1. De Bruijn sequences 陳柏澍 Novembers 2005

2. 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 0000 0001 0011

3. 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 ）.

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

5. 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.

6. 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.

7. Rotating drum problem 000 001100 010 101 011110 111 0000 0001 1000 1001 0010 0100 0011 1100 0101 1010 0111 1110 1111 0110 1101 1011

8. 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.

9. Rotating drum problem 000 001100 010 101 011110 111 0000 0001 1000 1001 0010 0100 0011 1100 0101 1010 0111 1110 1111 0110 1101 1011 000 001011 110001 111 100 101 010111 011110101010 100000 De Bruijn sequence ： 0000111100101101 0000 0001 0011 0111 1110 1111 1100100100100101 10110110110110100100 1000 Such a closed path is also called a complete cycle. De Bruijn graph

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

11. 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

12. 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.

13. 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.

14. 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.

15. 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.

16. Doubled graph of De Bruijn graph

17. Clearly,. Example ： 01 01 10 00 11 0011 01 10 000 111 011 110 100 001 101010

18. 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.

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

20. How many different ways Proof ： Induction on （ a ） If then has one vertex and two loops. Then which has one complete cycle. 0 1 01 10 01 00 11

21. How many different ways

24. Example ： 2 A

25. How many different ways

29. We just treat the third class here, the other two are similar. 1 3 2 4

30. How many different ways 1 3 2 4

31. 1 2 3 4 3 4 1 2