Journal of Statistical Mechanics: Theory and Experiment
Volume 2009
February 2009
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Iwan Jensen J. Stat. Mech. (2009) P02065 doi:10.1088/1742-5468/2009/02/P02065
A parallel algorithm for the enumeration of benzenoid hydrocarbons
Iwan Jensen
Show affiliations| Table 1. Update rules for Case 0 and Case 1. |
| Case 0 | Case 1 | ||||
| Input | Output | Input | Output | ||
| `00' | `00' | `12' | `00' | `00' | `12' |
| `01' | `01' | `10' | `01' | `01' | `10' |
| `02' | `02' | `20' | `02' | `02' | `20' |
| `10' | `01' | `10' | `10' | `01' | `10' |
| `20' | `02' | `20' | `20' | `02' | `20' |
| `11' | ` ' |
`11' | `11' | `' |
` ' |
| `12' | Acc | `12' | `12' | Acc | — |
| `21' | `00' | `21' | `21' | `00' | `12' |
| `22' | `' |
`22' | `22' | `' |
`' |
| Table 2. Number of fixed benzenoids Bh of size h ≥ 36. |
| h | Bh |
| 36 | 352 506 828 543 839 738 006 802 |
| 37 | 1 771 125 269 041 561 567 830 953 |
| 38 | 8 905 113 919 188 230 264 955 009 |
| 39 | 44 804 571 829 235 959 198 699 855 |
| 40 | 225 570 974 088 699 920 561 748 746 |
| 41 | 1 136 340 745 302 289 809 680 018 862 |
| 42 | 5 727 773 558 054 438 208 070 950 886 |
| 43 | 28 887 056 504 374 868 913 302 241 736 |
| 44 | 145 763 914 212 751 560 334 802 981 991 |
| 45 | 735 894 997 233 174 457 602 406 978 869 |
| 46 | 3 716 988 842 355 112 053 567 240 722 854 |
| 47 | 18 783 102 592 560 998 779 533 576 292 617 |
| 48 | 94 958 908 613 774 943 408 509 332 060 260 |
| 49 | 480 273 434 248 924 455 452 231 252 618 009 |
| 50 | 2430 068 453 031 180 290 203 185 942 420 933 |
| Table 3. Number of processors with total CPU time and actual running time (in the format hh:mm) as well and memory use for the parallel algorithm for enumerating benzenoids of maximal size 43 at width 22. |
| Proc. | Total time | Run time | Max Conf | Min Conf | Max Term | Min Term |
| 1 | 60:13 | 60:20 | 107 350 066 | 207 111 142 | ||
| 2 | 61:53 | 30:59 | 52 982 622 | 52 435 395 | 102 711 198 | 102 666 398 |
| 4 | 62:28 | 15:38 | 26 389 619 | 26 183 924 | 51 559 593 | 51 025 667 |
| 8 | 63:17 | 7:55 | 13 289 367 | 13 078 219 | 26 179 885 | 25 492 182 |
| 16 | 69:28 | 4:22 | 6 725 270 | 6 486 246 | 13 245 615 | 12 717 598 |
| 32 | 69:05 | 2:10 | 3 440 269 | 3 274 193 | 6 871 820 | 6 347 966 |
| 64 | 71:33 | 1:08 | 1 768 626 | 1 616 220 | 3 839 775 | 3 191 842 |
| Table 4. Estimates for the critical point qc = 1/κ and critical exponent –1–θ as obtained from second-and third-order differential approximants with L being the degree of the inhomogeneous polynomial. |
| Second-order approximants | Third-order approximants | |||
| L | qc = 1/κ | –1–θ | qc = 1/κ | –1–θ |
| 0 | 0.193 725 984 74(16) | –0.000 001 36(87) | 0.193 725 984 40(23) | –0.000 000 55(37) |
| 2 | 0.193 725 984 48(24) | –0.000 000 77(43) | 0.193 725 984 286(90) | –0.000 000 36(16) |
| 4 | 0.193 725 984 40(11) | –0.000 000 56(42) | 0.193 725 984 36(22) | –0.000 000 51(39) |
| 6 | 0.193 725 984 43(27) | –0.000 000 68(51) | 0.193 725 984 16(16) | –0.000 000 09(41) |
| 8 | 0.193 725 984 41(32) | –0.000 000 52(93) | 0.193 725 984 182(83) | –0.000 000 13(21) |
| 10 | 0.193 725 984 44(19) | –0.000 000 69(38) | 0.193 725 984 205(94) | –0.000 000 20(23) |
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A parallel algorithm for the enumeration of benzenoid hydrocarbons
Iwan Jensen J. Stat. Mech. (2009) P02065
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