A surprisingly simple de Bruijn sequence construction

Pick any length n binary string b1b2 · · · bn and remove the first bit b1. If b2b3 · · · bn1 is a necklace then append the complement of b1 to the end of the remaining string; otherwise append b1. By repeating this process, eventually all 2 binary strings will be visited cyclically. This shift rule leads to a new de Bruijn sequence construction that can be generated in O(1)-amortized time per bit. 1 A new de Bruijn sequence construction A de Bruijn sequence of order n is a cyclic sequence of length 2n where each substring of length n is a unique binary string. As an example, the cyclic sequence 0000100110101111 of length 16 is a de Bruijn sequence for n = 4. The 16 unique substrings of length 4 when considered cyclicly are: 0000, 0001, 0010, 0100, 1001, 0011, 0110, 1101, 1010, 0101, 1011, 0111, 1111, 1110, 1100, 1000. As illustrated in this example, a de Bruijn sequence of order n induces a very specific type of cyclic order of the length n binary strings: the length n − 1 suffix of a given binary string is the same as the length n− 1 prefix of the next string in the ordering. The number of unique de Bruijn sequences for a given n is 22 n−1−n [3]; however, only a few efficient constructions are known. In particular, there are . a shift generation approach based on primitive polynomials by Golomb [9], . three different algorithms to generate the lexicographically smallest de Bruijn sequence (also known as the Ford sequence): a Lyndon word concatenation algorithm by Fredricksen and Maiorana [8], a successor rule approach by Fredricksen [5], and a block concatenation algorithm by Ralston [12], . a lexicographic composition concatenation algorithm by Fredricksen and Kessler [7], . three different pure cycle concatenation algorithms by Fredricksen [6], Etizon and Lempel [4], and Huang [10] respectively, and . cool-lex based constructions by Sawada, Stevens and Williams [13] and Sawada, Williams and Wong [14]. Each algorithm requires only O(n) space and generates their de Bruijn sequences in O(n) time per bit, except the pure cycle concatenation algorithm by Etizon and Lempel which requires O(n2) space. The ∗School of Computer Science, University of Guelph, Canada. Research supported by NSERC. email: [email protected] †Division of Science, Mathematics, and Computing, Bard College at Simon‘s Rock, USA. email: [email protected] ‡School of Computer Science, University of Guelph, Canada. email: [email protected]