De Bruijn Sequence and Universal Cycle Constructions

Given an alphabet of size $k$, a de Bruijn (DB) sequence of order $n$ is a circular string of length $k^n$ where every $k$-ary string of length $n$ appears exactly once as a substring.

Example $n=4$ and $k=2$

0000101101001111 is a DB sequence where the 16 unique substrings of length 4 visited in order are:

  0000 0001 0010 0101 1011 0110 1101 1010
  0100 1001 0011 0111 1111 1110 1100 1000

This inspiration for this project is to update Fredricksen's survey [SIAM Review, Vol 24 (1982)] to include new results and constructions from the past 40 years. The following different construction methods are presented:

  • Euler cycles based on the de Bruijn graph
  • Linear feedback shift registers (LFSRs)
  • Recursive methods
  • Greedy methods
  • Shift rules based on cycle joining
  • Concatenation approaches
Additionally, there is an algorithm to decode the lexicographically smallest DB sequence (the Granddaddy) and a section on properties including an analysis of DB sequence discrepancies.

More generally, a universal cycle of order $n$ for a set $\mathbf{S}$ of length $n$ strings is a circular string of length $|\mathbf{S}|$ where every string in $\mathbf{S}$ appears exactly once as a substring. Universal cycle constructions are provided for:

  • Permutations (shorthand)
  • Multiset permutations (strings with fixed content)
  • Weak orders - both height and rank representations
  • Orientable sequences

Use the following to cite this project:

  • De Bruijn Sequence and Universal Cycle Constructions (2021). http://debruijnsequence.org.
About this project

This project was initiated in 2019 as a spinoff off from The Combinatorial Object Server. It is dedicated to presenting the many different ways de Bruijn sequences and their relatives can be constructed. Implementations of each algorithm are available for download, and an overview of each algorithm is presented.
Contact

For comments, contributions, or corrections please contact:

Joe Sawada
jsawada@uoguelph.ca

Last updated Nov 30, 2021