PARALLEL GENETIC ALGORITHMS IMPLEMENTED ON TRANSPUTERS

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1 PARALLEL GENETIC ALGORITHMS IMPLEMENTED ON TRANSPUTERS Viktor Nìmec, Josef Schwarz Technical University of Brno Faculty of Engineering and Computer Science Department of Computer Science and Engineering CZ Brno, Bozetechova 2 nemecv@dcse.fee.vutbr.cz schwarz@dcse.fee.vutbr.cz Abstract: This paper deals with a course-grain parallel genetic algorithm for solving the placement and TSP problem as well. Both of them are typical tasks from the field of the layout design of VLSI circuits. It can be proved that both are NP problems. A TRANSGEN program was created in OCCAM language, which enables wide class experiments via extensive menu for setting main parameters of genetic optimization. The main attention was paid to the adaptation of main genetics operators including the migration operator usable for placement optimization. To test TRANSGEN algorithm some experiments were done. A few circuits were used to be illustrated the behaviour of the algorithm. From this experiments we have concluded that the achieved linear speedup of the parallel version of the genetic program is effected by a better covergence of the itaration process. Key words: optimalization, layout, TSP, parallel genetic algorithm, transputers, granularity INTRODUCTION Placement is a cruical task in the layout of VLSI circuits and PCBs (printed circuit boards). It has a great impact on the cost and routability. It can be characterizied by an assigment of circuit elements(modules) to locations on a chip or PCB (printed circuit boards). Let E={e 1,...e n } be a set of circuit elements and set of nets, S={s 1,...,s k }, where a net is a set of elements to be interconnect. A set of locations L={l 1,...l n } is given. The locations (slots) are organized upon the layout styles. To simplify the problem the set of slots forms a regular structure of locations with c columns and r rows with unit distance of neighbour location. The objective function is minimal total sum of wires, which realize the nets. To precise the problem the hypergraph is used for the representation of the circuit and the position graph is used for the slots representation. Many different techniques are used to solve the placement problem [1]. Well known among them are the simulated annealing and decomposition methods. The genetic algorithm seems to be a new powerful optimization approach, which is based on the idea of optimizing by simulating the natural process of evolution. From the nature of genetic algorithm flows the high suitability for the parallelization of the simple sequential genetic algorithm. We have developed PGA -a parallel genetic algorithm on transputers T800, T9000 using the basic group of four genetic operators - selection (reproduction), crossover, mutation and migration.

2 Two possible implementations are presented in the paper - fine-grain and coarse-grain one. Experiments showed that perfect domain decomposition is the most advantageous method for implementing such genetic algorithm. Our approach was tested on travelling salesman (TSP) and genetic placement (GP) problems. TRANSPUTER ENVIRONMENT 1. Processor description We will describe a processor architecture the genetic algorithms have been implemented on and parallel principles of our approach in this section. We have worked with transputers - parallel processors manufactured by INMOS company. The transputer is a microcomputer with its own local memory and with links for connecting one transputer to another. It is possible to execute unlimited number of parallel processes on one processor because of its internal architecture (in this case we will obtain a pseudoparallel structure). A transputer network can be built by interconnecting more processors (transputers) via links. Processes executed on different nodes of this network runs in parallel and can communicate between themselves using channels. The transputer is then classed among MIMD (multiple instruction streams over multiple data streams) architectures. A language OCCAM is used for programing the transputers. The language OCCAM allows user to write parallel programms easily. The language OCCAM contains construct PAR for executing parallel processes and two special operators? and! are used for communication between processes. We have used this language for implementing the parallel genetic algorithms. 2. Algorithm implementation A genetic algorithm in general has three main parts: initialization (creation of initial population - this step is made only once at the beginning of the algorithm), reproduction and evaluation (these two steps are repeated in a loop). There can be more approaches to parallelization of such algorithm. Fine-granularity approach exploits parallel aspects of genetic algorithms to maximum. Evaluation of individuals can be decomposed into a number of parallel modules equivalent to a size of a generation. Each module evaluates one individual. Reproduction can be done in separate modules as well. Each module receives two individuals (parents) as parameters then performs the crossover and it returns a new individual (offspring) as the result. According to the terminology introduced in [2] these modules are called homes. The set of homes represents the colony. So far we have described a standard genetic algorithm with some parallel improvements. The colony represents the whole generation. Such decomposition may improve the speed of the algorithm when implemented on parallel processors, but it doesn t improve the efficiency of the algorithm. Experiments showed that this approach is not so advantageous because of large amount of communication between all modules. When the size of colony is great the communication overhead significantly slowes down the computations. The parallelizing of evaluation and the reproduction is not very promising then. We present another simple but effective parallelization technique here. We have used the coarse - granularity approach which brings a new aspect to the decomposition of the genetic algorithms. The main idea consists in dividing the generation of individuals into more colonies which exchange genetic material in certain intervals. This parallelization technique avoids the communication with trivial parallelization schemes. The distributed placement procedure (colony) is in fact a basic sequential genetic algorithm. This algorithm evaluates and reproduces certain amount of individuals in sequential way. These procedures are mapped onto a transputer network and executed in parallel. The procedures form a ring topology which is deadlock free. A new genetic operator migration is introduced. Migration transfers genetic individuals from one environment to another. Implementation of a genetic algorithm as described above falls into the perfect domain decomposition category. Very little communication overhead and the balanced computation results in almost 100% efficiency. This was also proved by our experiments.

