Constructing Maximum-Rotating de Bruijn Sequences Using their Necklace Trees

Author

Oscar Hernandez

Date of Award

2017

First Advisor

Aaron Williams

Second Advisor

Kenneth Knox

Abstract

A de Bruijn sequence of order n is a binary string of length 2n which, when viewed cyclically, contains every binary string of length n exactly once as a substring. An example for n = 6 is shown below. It is well-known that de Bruijn sequences of order n are in one-to-one correspondence with labeled Eulerian cycles on de Bruijn graphs of order n. Unfortunately, constructing de Bruijn sequences by finding those Eulerian cycles is impractical, and this thesis includes a discussion on alternate constructions including greedy algorithms, successor rules, and necklace concatenation algorithms. We present a simple framework for constructing de Bruijn sequences satisfying the \maximum- rotation" property. There are four canonical applications of this framework, all of which have easy proofs of correctness. Three of them produce previously studied de Bruijn sequences, now under a unifying construction, and this serves as the first proof of correctness of the fourth. The framework is based on the correspondence between maximum-rotating de Bruijn sequences and spanning trees of a graph related to the de Bruijn graph. Individual de Bruijn sequences can be measured in various ways, and one such measurement that dates back to 1970 is the number of \rotation" used in creating the sequence. In this thesis, we provide the first characterization of the de Bruijn sequences that use the maximum number of rotations. The characterization involves the spanning trees of a particular graph of necklace equivalence classes, and it provides a simple new framework for constructing individual de Bruijn sequences with this \maximum-rotation" property. We discuss the four canonical de Bruijn sequences that are consequences of this new framework, all of which have simple proofs of correctness. Three of the four canonical de Bruijn sequences had previously been discovered using dierent approaches, and this thesis provides the first unifying construction for them. Furthermore, the fourth construction had not been previously discovered.

Recommended Citation

Hernandez, Oscar, "Constructing Maximum-Rotating de Bruijn Sequences Using their Necklace Trees" (2017). Senior Theses. 1120.
https://digitalcommons.bard.edu/sr-theses/1120