*3.2.8. Experimental results*

We implemented the proposed algorithm on a 2.4GHz Intel Pentium IV with 512MB memory using MATLAB 7.2.0.232 (R2006a) in WINDOWS operating system. We applied the proposed algorithm to the relocation of n300a, n200a, and n100a, which are distributed in GSRC benchmarks in [17].

Mean Field Annealing Based Techniques for Resolving VLSI Automatic Design Problems 21

Although the use of SA provides for escaping from the local minima, it results in an excessive computation time requirement that has hindered experimentation with the Boltzmann machine. In order to overcome this major limitation of the Boltzmann machine, a mean field approximation may be used. In mean field network, the binary state stochastic neurons of the Boltzmann machine are replaced by deterministic analogue neurons. A simple formulation of the Traveling Salesman Problems energy function is described which, in combination with a normalized Hopfield-Tank neural network, eliminates the difficulty in finding valid tours[1]. This technique, as the one of the bases of MFA algorithm, is applicable to many other optimization problems involving n-way decisions (such as VLSI layout and resource allocation) and is easily implemented in a VLSI neural network. The solution quality is shown to be dependent on the formation of elements of the problem configuration which are influenced by the constraint penalties and the temperature as what is borrowed from SA technique. The applied algorithm for local relocation problem is modified form of which is applied for cell placement problem. The cooling schedule has three stages that the final stage is very fast cooling with decreasing factor 0.65 that may be what you mean *quenching*. Otherwise other two stages with decreasing factors 0.95 and 0.8 are not so fast and have *annealing* essence. For more information about this topic, one can

*Electrical Engineering Department, Engineering Faculty- Razi University, Kermanshah, Iran* 

[1] VandenBout, D. E. & Miller, T. K. (1989). Improving the performance of the Hopfield-Tank neural network through normalization and annealing, *Biological Cybernetics*,

[2] Peterson, C.& Soderberg, B. (1989). A new method for mapping optimization problems on to neural networks, *International Journal of Neural Systems,* vol.1 (3), pp. 3–22. [3] Takahashi, Y. (1997). Mathematical improvement of the Hopfield model for TSP, feasible solutions by synapse dynamical systems. *Neurocomputing,* vol. 15 pp. 15–43. [4] Gislen, L.; Peterson, C. & Soderberg, B. (1992).Complex scheduling with Potts neural

[5] Bultan, T. & Aykanat, C. (1992). A new mapping heuristic based on mean field

[6] Ohlsson, M.; Peterson, C. & Soderberg, B. (1993). Neural networks for optimization problems with inequality constraints - the knapsack problem, *Neural Computation,* vol.5

[7] Ohlsson, M. & Pi, H. (1997).A study of the mean field approach to knapsack problems,

annealing, *Journal of Parallel and Distributed Computing,* vol. 16, pp. 292–305.

refer to [1].

**Author details** 

**5. References** 

(2), pp. 331–339.

Gholam Reza Karimi and Ahmad Azizi Verki

Volume 62, Number 2, Pages 129-139.

*Neural Networks,* vol. 10(2), pp. 263–271.

networks," *Neural Computation*, vol. 4, pp. 805–831.

For every benchmark five different problems were resolved using our proposed algorithm and maximum and average runtime of 10 runs of them are presented in Table 1. Results show that our MFA based algorithm is faster than SA-based proposed method in SA-based relocation method in [18] because the number of displacements is limited to the number of movable modules of problem and the problem is local relocation. Actually relocation range reflects on number of displacements and also similarity of resultant placement with model placement.

Results show runtimes of our proposed algorithm almost do not depend on the size of benchmark circuit in compare to the method represented in SA-based proposed method, actually size of local relocation range and numbers of movable modules of each problem are the main parameters here. Also feasibility of local relocation solution, to guarantee the similarity of resultant placement with model placement depends on the existence of enough dead space near additional module so that the relocation rage becomes limited and small.


**Table 1.** MFA Local Relocation results for GSRC benchmarks
