1 Introduction
2 LT coding method
2.1 Encoding and decoding of LT codes
2.2 Degree distribution
3 Optimization problem
3.1 Variables design based on sparse degree distribution
3.2 Objective formulation
4 ECSO-based solution
4.1 Preliminaries of typical CSO
4.1.1 Division and classification
4.1.2 Hierarchy and relationship
4.2 Enhanced CSO algorithm
4.2.1 Substitution of bottom individuals
4.2.2 Revision of chicks’ update equation
4.2.3 Introduction of DE
4.3 Procedure of ECSO-based approach
5 Simulation results
5.1 Simulations setup
Function | ID | Bounds | Optimum |
---|---|---|---|
High conditioned elliptic | F1 | [− 100,100] | 0 |
Penalized | F2 | [− 50,50] | 0 |
Rosenbrock | F3 | [− 30,30] | 0 |
Ackley | F4 | [− 32,32] | 0 |
Griewank | F5 | [− 600,600] | 0 |
Sphere | F6 | [− 100,100] | 0 |
Step | F7 | [− 100,100] | 0 |
Schwefel’s P1.2 | F8 | [− 100,100] | 0 |
Rastrigin | F9 | [− 5.12,5.12] | 0 |
Axis parallel hyer-ellipsoid | F10 | [− 5.12,5.12] | 0 |
Schwefel’s P2.22 | F11 | [− 100,100] | 0 |
Quartic | F12 | [− 1.28,1.28] | 0 |
5.2 Testing on the benchmark functions
Type | APSO | DE | CSO | SCSO | RCSO | ICSO | ECSO |
---|---|---|---|---|---|---|---|
ω
| 0.9–0.4 | N/A | N/A | N/A | N/A | N/A | N/A |
c1
| 2 | N/A | N/A | N/A | N/A | N/A | 2 |
c2
| 2 | N/A | N/A | N/A | N/A | N/A | 2 |
c3
| N/A | N/A | N/A | N/A | 0.4 | N/A | 0.4 |
G
| N/A | N/A | 5 | 5 | 5 | 5 | 5 |
N
R
| N/A | N/A | 15 | 15–25 | 15 | 15 | 15–25 |
N
H
| N/A | N/A | 65 | 65–55 | 65 | 65 | 65–55 |
N
C
| N/A | N/A | 25 | 25 | 25 | 25 | 25 |
N
M
| N/A | N/A | 30 | 30 | 30 | 30 | 30 |
N
W
| N/A | N/A | N/A | 15–25 | N/A | N/A | 15–25 |
CR
| N/A | 0.8 | N/A | N/A | N/A | 0.8 | 0.8 |
M
| N/A | 0.5 | N/A | N/A | N/A | 0.5 | 0.5 |
Function | Algorithm | Standard deviation | Mean | Best | Worst |
---|---|---|---|---|---|
F1 | APSO | 5.9492E+04 | 8.7705E+03 | 7.7646E−09 | 5.3110E+05 |
DE | 7.4107E+03 | 1.3125E+03 | 1.0849E−29 | 6.2729E+04 | |
CSO | 7.6889E+02 | 1.0121E+02 | 4.7242E−51 | 7.4324E+03 | |
BCSO | 6.6555E+02 | 8.7229E+01 | 1.5939E−56 | 6.5051E+03 | |
DCSO | 9.7958E+02 | 1.2479E+02 | 2.3529E−13 | 9.7158E+03 | |
CCSO | 3.7588E+02 | 5.5510E+01 | 5.1173E−49 | 3.5087E+03 | |
ECSO | 1.1425E+02 | 1.3967E+01 | 1.7777E−76 | 1.1238E+03 | |
F2 | APSO | 4.8242E+08 | 3.0668E+08 | 2.9954E+00 | 1.6746E+09 |
DE | 1.4118E+08 | 6.6593E+07 | 5.6825E+00 | 8.1316E+08 | |
CSO | 5.2294E+07 | 1.3454E+07 | 4.4328E−03 | 3.9428E+08 | |
BCSO | 5.7113E+07 | 1.3255E+07 | 4.6421E−03 | 4.2143E+08 | |
DCSO | 3.5179E+07 | 6.6830E+06 | 1.3967E+00 | 2.9323E+08 | |
CCSO | 3.8502E+07 | 6.9098E+06 | 4.8575E−03 | 3.6967E+08 | |
ECSO | 3.8038E+07 | 5.8436E+06 | 9.9838E−02 | 3.6491E+08 | |
F3 | APSO | 3.8828E+11 | 2.5392E+12 | 1.8276E+12 | 3.0721E+12 |
DE | 3.6839E+11 | 1.8362E+11 | 8.6652E+06 | 2.1543E+12 | |
