Shuffle Exchange Network and de Bruijn

1
Shuffle Exchange Network and de Bruijns Graph

• Shuffle Exchange graph

010
011
000
001
110
111
100
101
Merge exchange into a single node
01
1
1
00
0
0
1
1
11
0
0
10
De Bruijn Graph (label shift left and add the
label)
2
Same Graph, Another labeling on edges
01
1
1
0
0
00
11
0
0
1
1
10
node x1x0 ? x0 (x1 ? label)
3
f is either 0 or 1 For 0 shift 1
complement
?
f
Note that each complete cycle of shift register
corresponds to a HC of de Bruijns Graph
4
?
0
0
.
001
000
5
x3 x 1 is irreducible
001 011 111 110 101 010 100 001 gt DeBruijn sequen
ce
001
011
000
101
010
111
100
110
x3 x2 1 is irreducible
001 010 101 011 111 110 100 001
?
0
0
1
6
Shift Register
x3 x2 x 1 ? (x21)(x1) not irreducible
?
?
001 011 110 100 gt degenerated cycle
0
0
1
001
011
000
101
010
111
100
110
7
For 4 bit
?
0001 0011 0111 1111 1110 1101 1010
0101 1011 0110 1100 1001 0010
0100 1000
8
Cycle decomposition based n
?
?
001
011
100
110
011
011 ? 101 ? 010
9
1
1
1
0
1
000
0
0
0
Conventional labeling
1
1
1
0
1
00
0
0
0
10
001
011
000
101
010
111
100
110