5. Experimental results and discussion

Two coordinate sets of 15 terminal nodes are randomly created based on varied distribution topology of terminal nodes in VLSI system on a defined two dimensional 100 100 search space. Coordinate sets for nearly Uniform distribution and Bivariate distribution are graphically represented in Figures 4 and 5 respectively.

5.1. Experiment A

Experiments on PSO-W as well as on the modified algorithms PSO-ST [20] and PSO-SAAC are performed separately for two said coordinate sets to interconnect the terminal points for each set and to return the minimum cost of interconnection correspondingly. The minimum interconnection VLSI global routing cost, the average interconnection VLSI global routing cost over the 25 simulations of the algorithms and the corresponding standard deviations are recorded.

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For nearly uniform distribution of terminal nodes in VLSI layout, PSO-SAAC works best in compared to the other two algorithms. In SET 1 '338' is achieved as the minimum interconnection global cost value for PSO-SAAC, given in Figure 6. From Table 1 it can be analyzed that for bivariate distribution of terminal nodes in VLSI layout, self-tuned acceleration constant controlling mechanism for PSO-ST outruns the other two algorithms. In random uniform distribution environment, PSO-ST generates lowest minimum interconnection cost as 253,

Test case gbest value PSO-W PSO-ST PSO-SAAC SET 1 Average 354.5 348.4 341.7

SET 2 Average 256.7 254.9 257.3

Table 1. Comparison of PSO-W with PSO-ST and PSO-SAAC.

Figure 6. Minimum 'cost' Steiner tree obtained for PSO-SACC in SET 1.

Minimum 350 343 338

Minimum 253 253 255

The results of average gbest and minimum gbest are summarized in Table 1.

Experiments on all the algorithms are performed 25 times for each of these coordinate sets of varied distribution topologies in VLSI system. The population size of the swarms has been set as 100 and maximum iteration of 75 is used for all the algorithms.

Figure 4. Nearly uniform distribution of terminal nodes on 100 100 search space.

Figure 5. Bivariate distribution of terminal nodes on 100 100 search space.

## 5.1. Experiment A

5. Experimental results and discussion

72 Particle Swarm Optimization with Applications

cally represented in Figures 4 and 5 respectively.

as 100 and maximum iteration of 75 is used for all the algorithms.

Figure 4. Nearly uniform distribution of terminal nodes on 100 100 search space.

Figure 5. Bivariate distribution of terminal nodes on 100 100 search space.

Two coordinate sets of 15 terminal nodes are randomly created based on varied distribution topology of terminal nodes in VLSI system on a defined two dimensional 100 100 search space. Coordinate sets for nearly Uniform distribution and Bivariate distribution are graphi-

Experiments on all the algorithms are performed 25 times for each of these coordinate sets of varied distribution topologies in VLSI system. The population size of the swarms has been set Experiments on PSO-W as well as on the modified algorithms PSO-ST [20] and PSO-SAAC are performed separately for two said coordinate sets to interconnect the terminal points for each set and to return the minimum cost of interconnection correspondingly. The minimum interconnection VLSI global routing cost, the average interconnection VLSI global routing cost over the 25 simulations of the algorithms and the corresponding standard deviations are recorded. The results of average gbest and minimum gbest are summarized in Table 1.

For nearly uniform distribution of terminal nodes in VLSI layout, PSO-SAAC works best in compared to the other two algorithms. In SET 1 '338' is achieved as the minimum interconnection global cost value for PSO-SAAC, given in Figure 6. From Table 1 it can be analyzed that for bivariate distribution of terminal nodes in VLSI layout, self-tuned acceleration constant controlling mechanism for PSO-ST outruns the other two algorithms. In random uniform distribution environment, PSO-ST generates lowest minimum interconnection cost as 253,


Table 1. Comparison of PSO-W with PSO-ST and PSO-SAAC.