CSO | 1.5036E+11 | 4.1156E+10 | 8.3924E+00 | 9.5963E+11 | |
BCSO | 1.2359E+11 | 2.4851E+10 | 8.0768E+00 | 1.1022E+12 | |
DCSO | 1.0919E+11 | 2.3546E+10 | 8.9219E+00 | 9.4549E+11 | |
CCSO | 1.0876E+11 | 2.2786E+10 | 8.2943E+00 | 9.5715E+11 | |
ECSO | 1.0614E+11 | 1.7426E+10 | 7.5968E+00 | 9.7048E+11 | |
F4 | APSO | 7.9438E+00 | 8.5669E+00 | 9.4325E−01 | 2.0735E+01 |
DE | 4.8955E+00 | 1.4568E+01 | 4.9839E+00 | 2.0421E+01 | |
CSO | 5.3134E+00 | 2.9588E+00 | 9.1020E−06 | 1.9524E+01 | |
BCSO | 4.9538E+00 | 2.4553E+00 | 1.0627E−06 | 1.9181E+01 | |
DCSO | 4.9357E+00 | 2.8540E+00 | 6.1677E−04 | 1.9620E+01 | |
CCSO | 5.0396E+00 | 2.7284E+00 | 7.3455E−06 | 1.9609E+01 | |
ECSO | 3.6984E+00 | 1.3313E+00 | 1.8692E−07 | 1.9516E+01 | |
F5 | APSO | 1.2130E+00 | 1.9364E+00 | 3.5936E−01 | 4.3676E+00 |
DE | 5.9648E−01 | 1.3952E+00 | 6.9788E−01 | 3.4824E+00 | |
CSO | 4.8664E−01 | 3.1842E−01 | 2.9110E−05 | 2.3938E+00 | |
BCSO | 4.8423E−01 | 2.7001E−01 | 8.1348E−10 | 2.3181E+00 | |
DCSO | 4.8122E−01 | 2.6827E−01 | 2.3956E−08 | 2.2155E+00 | |
CCSO | 4.8224E−01 | 3.0687E−01 | 2.4610E−06 | 2.4380E+00 | |
ECSO | 4.0396E−01 | 1.7115E−01 | 1.3262E−14 | 2.2312E+00 | |
F6 | APSO | 2.2211E−01 | 1.0988E−01 | 2.8881E−03 | 1.8355E+00 |
DE | 3.4216E−01 | 3.0496E−01 | 3.1631E−03 | 1.5045E+00 | |
CSO | 1.3665E−01 | 3.4548E−02 | 1.6614E−14 | 1.0647E+00 | |
BCSO | 1.2132E−01 | 3.0265E−02 | 8.0612E−16 | 9.2391E−01 | |
DCSO | 1.2599E−01 | 3.3452E−02 | 4.3039E−12 | 1.0142E+00 | |
CCSO | 1.1833E−01 | 3.0734E−02 | 5.6060E−15 | 8.9365E−01 | |
ECSO | 1.0101E−01 | 2.2448E−02 | 4.6118E−18 | 7.9359E−01 |
Function | Algorithm | Standard deviation | Mean | Best | Worst |
---|---|---|---|---|---|
F7 | APSO | 1.7199E+01 | 4.3650E+00 | 0.0000E+00 | 1.0870E+02 |
DE | 2.2119E+01 | 2.0320E+01 | 0.0000E+00 | 9.8650E+01 | |
CSO | 9.4792E+00 | 3.0670E+00 | 0.0000E+00 | 6.4400E+01 | |
BCSO | 9.4334E+00 | 2.9095E+00 | 0.0000E+00 | 6.6550E+01 | |
DCSO | 9.0134E+00 | 4.0845E+00 | 4.5000E−01 | 6.4250E+01 | |
CCSO | 8.0428E+00 | 2.4880E+00 |
0.0000E+00
| 5.9200E+01 | |
ECSO | 6.7709E+00 | 1.6255E+00 | 0.0000E+00 | 5.2950E+01 | |
F8 | APSO | 5.7047E+03 | 8.2521E+03 | 2.2036E+03 | 2.5229E+04 |
DE | 3.4683E+03 | 3.8976E+03 | 5.5399E+02 | 2.2475E+04 | |
CSO | 3.4032E+03 | 2.4246E+03 | 3.3888E+02 | 2.1273E+04 | |
BCSO | 3.0753E+03 | 2.3294E+03 | 2.7688E+02 | 1.6230E+04 | |
DCSO | 2.6553E+03 | 1.1508E+03 | 5.3335E−05 | 1.6955E+04 | |
CCSO | 3.1754E+03 | 1.6036E+03 | 1.2732E+02 | 2.2657E+04 | |
ECSO | 2.6438E+03 | 1.1180E+03 | 3.9454E−07 | 1.6007E+04 | |
F9 | APSO | 2.7305E+02 | 1.5363E+02 | 1.8616E+01 | 1.1794E+03 |
DE | 2.0441E+02 | 2.5411E+02 | 5.7273E+01 | 1.0046E+03 | |
CSO | 1.0097E+02 | 4.0486E+01 | 9.6601E−06 | 6.2801E+02 | |
BCSO | 1.0187E+02 | 3.6269E+01 | 1.2021E−08 | 6.5835E+02 | |
