A simple shift rule for k-ary de Bruijn sequences

A k-ary de Bruijn sequence of order n is a cyclic sequence of length k in which each k-ary string of length n appears exactly once as a substring. A shift rule for a de Bruijn sequence of order n is a function that maps each length n substring to the next length n substring in the sequence. We present the first known shift rule for k-ary de Bruijn sequences that runs in O(1)-amortized time per symbol using O(n) space. Our rule generalizes the authors’ recent shift rule for the binary case (A surprisingly simple de Bruijn sequence construction, Discrete Mathematics 339, pages 127-131). 1 A new de Bruijn sequence construction A k-ary de Bruijn sequence is a cyclic sequence of length kn in which each k-ary string of length n appears exactly once as a substring. As an example, the cyclic sequence 111222333232212312113213313 is a 3-ary de Bruijn sequence for n = 3; the 27 unique length 3 substrings when considered cyclicly are: 111, 112, 122, 222, 223, 233, 333, 332, 323, 232, 322, 221, 212, 123, 231, 312, 121, 211, 113, 132, 321, 213, 133, 331, 313, 131, 311. As illustrated in this example, a k-ary de Bruijn sequence of order n induces a very specific type of cyclic order of k-ary strings of length n: the length n− 1 suffix of a given string is the same as the length n− 1 prefix of the next string in the ordering. The number of unique k-ary de Bruijn sequences for a given n and k is equal to k!k n−1 /kn [3]; however, only a few efficient constructions are known. In particular, there are . a Lyndon word concatenation algorithm by Fredricksen and Maiorana [11] that generates the lexicographically smallest de Bruijn sequence (also known as the Ford sequence), . a block concatenation algorithm by Ralston [16], . a lexicographic composition concatenation algorithm by Fredricksen and Kessler [10], and . two different pure cycle concatenation algorithms by Fredricksen [8], and Etzion and Lempel [5]. ∗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 and Information Systems, Northwest Missouri State University, USA. email: [email protected]