Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
International Conference on Futuristic Trends in Computing and Communication (ICFTCC-2015) PSO Algorithm with Self Tuned Parameter for Efficient Routing in VLSI Design Sudipta Ghosh1, Subhrapratim Nath2, Subir Kumar Sarkar3 1 Assistant Prof., Dept. of Electronics & Communication Engineering, Meghnad Saha institute of Technology, Kolkata, India 2 Assistant Prof., Dept. of Computer Science & Engineering, Meghnad Saha institute of Technology, Kolkata, India 3 Professor, Dept. of Electronics & Telecommunication Engineering, Jadavpur University, Kolkata, India. Abstract— Device size is scaled down to a large extent with the rapid advancement of VLSI technology. Consequently this has become a challenging area of research to minimize the interconnect length, which is a part of VLSI physical layer design. VLSI routing is broadly classified into 2 categories: Global routing and detailed routing. The Rectilinear Steiner Minimal Tree (RMST) problem is one of the fundamental problems in Global routing arena. Introduction of the metaheuristic algorithms, like particle swarm optimization, for solving RMST problem in global routing optimization field achieved a magnificent success in wire length minimization in VLSI technology. In this paper, we propose a modified version of PSO algorithm which exhibits better performance in optimization of RMST problem in VLSI global routing. The modification is applied in PSO algorithm for controlling acceleration coefficient variables by incorporating a self tuned mechanism along with usual optimization variables, resulting in high convergence rate fot finding the best solution in search space. Keywords— Global routing, RSMT, Meta-heuristic, PSO. I. INTRODUCTION Advancement in IC process technology in nano-meter regime leads to the fabrication of billions of transistors in a single chip. The number of transistors per die will still grow drastically in near future, which increases complexity and thereby imposes enormous challenges in VLSI for physical layer design, especially in routing. In order to handle this complexity, global routing followed by detailed routing is adopted. The primary objectives of global routing in wire length reduction, is becoming very crucial in modern chip design. The only way to minimize the length of interconnects in VLSI physical layer design technology is to address the problem of Rectilinear Steiner Minimal Tree [1]. To solve this NP complete problem, meta-heuristic algorithms [2] like particle swarm optimization is adopted. It is a robust optimization technique, introduced in 1995 by Eberhert and Kenedy [3]. The PSO approach in VLSI routing is first implemented by the authors Dong et al. at 2009. Various improvements over original PSO algorithm have been made to make this algorithm more efficient. The introduction of linearly decreasing inertia weight by Shi and Eberhart [4] increases the convergence rate of the algorithm. Innovation of a self-adaptive inertia weight function [5] enhances the convergence rate of the PSO algorithm for multi-dimensional problem. This paper proposes further improvement of the existing PSO algorithm which modifies the acceleration coefficients of PSO algorithm yielding accurate results and ISSN: 2348 – 8549 thereby establishing better and efficient exploration and exploitation in the search space. This motivates us to apply the algorithm suitably to address the RMST problem in minimization of interconnect length in the field of VLSI routing optimization. The paper is organized as follows. In section II brief outline of PSO algorithm is given. In section III proposed algorithm of PSO is described in steps followed by experiments and results in section IV. Finally the paper concludes with section V. II. BASIC PSO ALGORITHM An PSO is a kind of evolutionary computation technique. More specifically it is a meta-heuristic algorithm, derived from the collective intelligence exhibited by swarm of insects, schools of fishes or flock of birds etc, implemented to revolve many kinds of optimization problems. The basic PSO model [3] consists of a swarm ‘S’ containing ‘n’ particle (S=1, 2, 3…., n) in a ‘D’ – dimensional solution space. Each of the particles, having both position ‘Xi’ and velocity vector ‘Vi’ of dimension ‘D’ respectively, are given by, Xi = (x1, x2,…..xn) ; Vi = (v1, v2, ……vn); The variable ‘i’ stands for the i-th particle. Position ‘Xi’ represents a possible solution to the optimization problem. Velocity vector ‘Vi’ represents the change of rate of position of the i-th particle in the next iteration. A particle updates its position and velocity through these two equations (1) and (2): V t+ 1 = Vt + c1r1 *(p i – X t) + c2r2 *(p g – X t) ...…(1) Xt+ 1= V t +1 + X t ……………………..………...(2) Here the constants ‘c1’ and ‘c2’ are responsible for the influence of the individual particle’s own knowledge (c1) and that of the group (c2), both usually initialized to 2. The variables ‘r 1’ and ‘r2’ are uniformly distributed random numbers defined by some upper limit, ‘r max’, that is a parameter of the algorithm [6]. ‘p i’ and ‘p g’ are the particle’s previous best position and the group’s previous best position. ‘Xt’ is the current position for the dimension considered .The particles are directed towards the previously known best points in the search space. The balance between the movement towards the local best and that towards the global best, i.e. between exploration and exploitation, is considered in the above equation by Shi and Eberhart in generating acceptable solution in the path of particles. Therefore the inertia weight ‘w’ is introduced [4] in equation (3) as follows : www.internationaljournalssrg.org Page 60 International Conference on Futuristic Trends in Computing and Communication (ICFTCC-2015) for the first time. The pseudo code for the algorithm is given below. V t+ 1 = w*Vt + c1r1 *(p i – X t) + c2r2 *(p g – X t) .…(3) III. PAGE CONTROLLING PARAMETER OF PSO FOR OPTIMIZATION IN VLSI GLOBAL ROUTING While optimizing RSMT problem for wire length minimization in VLSI physical layer design with the help of particle swarm optimization algorithm, we have considered controlling of parameters to facilitate maximum convergence and prevent an explosion of swarm. The following parameters in PSO algorithm are controlled. D. Pseudo Code - - A. Selection of max velocity Larger upsurge or diminution of particle velocities leads to divergence of swarm due to un-inhibited increase the magnitude of particle velocity, |Vi,j(t+1)|,especially for enormous search space. The max velocity is limited by : Vi,j(t+1) = (Xj(max) - Xj(min)) / k where Vi,j(t+1) is the velocity for next iteration. Xj(max) and Xj (min) is the max and min position value respectively, found so far by the particles in the j-th dimension and K is a user defined parameter that controls the particles’ steps in each dimension of the search space with K=2 [7]. B. Inertia weight The dimension size of the search space affects the performance of PSO. In complex high dimensional condition, basic PSO algorithm constricted into local optima, leading premature convergence. Hence large inertia weight is required. Smaller inertia weight is utilized for small dimension size of search space to strengthen local search capability, assuring high rate of convergence [8]. A self adaptive inertia weight function is used in our algorithm, which relates inertia weight, fitness, swarm size and dimension size of the search space. W = [3 − exp ( − S / 200) + (R / 8 * D) 2] −1 where S is the swarm size, D is the dimension size and R is the fitness rank of the given particles [5]. In our algorithm D is taken as 100. C. Proposed Modification: Self-tuned Acceleration coefficient The acceleration coefficient determines the scaled distribution of random cognitive component vector and social component vector i.e. the movement of each particle towards its individual and global best position respectively. Using smaller value limits the movement of the particle, while using larger for the coefficient may cause the particle to diverge. For c1 = c2 > 0, particles are attracted to the average of ‘pbest’ i.e. particles’ local best position and ‘gbest’ i.e. particles’ global best position value. A good starting point proposed to be c 1= c2 =2 is used in algorithm [9], [10] for acceleration constant. Again c1= c2 = 1.49 generates good results for convergence as proposed Shi and Eberhart [11], [12]. In our proposed modification, we have introduced self tuned acceleration coefficients, linearly decreasing over time in the range of 2 to 1.4. This is introduced in VLSI routing optimization problem ISSN: 2348 – 8549 - Preprocess Block The search space for the problem is defined for a fixed dimension. Within the search space the user defined terminal node in form of coordinates are represented as ‘1’ to form the required matrix. The weight of the path between the nodes, including the Steiner nodes, is calculated to form the objective functions for n-particles. Cost of each objective function is calculated by Prim’s algorithm. Minimal cost along with the corresponding objective function is identified amongst these results. PSO Block Initialization for First Iteration - Each of the objective function is assigned for position and velocity vector - Each objective function represents local-best for each particle of ‘n’ population. - The objective function with minimal cost is assigned as global-best for the swarm of particles. Execution of the PSO Block While (termination criteria is not met) - Velocity and position is updated for ‘n’ number of population using the equation no (3) and (2). - Fitness values (present pi) of ‘n’ particles are evaluated by Prim’s algorithm. - Each pi is compared with previous pi, - If pi(present) > pi(previous), then pbest pi(present); Else retain the previous value. - New pg minimal of all pi s (present). - If pg(present) > pg(previous), then gbest pg(present); Else retain the previous value. Controlling inertia constant(w) - Calculate the self-adaptive inertia weight of each particle within the pop using equation no (3) in terms of fitness, population (swarm size) and dimension of the search space. Controlling acceleration constant (c1, c2) - Calculate self-tuned linearly decreasing acceleration constant. Check the termination criteria. If satisfied, stop execution and get the global best value for the swarm size. Otherwise execute the PSO block again www.internationaljournalssrg.org Page 61 International Conference on Futuristic Trends in Computing and Communication (ICFTCC-2015) IV. EXPERIMENTS AND RESULTS 3 sets of co-ordinates for 10 terminal nodes are randomly generated for connections in the defined search space of 100x100. The experiment executed 25 times for each set. The population size of swarm is taken 200. The maximum no of iteration is set as 100. Experimental co-ordinates are shown in table-1. TABLE I PSO with Self tuned c1,c2 224 253.2 Experiment No. 3 PSO with c1=c2=2 219 223.31 PSO with Self tuned c1,c2 205 218.7 COORDINATE SET OF 10 TERMINAL NODES FOR 3 EXPERIMENTS . NO. 1 2 3 4 5 6 7 8 9 10 Coordinate Set 1 X 09 17 31 20 40 49 56 72 88 93 Y 53 01 38 89 67 95 70 29 63 100 Coordinate Set 2 X 84 93 60 76 64 17 95 54 21 29 Y 15 67 48 89 79 62 38 35 57 81 The comparison of the performances of existing PSO and self-tuned PSO algorithm are shown in the bar charts. As our proposed algorithm generates lower global best value as shown in figure 1, it implies that the cost of the Rectilinear Steiner Minimal Tree (RSMT), constructed by interconnecting the terminal nodes, has been reduced. The ‘mean value’ of the global best parameter i.e. average of the minimum cost has also been improved as shown in figure 2. So this algorithm can effectively handle the RSMT problem of graphs and thereby reduce the interconnect length to a great extent. Minimum 'gbest' value 350 308 Coordinate Set 3 X 44 42 97 61 89 91 80 69 31 56 Y 63 99 36 25 68 51 28 58 82 94 We first performed the experiment for PSO algorithm with acceleration coefficient taken as c1=c2=2 and then the experiment is again performed for the proposed PSO algorithm with self tuned acceleration coefficient for the same three coordinate sets. The result is tabulated in table-2. From the comparative analysis it is observed that our