3 DETAILS OF PGA ON THE PLACEMENT PROBLEM 1. String encoding. Several representations of the different complexity were considered for the string/placement. Finally an extended encoding was used. A serial number is associated with each symbol (node of hypergraph/element of circuit) in the string, which represents the node allocation (number of slot) in the placement. Thus, symbols in the string are position independent. When a node is moved along the string the interpretation of the string remains unchanged. It allows simply to change the slot ordering.. 2. Population size and offspring size. The population size N p and offspring size N o determines processing time and result quality. The both parameters are provided by the user. The parameter N p and N o are considered to be from 5 to 300. From experimental observation [4], it was found, that the relative small population of about 30 individuals/configutations gives the reasonable performance. The suitable number of generated offsprings N o is about 30% of population size [3]. In case of grain coarse PGA the size of the total population is about 300 and it is distributed into 2,5 or 10 subpopulation. Each population is allocated to each process and the standard GA is performed on each process. 3. Fitness function. The fitness function used in the procedure Evaluation is based upon the classical objective function of the minimal total sum of wires. The length of wires is estimated by the minimal semiperimeter of rectangle enclosing elements of each net (HSP). It is easy to compute and it is often used as standard measure of the net complexity. It can be proved that the sum of semiperimeters equals the sum of mincut values used as a criterion in decomposition algorithms (Breuer mincut algorithm). In case of standard cell placement it would be necessary to extend the simple objective function with penalty subfunction to effect the overlapping of standard cells. Fitness function is based on the objective function. We compute the sum of costs (total wire lengths) at first. The fitness function are then computed as the division of the sum by each cost. The smaller the cost the higher is its fitness function. Suitable parents for crossover are then selected according to their fitness function in the reproduction phase of our algorithm. 4. Crossover operator. The traditional crossover operator which is used typically for the bit string is not suitable for the placement problem because it produces very often irregular configuration (conflict crossing). We used the powerfull conflictless operator called PMX [2] (Partially Map Crossover). Usually the parents are chosen from the population on the basis of their fitness. The fitter individuals have a higher propability of being selected. In TRANSGEN the choise of parents is supported by special routine which enables to specify the probability function use the concept of roullete wheel. Morover there are posibble to use a to cut srossover operator with the specifyied average distance D c between them. 5. Mutation. It is possible to use two mutations operators together or separately. Both of them are based upon the classical random exchange of nodes. The number of nodes is fixed to value of two, but can be expanded to value of three. The first mutation operator is used for parents mutation.the mutation rate R mp is defined as percentage of the total number of symbols in the population, which are mutated in each generation. For the n - node/element placement problem and for the size N c of the population nn c R mp /2 pairwise interchanges are performed. Typicaly average value R mp = The mutation rate is set dynamically using some special procedure which allows to scan the relief of the fitness function of the whole population and to control so

