Table 2 Comparison of the maximum approximation ratios given by the EH algorithm with ours (Alg1)
From: A new 1.375-approximation algorithm for sorting by transpositions
n | Transposition diameter | Max. approx.ratio | Averageapprox. ratio | Average distance | Number of times EH exceeded the 1.375-approx. | Time to sort all permutationsa | ||||
|---|---|---|---|---|---|---|---|---|---|---|
EH | Alg1 | EH | Alg1 | EH | Alg1 | EH | Alg1 | |||
2 | 1 | 1.00 | 1.00 | 1.0 | 1.0 | 1.00 | 1.00 | 0 | \(<1s\) | \(<1s\) |
3 | 2 | 1.00 | 1.00 | 1.0 | 1.0 | 1.20 | 1.20 | 0 | \(<1s\) | \(<1s\) |
4 | 3 | 1.00 | 1.00 | 1.0 | 1.0 | 1.6086 | 1.6086 | 0 | \(<1s\) | \(<1s\) |
5 | 3 | 1.00 | 1.00 | 1.0 | 1.0 | 2.0924 | 2.0924 | 0 | \(<1s\) | \(<1s\) |
6 | 4 | \(1.333{\bar{3}}\) | 1.00 | 1.0004 | 1.0 | 2.6063 | 2.6050 | 0 | \(<1s\) | \(<1s\) |
7 | 5 | \(1.333{\bar{3}}\) | 1.25 | 1.0129 | 1.0113 | 3.1762 | 3.1704 | 0 | \(<1s\) | \(<1s\) |
8 | 6 | 1.5 | 1.25 | 1.0210 | 1.0183 | 3.7178 | 3.7076 | 2 | \(<2s\) | \(<2s\) |
9 | 6 | 1.5 | 1.25 | 1.0301 | 1.0256 | 4.2796 | 4.2603 | 20 | \(\approx 10s\) | \(\approx 13s\) |
10 | 7 | 1.5 | 1.25 | 1.0341 | 1.0282 | 4.8051 | 4.7772 | 110 | \(\approx 3m\) | \(\approx 2m\) |
11 | 8 | 1.5 | \(1.333{\bar{3}}\) | 1.0392 | 1.0321 | 5.3526 | 5.3157 | 440 | \(\approx 35m\) | \(\approx 30m\) |
12 | 9 | 1.5 | \(1.333{\bar{3}}\) | 1.0415 | 1.0336 | 5.8694 | 5.8248 | 1448 | \(\approx 8.5h\) | \(\approx 8.1h\) |