Revisiting the Prefer-same and Prefer-opposite de Bruijn sequence constructions

Abstract

We present a simple greedy algorithm to construct the prefer-same de Bruijn sequence and prove that it is equivalent to the more complex algorithm first stated by Eldert et al. without proof (Eldert et al., 1958 [3]), and later by Fredricksen (Fredricksen, 1982 [5]). Then we prove that the resulting sequence has the lexicographically largest run-length representation among all de Bruijn sequences. Furthermore, we prove that the sequence resulting from a prefer-opposite greedy construction has the lexicographically smallest run-length representation among all de Bruijn sequences.