4 called the saturation depth. The second mutation operator is applied to group of offspring. The typical mutation rate R mo = 0.01 is fixed and can be only changed a the beginning of optimization process. 6. Migration. The migration operator enables to combine the algorithms running on different processors into a single distributed genetic algorithm. The migration rate R mig specifies the epoch length that separates the interchange of the best choromozones among all the processes. We used elitism concept so as the only such a chromozone is accepted from the other process that is better than the current one. The effectivness of the migration depends namely on the migration rate. It has been experimentally observed that the suitable range of migration rate lies between 0,2 and 0,05. Concurently it can be used another mechanism of the exchages which is activated in a case only when the accepted chromozone is better then the current one by a fixed value named threshold point. 7. Selection. After generating offspring the procedure Selection is activated to choose the future generation from joined set of parents and offspring. In the TRANSGEN algorithm a competetive strategy is implemented - all the parents and offspring compete each other and the N c fittest individuals survive and move into the next generation. 8. Stopping criterion The standard way for finishing the optimization is the limitation of the number of generations N g. The another possibility is the time limitation. 9. Experiments The Table 1. shows the resulted minimal value of the objective function with diferent number of processes (N proc ). The total population size is set to be the same in all cases. N g is the number of generations. In case of RL25 experiment N g value specifies the number of generations needed to find the global optimum. In the second experiment IC67 the global optimum is not known. We have observed the value of objective function (HSP) reached in the fixed number of generations. The PGA using one process represent the standard GA. RL25 known optimum = 45 IC67 N proc N p N g HSP N proc N p N g HSP Table1: Results of RL25 and IC67 experiments DETAILS OF GA FOR TSP problem Although a main effort has been concentrated on genetic placement algorithm (see the previous section) we have made experiments with travelling salesman problem as well to test the versatility of our approach. We will describe only the differencies between TSP and GP algorithm here because the main principles of implementing a parallel genetic algorithm have already been mentioned. We have used ordered crossover method [2] during the reproduction phase. Obviously, the position of a gen in a chromozone is not significant in this case. A more important attribute is the order of genes. The ordered

5 e l u a v Parallel genetic algorithm implemented on transputers - Viktor Nìmec, Josef Schwarz crossover seems to be the most suitable method because it tends to preserve the order of genes in a chromozone. The problem was to find the shortest path between 15 cities. The distances between each two were given in a matrix. The results of our experiments are summarised in the table 2. The column N proc specifies number of colonies used. There were 30 individuals in each colony. The notation 4 (2+2) means that there were four colonies from which two were mapped on the first processor node and another two on the second. The second column specifies the time needed to find a global solution (the global solution was known). The column solution specifies the number of generations in which the solution was find. N proc Time [s] N g 1 (1+0) (1+1) (2+0) (2+2) Table 2: Results of experiments with TSP problem. Due to the order crossover method used the results were very good. This is a typical example of perfect domain decomposition. This example demonstrates the benefit of decomposing the algorithm into two processor nodes (see 2nd and 3rd row of table 2) P S H Nproc = 1 Nproc = Generation no. Graph 1: Optimalization curve for the IC67 placement problem PERFORMANCE The graph 1 demonstrates benefits of parallel approach to genetic algorithms. The generation size is fixed in both cases. If 10 logical processes (colonies) are used the optimalization procedure converges faster than in case when 1 logical process is used. Obviously, the solution for more processes is equal or even better than for one logical process (which in fact represents the simple GA).

6 The most important advantage is the considerable speedup of computations when more processors are available. In the above mentioned case the speedup coefficient can be theoretically equal to 10. The communication overhead and context switching leads to slight decrease of this value. The overall slowdown also depends on the type of decomposition used. The main attribute of the perfect domain decomposition (used in our PGA algorithm) is the low amount of communication between logical processes. We have proved that the speedup is almost linear in proportion to number of processors used for computations. CONCLUSION This paper described a new distributed placement algorithm for the PCB layout with set of the discrete slots. A new approach was presented by using independent genetic modules colonies. The TRANSGEN program is developed in OCCAM language and implemented on the transputers T800 and T9000. The parallel algorithm PGA preserves the result quality of the single algorithm with achieved speedup. The speedup is almost linear in proportion to number of physical processors (transputers) used. The migration operator introduces a new important attribute in solving genetic algorithms. In some cases there are better convergence in parallel version PGA. It is obvious that the TRANSGEN is a promising new tool that need futher investigation namely in the refinning of mutation procedure and in the chromozone representation. It is worth to examine a dynamically setting of migration rate according the population size and the depth of the saturation in each proceess. Finaly it is possible to implement futher version of PGA for the standard cell layout and for other CAD problems. This aim can be accomplished using the block architecture of PGA. ACKNOWLEDGEMENT This work was done under the support of the grant agency GAÈR contract number 102/94/1096. REFERENCES [1] Schwarz J.: Design Automatization of Composition and Placement of Integrated Circuit. PhD. Theses, Technical University of Brno, Facultyof Engineering and Computer Science, Department of Computer Science and Engineering, Brno, 1993, 136 pp., in Czech. [2] Goldberg E.,D.: Genetic Algorithms in Search, Optimization, and Machine Learning, Addison- Wesley, 1989 [3] Shahookar K., Mazumder P.: VLSI Cell Placement Techniques, ACM Computing Survays, Vol. 23, No. 2, June 1991, pp [4] INMOS: Transputer databook, Prentice-hall, 1989 [5] Dvorak, Vaclav: Concurent algorithms for message passing parallel algorithms, Teaching material developed in TEMPUS joint european project no. 0449, 1994