proposed PSO algorithm generates lower ‘gbest’ i.e. global best value compared to the previous algorithm [9],[10]. It is also seen from the results that the ‘mean value’ is also improved for our proposed algorithm with self tuned linearly decreasing acceleration coefficient than the existing PSO algorithm [9], [10]. TABLE II EXPERIMENTAL RESULTS . Minimum ‘gbest’ Value 300 Exp. No. 1 Exp. No. 2 PSO with c1 = c2 = 2 Exp. No. 3 PSO with self tuned c1 , c2 Fig. 1. Comparison of Minimum Cost obtained by Existing and Modified PSO algorithm Mean 'gbest' value 319.81 307.5 300 319.81 200 PSO with Self tuned c1,c2 295 307.5 150 ISSN: 2348 – 8549 205 150 308 Experiment No. 2 219 200 PSO with c1=c2=2 248 224 264.67 253.2 250 Experiment No. 1 PSO with c1=c2=2 248 250 350 Mean ‘gbest’ Value 295 223.31 218.7 Exp. No. 1 PSO with c1 = c2 = 2 Exp. No. 2 Exp. No. 3 PSO with self tuned c1, c2 264.67 Fig.2. Comparison of Mean Cost obtained by Existing and Modified PSO algorithm www.internationaljournalssrg.org Page 62 International Conference on Futuristic Trends in Computing and Communication (ICFTCC-2015) V. CONCLUSIONS Wire length minimization in VLSI technology can be achieved through global routing optimization using PSO algorithm. In our proposed algorithm a modification is incorporated to the existing PSO algorithm. The technique used here is to modify the acceleration coefficients in such a way that it can tune itself over the iteration process throughout the experiment. The experimental results exhibits a clear difference in the performance of the modified version from the existing one. It is seen that the convergence rate is high and also it can be applied for a large search space and having still good result which clearly establish the robustness and stability of the optimization algorithm. Therefore this algorithm can effectively be used in global routing optimization in VLSI physical layer design. The further scope of the work can be extended to examine the algorithm in obstacle avoiding routing environment. REFERENCES [1] J.-M. Ho, G. Vijayan, and C.K. Wong, “New algorithms for the rectilinear Steiner tree problem”, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 1990, pp. 185-193. [2] Xin-She Yang, Nature-Inspired Metaheuristic Algorithms. Luniver Press, UK, 2008. ISSN: 2348 – 8549 [3] R.C.Eberhart and J.Kennedy, “A new optimizer using particles swarm theory”, Proeedings of Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, 1995, pp. 39-43. [4] Y.H. Shi and R.C.Eberhart, “Empirical study of particle swarm optimization”, Proceedings of IEEE Congress on Evolutionary Computation, Washington DC, 1999, pp. 1945-1950. [5] Dong Chen, Wang Gaofeng, Chen Zhenyi and Yu Zuqiang, “A Method of Self-Adaptive Inertia Weight For PSO”, Proceedings of IEEE International Conference on Computer Science and Software Engineering, Dec 2008, vol. 1, pp. 1195-1198 [6] A.Carlisle and G.Dozier, “An off-the-shelf PSO”, Proceedings of the Workshop on Particle Swarm Optimization, Indianapolis, IN.2001 [7] A. Rezaee Jordehi, and J. Jasni, “Parameter selection in particle swarm optimization: a survey”, Journal of Experimental & Theoretical Artificial Intelligence, 2013, 25(4). pp. 527-542. [8] F.Van Den Bergh and A.P.Engelbrecht, “Effects of swarm size on cooperative particle swarm optimizers”, Proceedings of the Genetic and Evolutionary Computation Conference, San Francisco, California, 2001, pp.892-899. [9] R. Eberhart, Y. Shi, and J. Kennedy, Swarm Intelligence. San Mateo, CA: Morgan Kaufmann, 2001. [10] J. Kennedy and R. Mendes. Population structure and particle swarm performance. In Proceedings of the IEEE Congress on Evolutionary Computation , 2002, vol 2, pp. 1671–1676. [11] Y.H. Shi and R.C.Eberhart, “A modified particle swarm optimizer”, Proceedings of IEEE World Congress on Computational Intelligence, 1998, pp. 69-73. [12] Rania Hassan, Babak Cohanim, Olivier de Weck, and Gerhard Venter, “A Comparison of Particle Swarm Optimization and the Genetic Algorithm”, American Institute of Aeronautics and Astronautics journal, 2005, 2055-1897. www.internationaljournalssrg.org Page 63