From: Efficient enumeration of monocyclic chemical graphs with given path frequencies
Entry | Â | Â | 2-Phase Algorithm | Our Algorithm | Â | |||||
---|---|---|---|---|---|---|---|---|---|---|
Formula | n | K | Tree | Time | Time/graph | Tree | Cycle | Time | Time/graph | Ratio |
 |  | 1 | 388,192 | 0.288 | 741.9E-9 | 388,192 | 1,786,467 | 14.056 | 6.5E-6 | 8.7 |
 |  | 2 | 614 | 0.021 | 34.2E-6 | 614 | 229 | 0.134 | 159.0E-6 | 4.6 |
C00062 | Â | 3 | 95 | 0.020 | 210.5E-6 | 95 | 0 | 0.039 | 410.5E-6 | 2.0 |
C6H14N2O4 | 12 | 4 | 1 | 0.008 | 8.0E-3 | 1 | 0 | 0.018 | 18.0E-3 | 2.3 |
 |  | 5 | 1 | 0.005 | 5.0E-3 | 1 | 0 | 0.006 | 6.0E-3 | 1.2 |
 |  | 6 | 1 | 0.006 | 6.0E-3 | 1 | 0 | 0.006 | 6.0E-3 | 1.0 |
 |  | 7 | 1 | 0.004 | 4.0E-3 | 1 | 0 | 0.004 | 4.0E-3 | 1.0 |
 |  | 1 | 1,708 | 0.007 | 4.1E-6 | 1,708 | 12,626 | 0.050 | 3.5E-6 | 0.9 |
 |  | 2 | 50 | 0.004 | 80.0E-6 | 50 | 1,085 | 0.043 | 37.9E-6 | 0.5 |
C00095 |  | 3 | 0 | 0.046 | – | 0 | 286 | 0.046 | 160.8E-6 | – |
C6H12O6 | 12 | 4 | 0 | 0.006 | – | 0 | 19 | 0.035 | 1.8E-3 | – |
 |  | 5 | 0 | 0.004 | – | 0 | 7 | 0.030 | 4.3E-3 | – |
 |  | 6 | 0 | 0.004 | – | 0 | 5 | 0.028 | 5.6E-3 | – |
 |  | 7 | 0 | 0.004 | – | 0 | 5 | 0.013 | 2.6E-3 | – |
 |  | 1 | 5,446,987 | 7.690 | 1.4E-6 | 5,446,987 | 31,395,098 | 217.681 | 5.9E-6 | 4.2 |
 |  | 2 | 373 | 0.022 | 59.0E-6 | 373 | 71 | 0.171 | 385.1E-6 | 6.5 |
C03343 | Â | 3 | 187 | 0.023 | 123.0E-6 | 187 | 25 | 0.071 | 334.9E-6 | 2.7 |
C16H22O4 | 15 | 4 | 101 | 0.022 | 217.8E-6 | 101 | 9 | 0.042 | 381.8E-6 | 1.8 |
 |  | 5 | 51 | 0.022 | 431.4E-6 | 51 | 6 | 0.059 | 1.0E-3 | 2.4 |
 |  | 6 | 43 | 0.009 | 209.3E-6 | 43 | 0 | 0.036 | 837.2E-6 | 4.0 |
 |  | 7 | 28 | 0.013 | 464.3E-6 | 28 | 0 | 0.020 | 714.3E-6 | 1.5 |
 |  | 1 | 2,926,382 | 2.878 | 983.5E-9 | 2,926,382 | 23,965,432 | 146.669 | 5.5E-6 | 5.5 |
 |  | 2 | 41,468 | 1.035 | 25.0E-6 | 41,468 | 213,820 | 37.792 | 148.0E-6 | 5.9 |
C00645 | Â | 3 | 491 | 0.562 | 1.1E-3 | 491 | 4,482 | 6.281 | 1.3E-3 | 1.1 |
C8H15NO6 | 15 | 4 | 0 | 0.523 | – | 0 | 73 | 4.168 | 57.1E-3 | – |
 |  | 5 | 0 | 0.374 | – | 0 | 5 | 2.320 | 464.0E-3 | – |
