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Table 1 Comparison of 2-Phase algorithm and our algorithm in chemical graphs (I)

From: Efficient enumeration of monocyclic chemical graphs with given path frequencies

Entry

  

2-Phase Algorithm

Our Algorithm

 

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

 

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

 

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

 

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

 

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

 

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