Figure 6. Minimum 'cost' Steiner tree obtained for PSO-SACC in SET 1.

given in Figure 7. It is also seen that acceleration constant tuning mechanism of PSO improves the average interconnection cost of the VLSI global best parameter.

coordinate set which have been considered. The results of minimum interconnect cost, average

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The Minimum Rectilinear Spanning Tree (RMST) for minimum interconnect cost generated for the said VLSI topologies in case of all algorithms are shown in Figures 8, 9, and 10. It reveals that the algorithm PSO-MU generate lowest minimum global best value as well as minimum mean value in all two coordinate sets. This indicates that this algorithm PSO-MU, in comparison to PSO-W and PSO-C, ensure efficient VLSI global routing cost minimization and better

It is observed from Table 3 that for Coordinate Set 1, gbest value of PSO –MU is found to be 329 where execution time of this algorithm is much greater than PSO-W. Runtime of PSO-C is found to be 101.51 compared to 85.48 for PSO-MU algorithm. This implies that PSO-MU

> Minimum 350 345 329 System time 52.825 101.51 85.48

> Minimum 253 254 248 System time 49.05 86.01 66.96

Test case gbest value PSO-W PSO-C PSO-MU SET 1 Average 354.5 350.4 336.8

SET 2 Average 256.7 256 250.4

Table 3. Comparison of PSO variants over average, minimum gbest value and system time.

Figure 8. Minimum 'interconnection cost' Steiner Tree obtained for PSO-W in SET 1.

cost and average execution time of all algorithms are recorded on Table 3.

convergence.

So, it can be safely stated that for nearly uniform distribution PSO-SAAC and for increased random bivariate distributions PSO-ST reduces the cost of RMST, constructed by interconnecting the terminal nodes. So RSMT problem of graphs can be effectively managed and thereby the VLSI interconnect length is reduced to a great extent.

It is also observed that from Table 2, that standard deviation value for PSO-SAAC is lowest for SET 1 whereas PSO-ST achieves lowest standard deviation value for SET 2. This implies that for nearly uniform and less random distribution, self-adaptive mechanism of PSO ensures more consistency while self-tuned mechanism of PSO is more consistent in case of highly random distribution of terminal nodes in the defined search space.

## 5.2. Experiment B

The experiments are performed first on weighted PSO (PSO-W) and then on PSO with constriction factor (PSO-C) and lastly on PSO with mutation algorithm (PSO-MU) for all two

Figure 7. Minimum 'cost' Steiner tree obtained for PSO-ST in SET 2.


Table 2. Standard deviation of gbest values.

coordinate set which have been considered. The results of minimum interconnect cost, average cost and average execution time of all algorithms are recorded on Table 3.

given in Figure 7. It is also seen that acceleration constant tuning mechanism of PSO improves

So, it can be safely stated that for nearly uniform distribution PSO-SAAC and for increased random bivariate distributions PSO-ST reduces the cost of RMST, constructed by interconnecting the terminal nodes. So RSMT problem of graphs can be effectively managed

It is also observed that from Table 2, that standard deviation value for PSO-SAAC is lowest for SET 1 whereas PSO-ST achieves lowest standard deviation value for SET 2. This implies that for nearly uniform and less random distribution, self-adaptive mechanism of PSO ensures more consistency while self-tuned mechanism of PSO is more consistent in case of highly

The experiments are performed first on weighted PSO (PSO-W) and then on PSO with constriction factor (PSO-C) and lastly on PSO with mutation algorithm (PSO-MU) for all two

Test case PSO-W PSO-ST PSO-SAAC

SET 1 7.77 0.71 5.41 SET 2 1.94 1.88 4.12

the average interconnection cost of the VLSI global best parameter.

and thereby the VLSI interconnect length is reduced to a great extent.

random distribution of terminal nodes in the defined search space.

Figure 7. Minimum 'cost' Steiner tree obtained for PSO-ST in SET 2.

Table 2. Standard deviation of gbest values.

5.2. Experiment B

74 Particle Swarm Optimization with Applications

The Minimum Rectilinear Spanning Tree (RMST) for minimum interconnect cost generated for the said VLSI topologies in case of all algorithms are shown in Figures 8, 9, and 10. It reveals that the algorithm PSO-MU generate lowest minimum global best value as well as minimum mean value in all two coordinate sets. This indicates that this algorithm PSO-MU, in comparison to PSO-W and PSO-C, ensure efficient VLSI global routing cost minimization and better convergence.

