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Merging Shuffle Exchange Graphs in De Bruijn Graphs: A Study of Labeling Techniques

This document explores the merging of Shuffle Exchange Graphs within De Bruijn Graphs, focusing on label manipulation techniques. The core method involves combining multiple states into a single node while preserving the structure of the graph. We examine different labeling strategies, including edge labeling adjustments and the implications on cycle decomposition. Additionally, the analysis highlights irreducibility conditions of the generated graphs and cycles, offering insights into the interplay between shift registers and labeling transformations.

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Merging Shuffle Exchange Graphs in De Bruijn Graphs: A Study of Labeling Techniques

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  1. 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)

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

  3. 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

  4. 0 0 . 001 000

  5. 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

  6. 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

  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. 001 011 000 010 101 111 100 110

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