Table 1 Examples of natural and Gray labeling.
From: Efficient bit labeling in factorization machines with annealing for traveling salesman problem
Route | Natural | Gray | ||||||
|---|---|---|---|---|---|---|---|---|
\(\textbf{r}\) | \(m(=\underline{m})\) | \(\underline{m}(\ne m)\) | \(\textbf{b} (= \underline{\textbf{b}})\) | \(\underline{\textbf{b}} (\ne \textbf{b})\) | \(|\mathcal {S}| (= |\underline{\mathcal {S}}|)\) | \(|\underline{\mathcal {S}}| (\ne |\mathcal {S}|)\) | \(\textbf{b} (= \underline{\textbf{b}})\) | \(\underline{\textbf{b}} (\ne \textbf{b})\) |
(1,2,3,4) | 0 | 24 | 00000 | 11000 | (0,0,0) | (0,3,0) | 00000 | 01000 |
(1,2,4,3) | 1 | 25 | 00001 | 11001 | (0,0,1) | (0,3,1) | 00001 | 01001 |
(1,3,2,4) | 2 | 26 | 00010 | 11010 | (0,1,0) | – | 00100 | – |
(1,3,4,2) | 3 | 27 | 00011 | 11011 | (0,1,1) | – | 00101 | – |
(1,4,2,3) | 4 | 28 | 00100 | 11100 | (0,0,2) | (0,3,2) | 00011 | 01011 |
(1,4,3,2) | 5 | 29 | 00101 | 11101 | (0,1,2) | – | 00111 | – |
(2,1,3,4) | 6 | 30 | 00110 | 11110 | (1,0,0) | (1,3,0) | 10000 | 11000 |
(2,1,4,3) | 7 | 31 | 00111 | 11111 | (1,0,1) | (1,3,1) | 10001 | 11001 |
(2,3,1,4) | 8 | – | 01000 | – | (1,1,0) | – | 10100 | – |
(2,3,4,1) | 9 | – | 01001 | – | (1,1,1) | – | 10101 | – |
(2,4,1,3) | 10 | – | 01010 | – | (1,0,2) | (1,3,2) | 10011 | 11011 |
(2,4,3,1) | 11 | – | 01011 | – | (1,1,2) | – | 10111 | – |
(3,1,2,4) | 12 | – | 01100 | – | (0,2,0) | – | 01100 | – |
(3,1,4,2) | 13 | – | 01101 | – | (0,2,1) | – | 01101 | – |
(3,2,1,4) | 14 | – | 01110 | – | (1,2,0) | – | 11100 | – |
(3,2,4,1) | 15 | – | 01111 | – | (1,2,1) | – | 11101 | – |
(3,4,1,2) | 16 | – | 10000 | – | (0,2,2) | – | 01111 | – |
(3,4,2,1) | 17 | – | 10001 | – | (1,2,2) | – | 11111 | – |
(4,1,2,3) | 18 | – | 10010 | – | (0,0,3) | (0,3,3) | 00010 | 01010 |
(4,1,3,2) | 19 | – | 10011 | – | (0,1,3) | – | 00110 | – |
(4,2,1,3) | 20 | – | 10100 | – | (1,0,3) | (1,3,3) | 10010 | 11010 |
(4,2,3,1) | 21 | – | 10101 | – | (1,1,3) | – | 10110 | – |
(4,3,1,2) | 22 | – | 10110 | – | (0,2,3) | – | 01110 | – |
(4,3,2,1) | 23 | – | 10111 | – | (1,2,3) | – | 11110 | – |