 |  | 6 | 0 | 0.135 | – | 0 | 3 | 0.430 | 143.3E-3 | – |
 |  | 7 | 0 | 0.121 | – | 0 | 1 | 0.328 | 328.0E-3 | – |
 |  | 1 | 167,172,180 | 238.554 | 1.4E-6 | ≥ 1,594,520 | ≥ 33,962,677 | T. O. | 50.7E-6 | 35.5 |
 |  | 2 | 210 | 1.232 | 5.9E-3 | 210 | 641 | 1.888 | 2.2E-3 | 0.4 |
C00837 |  | 3 | 0 | 0.853 | – | 0 | 4 | 1.206 | 301.5E-3 | – |
C8H18N6O4 | 18 | 4 | 0 | 0.445 | – | 0 | 2 | 0.523 | 261.5E-3 | – |
 |  | 5 | 0 | 0.389 | – | 0 | 1 | 0.596 | 596.0E-3 | – |
 |  | 6 | 0 | 0.298 | – | 0 | 1 | 0.367 | 367.0E-3 | – |
 |  | 7 | 0 | 0.285 | – | 0 | 1 | 0.395 | 395.0E-3 | – |
 |  | 1 | 62,234,720 | 321.155 | 5.2E-6 | ≥ 4,812,773 | ≥ 40,426,928 | T. O. | 39.8E-6 | 7.7 |
 |  | 2 | 884 | 0.310 | 350.7E-6 | 884 | 180 | 1.824 | 1.7E-3 | 4.9 |
C07178 | Â | 3 | 22 | 0.026 | 1.2E-3 | 22 | 4 | 0.099 | 3.8E-3 | 3.2 |
C21H28N2O5 | 18 | 4 | 1 | 0.004 | 4.0E-3 | 1 | 0 | 0.005 | 5.0E-3 | 1.3 |
 |  | 5 | 1 | 0.004 | 4.0E-3 | 1 | 0 | 0.005 | 5.0E-3 | 1.3 |
 |  | 6 | 1 | 0.004 | 4.0E-3 | 1 | 0 | 0.005 | 5.0E-3 | 1.3 |
 |  | 7 | 1 | 0.005 | 5.0E-3 | 1 | 0 | 0.005 | 5.0E-3 | 1.0 |
 |  | 1 | ≥ 1,208,446,991 | T. O. | 1.5E-6 | ≥ 7,009,856 | ≥ 47,008 | T. O. | 255.1E-6 | 171.2 |
 |  | 2 | 27,312,856 | 965.131 | 35.3E-6 | ≥ 337,989 | ≥ 3,593,865 | T. O. | 458.1E-6 | 13.0 |
C00270 |  | 3 | 156,073 | 391.611 | 2.5E-3 | ≥ 1,546 | ≥ 234,187 | T. O. | 7.6E-3 | 3.0 |
C11H19NO9 | 21 | 4 | 0 | 299.393 | – | ≥ 0 | ≥ 165 | T. O. | 10.9E+0 | – |
 |  | 5 | 0 | 208.268 | – | ≥ 0 | ≥ 7 | T. O. | 257.1E+0 | – |
 |  | 6 | 0 | 19.361 | – | 0 | 2 | 85.109 | 42.6E+0 | – |
 |  | 7 | 0 | 9.720 | – | 0 | 2 | 30.301 | 15.2E+0 | – |
 |  | 1 | ≥ 664,049,939 | T. O. | 2.7E-6 | ≥ 4,621,297 | ≥ 33,216,732 | T. O. | 47.6E-6 | 17.5 |
 |  | 2 | 164,885 | 34.357 | 208.4E-6 | 164,885 | 1,425 | 213.810 | 1.3E-3 | 6.2 |
C03690 | Â | 3 | 32,995 | 15.978 | 484.3E-6 | 32,995 | 179 | 66.612 | 2.0E-3 | 4.1 |
C24H38O4 | 23 | 4 | 3,884 | 2.265 | 583.2E-6 | 3,884 | 17 | 5.383 | 1.4E-3 | 2.4 |
 |  | 5 | 1,237 | 1.490 | 1.2E-3 | 1,237 | 13 | 3.466 | 2.8E-3 | 2.3 |
 |  | 6 | 559 | 0.773 | 1.4E-3 | 559 | 0 | 1.554 | 2.8E-3 | 2.0 |
 |  | 7 | 177 | 0.445 | 2.5E-3 | 177 | 0 | 0.617 | 3.5E-3 | 1.4 |