theorems & Definitions (24)

• Theorem 1
• Remark 2
• Theorem 3
• Remark 5
• Theorem 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10
• Lemma 11