Shuffle Exchange Network and de Bruijn’s Graph

Shuffle Exchange Network and de Bruijn’s Graph 010 011 Shuffle Exchange graph 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)

Same Graph, Another labeling on edges 01 1 1 0 0 00 0 0 11 1 1 10 node x1x0 x0 (x1  label)

1 001 011 1 1 0 0 1 000 010 101 0 111 0 1 0 0 1 1 100 110 1 f is either 0 or 1 For 0: shift 1: complement  f Note that each complete cycle of shift register corresponds to a HC ofde Bruijns Graph

 0 0 . 001 000

Shift Register  0 0 1 001 011 000 010 101 111 100 110 x3 + x + 1 is irreducible 001 011 111 110 101 010 100 001 => DeBruijn sequence 001 011 000 010 101 111 100 110 x3 + x2 + 1 is irreducible 001 010 101 011 111 110 100 001  0 0 1

Shift Register x3 + x2 + x + 1 ? = (x2+1)(x+1) not irreducible   001 011 110 100 => degenerated cycle 0 0 1 001 011 000 010 101 111 100 110

For 4 bit  0001 0011 0111 1111 1110 1101 1010 0101 1011 0110 1100 1001 0010 0100 1000

Cycle decomposition based n   001 011 100 110 011 011  101  010

1 1 1 0 1 000 0 0 0 Conventional labeling 1 1 1 0 1 00 0 0 0

001 011 000 010 101 111 100 110

Shuffle Exchange Network and de Bruijn’s Graph