Multi de Bruijn Sequences

TL;DR

This paper introduces and analyzes multi de Bruijn sequences, which contain every k-mer exactly m times, generalizing classical de Bruijn sequences and deriving formulas for their enumeration.

Contribution

It generalizes de Bruijn sequences to the multi-occurrence case, providing formulas and enumeration methods for these sequences and related multisets.

Findings

Formulas for counting multi de Bruijn sequences.

Extension of the Burrows-Wheeler Transform for enumeration.

Generalization of classical de Bruijn sequence results.

Abstract

We generalize the notion of a de Bruijn sequence to a "multi de Bruijn sequence": a cyclic or linear sequence that contains every k-mer over an alphabet of size q exactly m times. For example, over the binary alphabet {0,1}, the cyclic sequence (00010111) and the linear sequence 000101110 each contain two instances of each 2-mer 00,01,10,11. We derive formulas for the number of such sequences. The formulas and derivation generalize classical de Bruijn sequences (the case m=1). We also determine the number of multisets of aperiodic cyclic sequences containing every k-mer exactly m times; for example, the pair of cyclic sequences (00011)(011) contains two instances of each 2-mer listed above. This uses an extension of the Burrows-Wheeler Transform due to Mantaci et al, and generalizes a result by Higgins for the case m=1.