De Bruijn sequences 陳柏澍 Novembers 2005

Each of the segments is one of two types, denoted by 0 and 1. Any four consecutive segments uniquely determine the position of the drum. Rotating drum problem

Rotating drum problem The problem above means that the 16 possible quadruples of consecutive 0 ’ s and 1 ’ s on the drum should be the binary representations of the integers 0 to 15 （ in other words, all distinct ）.

Rotating drum problem The first, can this be done ？ The second, if yes, in how many different ways ？

Rotating drum problem Both questions were treated by N. G. de Bruijn （ 1946 ） and for this reason the graphs described later and the corresponding circular sequence are often called De Bruijn graphs and De Bruijn sequences, respectively.

Rotating drum problem Consider a diagraph by taking all 3-bit binary words as vertices and joining the vertex by a directed edge to and. The arc is labeled.

Rotating drum problem

Rotating drum problem Clearly, the graph is strong connected and every vertex has in-degree 2 and out-degree 2. So the graph is Eulerian. The Eulerian circuit with specified edge label forms the circular arrangement we desired.

Rotating drum problem De Bruijn sequence ： Such a closed path is also called a complete cycle. De Bruijn graph

A cyclic sequence is called De Bruijn sequence if following two properties are satisfied ： （ 1 ） （ 2 ） distinct n-dim vectors Example ： De Bruijn sequence

De Bruijn graph De Bruijn graph is a weighted diagraph that satisfies following two properties ： （ 1 ） （ 2 ） Connecting to with the directed edge having the weight

De Bruijn graph Lemma 1 ： is Eulerian. Proof ： Clearly, is strong connected and. Thus, it ’ s Eulerian. Lemma 2 ： All weights in are all distinct. Proof ： Trivial from the definition.

De Bruijn sequence Theorem 1 ： De Bruijn sequence exists. Proof ： By Lemma 1, there exists a Eulerian circuit C in. Suppose C passes through edges. Suppose the weight of each edge is.

De Bruijn sequence Let, thus is the desired De Bruijn sequence. It can be easily check that any n consecutive segments of the sequence above maps to the weight of the unique edge. By Lemma2, they are all distinct.

Doubled graph of De Bruijn graph Let De Bruijn graph. Define the doubled graph of as follows ： （ 1 ） each edge of corresponds to a vertex of.

Doubled graph of De Bruijn graph

Clearly,. Example ：

2-in 2-out graph 2-in 2-out graph is the diagraph with in- degree 2 and out-degree 2 for very vertex. Clearly, and are both 2-in 2-out graph.

How many different ways Theorem 2 ： Let be a 2-in 2-out graph on vertices with complete cycles. Then has complete cycles.

How many different ways Proof ： Induction on （ a ） If then has one vertex and two loops. Then which has one complete cycle

How many different ways

Example ： 2 A

How many different ways

We just treat the third class here, the other two are similar

How many different ways