It is observed from Table 3 that for Coordinate Set 1, gbest value of PSO –MU is found to be 329 where execution time of this algorithm is much greater than PSO-W. Runtime of PSO-C is found to be 101.51 compared to 85.48 for PSO-MU algorithm. This implies that PSO-MU


Table 3. Comparison of PSO variants over average, minimum gbest value and system time.

Figure 8. Minimum 'interconnection cost' Steiner Tree obtained for PSO-W in SET 1.

algorithm outperforms the performance of conventional PSO-W and PSO-C algorithm while reducing the Timing budget with respect to PSO-C algorithm in context to VLSI global routing. In order to analyze the consistency of these algorithms, the standard deviation (SD) values are calculated for all algorithms on each of the two coordinate sets and are recorded in the Table 4. SD value of PSO-C is found to be 0.71 and 2.25 for the two coordinate sets. These values are much lower compared to all other SD values of PSO-W and PSO-MU ensuring robustness although sacrificing system execution time of the algorithm independent of the distribution complexities of the search space in VLSI layout. This implies that PSO-C although generates higher value of global routing interconnection cost as well as system execution time compared to PSO-W and PSO-MU, it exhibits robustness of the algorithm throughout all varied distribu-

Test case PSO-W PSO-C PSO-MU SET 1 7.77 0.71 5.65 SET 2 1.94 1.88 3.83

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This chapter intends variants developed on Particle Swarm Optimization algorithm to resolve the global routing problem in VLSI domain. Simultaneously the controlling of acceleration constant in PSO has been verified for the VLSI routing problem. Lastly, a proportional analysis is done amongst the pre mentioned algorithms beside three variants of PSO, which have been recognized as decent routing algorithms in VLSI design. Researches are piloted to inspect the optimization property, rate of convergence, computational time and robustness of the algorithms including the ways by which algorithms work proficiently in problem space with

The outcomes demonstrates that from the standpoint of topologically dissimilar problem spaces of VLSI domain, the general performance of PSO-ST [20] is very agreeable, however PSO-SAAC executes finest in an approximately uniform distributed problem space. It has also been observed that the performance of PSO-C and PSO-MU are unhampered of the diverse distribution of VLSI global routing problem space. The performance of the algorithm PSO-MU preserves a balance between the optimization and convergence rate. Although PSO-MU is realized to be steady in random problem space [22], PSO-C is appeared to be the best algo-

Therefore the chapter indicated the exclusive merits and demerits of the PSO algorithm and its variants, well-matched for solving the wire-length minimization problem of global routing in VLSI physical design. It is projected that in the situation of VLSI global routing optimization, the paradigm of hybridization with essence of genetics can contest with the functioning of PSO conventional ones and can exhibit enhanced performance. Hence the global routing problem

tion topologies of the terminal nodes in the said VLSI layout.

dissimilar distributive topologies of VLSI layout.

rithm in the perspective of robustness.

6. Conclusion

Table 4. Standard deviation of gbest values.

Figure 9. Minimum 'interconnection cost' Steiner Tree obtained for PSO-C in SET 1.

Figure 10. Minimum 'interconnection cost' Steiner Tree obtained for PSO-MU in SET 1.


Table 4. Standard deviation of gbest values.

algorithm outperforms the performance of conventional PSO-W and PSO-C algorithm while reducing the Timing budget with respect to PSO-C algorithm in context to VLSI global routing.

In order to analyze the consistency of these algorithms, the standard deviation (SD) values are calculated for all algorithms on each of the two coordinate sets and are recorded in the Table 4. SD value of PSO-C is found to be 0.71 and 2.25 for the two coordinate sets. These values are much lower compared to all other SD values of PSO-W and PSO-MU ensuring robustness although sacrificing system execution time of the algorithm independent of the distribution complexities of the search space in VLSI layout. This implies that PSO-C although generates higher value of global routing interconnection cost as well as system execution time compared to PSO-W and PSO-MU, it exhibits robustness of the algorithm throughout all varied distribution topologies of the terminal nodes in the said VLSI layout.
