De Bruijn Cycles: Structure and Algorithms

• De Bruijn cycles are cyclic sequences in which every n-length word over a k-letter alphabet appears exactly once, serving as a fundamental combinatorial construct.
• A novel two-phase algorithm employs path decomposition (O(2^(n+1))) and conjugate pair joining to construct de Bruijn cycles from singular LFSRs.
• Detailed state diagram analysis reveals a rigid structure with primary cycles and perfect binary tree branches, enabling efficient sequence synthesis for diverse applications.

A de Bruijn cycle is a cyclic sequence of length $k^n$ on a $k$-letter alphabet in which each possible $n$-length word occurs exactly once as a consecutive substring (with wraparound). De Bruijn cycles are fundamental in combinatorics, with applications in coding theory, cryptography, bioinformatics, automata, and discrete mathematics. Their paper spans explicit constructions, algorithmic generation, algebraic characterizations, and extensions to universal cycles and generalized combinatorial classes. This article surveys rigorous advances in de Bruijn cycle theory with emphasis on state structure, cycle construction, algebraic techniques, partition problems, cycle joining, and algorithmic synthesis.

1. Structural Analysis of State Diagrams for Singular LFSRs

A focused class of singular linear feedback shift registers (LFSRs), with feedback function $F(x_1, x_2, \ldots, x_n) = x_{n-1} + x_n$ for $n \geq 3$, exhibits a highly restrictive state diagram $G_F$, foundational to the construction of new de Bruijn cycles (Wang et al., 2018). The structure is characterized as follows:

• $G_F$ has precisely two connected components: $G_1$, a loop at the all-zero state $[\mathbf{0}^n]$, and $G_2$, a 3-cycle $[0,1,1]$.
• All remaining states outside these cycles form branches with exact structure: $G_1 \setminus [0]$ is a perfect binary directed tree of depth $n-3$, and $G_2 \setminus [0,1,1]$ consists of three perfect binary directed trees of depth $n-3$.
• The adjacency graph for this LFSR class contains only two isolated vertices corresponding to $G_1$ and $G_2$.

This structure underpins the possibility of construction: every state ultimately is absorbed by one of the two primary cycles, and the remaining branch structure is predictable, enabling systematic traversal and manipulation.

2. Algorithmic Construction of de Bruijn Cycles from Singular LFSRs

Classic cycle joining approaches, which rely on state diagrams comprising unions of disjoint cycles, do not directly apply to the above singular LFSRs. The cited work (Wang et al., 2018) introduces a novel two-phase algorithm:

• Phase 1 – Path decomposition and branch removal: All non-cycle states are decomposed into directed paths from leaves (terminal tree vertices) down towards their ultimate cycle. Each path is recorded and, using companion relationships, wrapped into a branchless cycle by joining the tail of one path to the head of the next. This recursive decomposition ensures all branches are traversed and eliminated, yielding a collection of cycles without branching.
• Phase 2 – Cycle joining via conjugate pairs: Between pairs of cycles found in phase 1, suitable "conjugate pairs"—state pairs differing only in the first bit—are sought, typically occurring at points corresponding to transitions from branches to cycles. Swapping the successors at each such pair joins two cycles into one; iterating over all appropriate conjugate pairs ultimately produces a single cycle of length $2^n$, a de Bruijn cycle.
• The time complexity for path decomposition is $O(2^{n+1})$, and the conjugate pair identification is $O(2^{n-1})$.

Explicit combinatorial tables are provided for $n=6$ as an example, detailing paths, branch decompositions, and the ordered joining of cycles.

3. Mathematical Formulation and Key Theorems

The main theorems establish:

• For $F(x_1, ..., x_n) = x_{n-1} + x_n$, the state diagram contains just a loop and a 3-cycle, with every non-cycle state part of a finite-depth perfect binary tree (Theorems 1 and 2 (Wang et al., 2018)).
• Algorithm 1 gives a path decomposition strategy for arbitrary $n \geq 3$, while Algorithm 2 delineates a systematic search for all possible conjugate pairs, framed in algebraic combinatorics using binary vector representations and successor/predecessor relationships.
• Once all paths are exhausted and cycles joined at conjugate states, the result is a de Bruijn sequence covering all $2^n$ binary $n$-words precisely once.

For $n=6$, explicit conjugate pairs such as $((1,0,1,0,0,0), (0,0,1,0,0,0))$ are found, and cycle joining at these locations is shown to yield $5$ distinct order-$6$ de Bruijn cycles, as detailed in Table 3 (Wang et al., 2018).

4. Impact of State Diagram Structure on Algorithm Design

The rigid branching and absorption structure of singular LFSRs with $F(x_1, ..., x_n) = x_{n-1} + x_n$ enable:

• Systematic partitioning of state space into rooted tree branches leading to only two absorbing cycles.
• Algorithmic decomposition where all non-cycle states fall into a unique path, enabling deterministic enumeration and traversable removal.
• Predictable reconnection via companion and conjugate pairs, allowing deterministic recombination into branchless cycles, and subsequent strategic cycle joining.
• Application of this methodology demonstrates, for the first time, direct de Bruijn cycle construction from a class of singular LFSR state diagrams where the classic cycle joining method would otherwise be inapplicable.

The approach further illustrates the power of exploiting precise state diagram analysis to extend classic construction techniques to otherwise singular or degenerate settings.

5. Relation to Broader de Bruijn Sequence Literature

The approach is orthogonal to methods for nonsingular LFSRs or general cycle-joining constructions based on arbitrary characteristic polynomials (Chang et al., 2016, Chang et al., 2016). In contrast to these, which require enumeration of cycle structures or heavy algebraic computation (often involving cyclotomic numbers or Zech's logarithms), the path decomposition and conjugate-pair joining method leverages the sparsity and geometry of the singular case. The explicit mapping between leaves, paths, and cycles also provides a constructive path for analyzing how modified state diagrams can still be harnessed for sequence construction.

This contributes a foundational new class of de Bruijn sequences with potentially distinct statistical and combinatorial properties, and provides a blueprint for extending de Bruijn construction into other nonstandard state diagrams arising from singular feedback functions or nonclassical LFSRs.

6. Theoretical and Practical Significance

• This method demonstrates, rigorously for the first time, that singular state diagram LFSRs can directly yield de Bruijn cycles, closing a gap in the construction landscape (Wang et al., 2018).
• The explicit state diagram decomposition suggests further analysis of robustness, statistical balance, and implementation complexity for de Bruijn cycles beyond the classic product-of-primes or necklace-based families.
• The rigidity and predictability of the structural decomposition should support practical implementation of de Bruijn sequence generators with known transition graphs, possible hardware optimizations, and provable security properties in cryptographic contexts.

A plausible implication is that similar algorithmic decompositions could be abstracted for other classes of degenerate or highly structured automata where standard cycle joining or necklace concatenation fail, broadening the class of constructible de Bruijn-like cycles.

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Main Step	Description	Complexity
Path decomposition (Algorithm 1)	Partition into rooted tree paths	$O(2^{n+1})$
Branch removal	Wrap and stitch paths into branchless cycles	TaS above
Conjugate pair search (Alg. 2)	Identify joining points across cycles	$O(2^{n-1})$
Cycle joining	Swap successors at conjugate pairs	Linear in pairs

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This algorithmic framework and state-diagram analysis represent a significant extension of the constructional theory of de Bruijn cycles, establishing new connections between combinatorial state geometry and sequence synthesis.