DCSO | 1.0786E+02 | 4.3412E+01 | 9.1331E−07 | 6.5675E+02 | |
CCSO | 1.0120E+02 | 3.9449E+01 | 6.0732E−07 | 6.7608E+02 | |
ECSO | 8.0768E+01 | 2.3935E+01 | 1.0119E−08 | 5.7983E+02 | |
F10 | APSO | 1.9719E+01 | 4.4414E+00 | 2.0279E−02 | 1.6054E+02 |
DE | 2.7524E+01 | 2.2175E+01 | 1.8682E−01 | 1.3731E+02 | |
CSO | 1.2493E+01 | 3.7259E+00 | 4.7947E−12 | 7.6476E+01 | |
BCSO | 1.1817E+01 | 3.1029E+00 | 8.9682E−13 | 8.2953E+01 | |
DCSO | 1.1784E+01 | 3.6059E+00 | 1.3398E−08 | 7.8886E+01 | |
CCSO | 1.0974E+01 | 2.9772E+00 | 1.4122E−12 | 8.3549E+01 | |
ECSO | 8.2519E+00 | 1.7539E+00 | 3.9075E−13 | 6.5495E+01 | |
F11 | APSO | 1.9944E+11 | 3.6391E+10 | 3.2077E+01 | 1.7823E+12 |
DE | 1.7313E+10 | 3.2944E+09 | 2.4898E+03 | 1.5493E+11 | |
CSO | 4.6866E+08 | 5.2398E+07 | 2.7157E−06 | 4.6870E+09 | |
BCSO | 3.4758E+08 | 3.5540E+07 | 7.7432E−08 | 3.4934E+09 | |
DCSO | 3.2962E+08 | 3.4301E+07 | 1.7245E−04 | 3.3122E+09 | |
CCSO | 1.0896E+09 | 1.2330E+08 | 2.9050E−06 | 1.0878E+10 | |
ECSO | 1.0777E+08 | 1.1629E+07 | 1.1224E−06 | 1.0810E+09 | |
F12 | APSO | 1.2648E+02 | 1.9405E+01 | 1.9131E−02 | 1.1248E+03 |
DE | 1.2314E+02 | 6.5162E+01 | 1.1084E−01 | 8.0601E+02 | |
CSO | 6.0829E+01 | 1.6762E+01 | 3.9479E−03 | 4.3591E+02 | |
BCSO | 5.5154E+01 | 1.1477E+01 | 4.2233E−03 | 4.5967E+02 | |
DCSO | 3.5858E+01 | 9.0720E+00 | 2.1421E−03 | 2.6972E+02 | |
CCSO | 4.2886E+01 | 9.6930E+00 | 2.5033E−03 | 3.6524E+02 | |
ECSO | 4.0757E+01 | 7.0128E+00 | 2.1833E−03 | 3.6695E+02 |
5.3 Comparison and analysis on the degree distribution optimization
Algorithm | Standard deviation | Mean | Best | Worst | ||
---|---|---|---|---|---|---|
f
|
f
|
f
|
Overhead
|
f
|
Overhead
| |
PSO-G | 0.4129 | 38.3575 | 37.8000 | 0.1813 | 39.8000 | 0.2438 |
CSO | 0.3382 | 37.6063 | 36.6250 | 0.1445 | 38.3750 | 0.1992 |
ECSO | 0.3477 | 37.4281 | 36.5000 | 0.1406 | 38.0000 | 0.1875 |
Algorithm | Standard deviation | Mean | Best | Worst | ||
---|---|---|---|---|---|---|
f
|
f
|
f
|
Overhead
|
f
|
Overhead
| |
PSO-G | 0.8183 | 74.6711 | 70.2235 | 0.0972 | 77.3750 | 0.2090 |
CSO | 0.7815 | 74.0125 | 69.1326 | 0.0802 | 75.6250 | 0.1816 |
ECSO | 0.7602 | 73.3750 | 68.0000 | 0.0625 | 75.0000 | 0.1719 |
Algorithm | Standard deviation | Mean | Best | Worst | ||
---|---|---|---|---|---|---|
f
|
f
|
f
|
Overhead
|
f
|
Overhead
| |
PSO-G | 1.9759 | 149.0875 | 145.0000 | 0.1328 | 153.0000 | 0.1953 |
CSO | 1.3793 | 146.0437 | 142.7320 | 0.1151 | 148.7500 | 0.1621 |
ECSO | 1.4517 | 145.0875 | 141.0000 | 0.1016 | 147.2500 | 0.1504 |
d
| k = 32 | k = 64 | k = 128 |
---|---|---|---|
1 | 0.1826 | 0.0835 | 0.0824 |
2 | 0.3754 | 0.6660 | 0.4676 |
4 | 0.2886 | 0.0822 | 0.2673 |
8 | 0.0817 | 0.0596 | 0.0430 |
16 | 0.0717 | 0.0801 | 0.0667 |
32 | – | 0.0286 | 0.0423 |
64 | – | – | 0.0307 |