New Successor Rules to Efficiently Produce Exponentially Many Binary de Bruijn Sequences

TL;DR

This paper introduces new criteria for designing successor rules that efficiently generate exponentially many binary de Bruijn sequences, extending previous algorithms and demonstrating practical implementation with feedback shift registers.

Contribution

The paper presents novel general criteria for successor rule design, unifies previous algorithms, and provides efficient methods to generate exponentially many binary de Bruijn sequences.

Findings

Successfully generated exponential numbers of de Bruijn sequences

Proposed successor rules operate in O(n) time and memory

Validated methods through computational implementation

Abstract

We put forward new general criteria to design successor rules that generate binary de Bruijn sequences. Prior fast algorithms based on successor rules in the literature are then shown to be special instances. We implemented the criteria to join the cycles generated by a number of simple feedback shift registers (FSRs) of order $n$. These include the pure cycling register (PCR) and the pure summing register (PSR). For the PCR, we define a transitive relation on its cycles, based on their weights. We also extend the choices of conjugate states by using shift operations. For the PSR, we define three distinct transitive relations on its cycles, namely a run order, a necklace order, and a mixed order. Using the new orders, we propose numerous classes of successor rules. Each class efficiently generates a number, exponential in $n$, of binary de Bruijn sequences. Producing the next bit in each such sequence takes $O(n)$ memory and $O(n)$ time. We implemented computational routines to confirm the claims.