Genetic Placement Benjamin Kopp Ece556 fall Introduction. Motivation. Genie Specification and Overview

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1 Genetic Placement Benjamin Kopp Ece556 fall 2004 Originally proposed by James P. Cohoon and William D Paris 1987 IEEE Introduction Genetic algorithms are a state space search similar in nature to simulated annealing. A population of solutions is maintained, which forms the gene-pool from which all future solutions are created. Each iteration of the algorithm new solutions are created by selecting two current solutions and mating them together in a process called crossover. These new solutions are then added to the population, and a likewise number of solutions are then killed off to maintain the original population size. Solutions are rated on their fitness, with the most-fit solutions having the highest chance of both mating, and surviving to the next generation. Therefore over many generations, the best fit solutions will tend to dominate the population. Additionally, in every generation there is a small percent of the population which randomly mutates, introducing completely new solutions. This mutation prevents a lack of diversity and an early convergence to a local optima. Genie is a genetic algorithm designed to solve the placement problem of VLSI chip layout. It is an adaptation of genetic algorithm techniques employed in the AI field, and was among the first attempts at using genetic algorithms for VLSI placement. Motivation I chose to research the Genie algorithm described in the paper because of an interest in applying real-world ideas and methodologies to computer and AI problems. For the project I ended up implementing 2 versions, Genie 1 and Genie 2. The two variations, as proposed by the authors, deviate only in the selection algorithm used. Genie 1 uses a purely random selector, while Genie 2 uses a selector proportional to the fitness of the solution. For all trial runs, Genie 1 provided me with superior scores, so it is the algorithm used for my final results. Genie Specification and Overview Pseudo Code Overview Population P = Initialize(); While (generations < 10,000) { Offspring = Crossover(pp, tp in P); P = Selection(P,O); k=normal(mean K u *pop_size*num_nets, dev 1); For(i=1 to k) Mutate(s in P); If(P[0].cost < best_cost) Generations = 0; Generations++; } 1

2 As can be seen in the pseudo code, the high-level algorithm is very straight forward. The difficulty with genetic algorithms comes in the parameters and the individual functions. Genie is no exception. The quality of ones final solution is highly dependent upon finely tuned parameters, and correctly tailoring the functions for the specific topology of an individual circuit. The Authors had 4 different versions of the final algorithm that they switched between depending on the structure of the problem. I will be discussing only the methods that I chose to implement in my project, which mimics the setup that most consistently provided the authors with the best results. Data Structures: Module: contains an x,y coordinate pair, and a list of the connected nets. Net: contains a weight and a cross-list of modules connected to it. Solution: contains both a grid and a list representation of all modules, and a fitness cost associated with the solution. While the authors only suggest using a spatial representation for a solution, also maintaining a list by module ordering greatly speeds up performing the selection and mutation functions, as well as calculating the fitness. In addition to the performance benefits, it also aids in ease of coding, and readability. Therefore I believe the added memory usage is justified. These solutions were stored in a vector, and were kept sorted. Maintaining the sorted list allowed for faster best, and worst, fitness calculations, which were used in all selection functions, the cost function, and as a measure of progress for the algorithm as a whole. Functions: Population Constructor- Initialize() In the final algorithm, the population is created with 25% of the solutions generated randomly, and 75% generated using the following algorithm. 1. An unplaced net (Z) is randomly selected 2. All unplaced modules on that net are placed sequentially. 3. Working back from the last module placed on that net, a module is located that is connected to an unplaced net. If one is found, that net is selected, and the process repeats at step 2. If one is not found, but free nets remain, go back to step 1. When no free nets remain, continue to step Fold the resulting string into a grid representation, in row-major order as shown in figure 1. Fig 1. cell placement ordering 2

3 Crossover Operator- CrossOver(pp, tp) The crossover operator works on two parent solutions, a passing parent pp, and a target parent tp. A k x k box is copied from pp to to the same location in a copy of tp. The problem is this method can produce results that are invalid for a placement problem. This will occur when the modules in the two squares are different, in which case the copy will produce duplicate cells in the child solution. This is solved by copying all of the modules located in tp, and not in pp (tp-pp), to the location of the modules in pp, and not in tp (pp-tp). This is shown in figure 2. The variable k is a random normal variable with mean 3, and standard deviation 1. The Authors never mention varying the size of the selection box, however, for problems as small as modules, the proposed size is too large. One crossover operation could potentially copy the entire passing parents solution. This creates a very quickly converging population, which tends to yeild an inferior solution. I attempted shrinking the size of this box, but the results were equally poor. I have therefore concluded that an alternative crossover operator should be used for small problem sizes. I believe that the other crossover operator that they discarded would work very well for these small problems. This alternative crossover function works virtually identical to the mutation function, except the parents are deterministically selected. Unfortunately, I have not yet had the opportunity to verify this belief. Passing Parent Copy of Target Parent G I B F J L K D C H K J L E A J L K Fig. 2. Crossover Operation 3

4 Fitness Calculation Cost(solution) The fitness function used for Genie is a combination of the half-perimeter bounding rectangle for each net, and a weighted measure of the excess horizontal and vertical channel usage. To calculate the excess channel usage, the cut set of every channel is determined, and the average horizontal and vertical usage is calculated. A channel s cost is then increased whenever it s usage is more than the standard deviation above the average channel usage. While implementing this fitness function, I found that for the channel usage to be non-negligible, all modules should be given a width of 1 unit. Even with this assumption, the channel cost was orders of magnitude less than the half-perimeter bounding box. Selector- Select(Pop, Offspring) The selector function is the one operator that varies between Genie 1 and Genie 2. Genie 2 uses a weighted probability based on the fitness of a solution, calculated as: Genie 1 uses a purely random function to select between all solutions available in both the current population, and the new offspring. While this randomness does not fit with the idea of survival of the fittest, the rest of Genie is a strongly directed evolution algorithm. In other words, it aggressively pursues the best possible solutions in the selection, the crossover and the mutation functions. This causes the diversity caused by the randomness of Genie1 to be very beneficial, and ended up providing me with the best final solution. The choice function, which selects two parents for crossover, implements a version of the Genie 1 selector. The only difference is the choice function further constrains the selection to a module with better than average fitness. I feel that this constraint may actually reduce ones chance of returning the global optimum, as it essentially discards half of the population. Mutation Function- Mutate(solution) The mutation function used by genie was designed to optimize the cost of one net in the solution. This net is first randomly chosen, and a random module on the net is then selected. The selected module becomes the base. The farthest module on the net from the base is then located and moved to an adjacent location, Preferably one rectilinearly adjacent (see fig. 3). In figure 3, A has been selected as base. Module D is the farthest away on net Z. In this example, location (1) is preferred because dy < dx. If (1) is occupied by a module also on Z, then (2) is preferred as it is still rectilinearly adjacent. If (2) is also occupied by a module on Z, then the module is placed in (3). 2 3 D A 1 fig. 3. adjacency preference 4

5 This operator seems to work very well for decreasing the cost of the solution. It is very rare that the resulting solution is worse than the original. However, I feel that this is not necessarily a good thing. The purpose of a mutation is to introduce completely new solution possibilities into the population. This will greatly increase the population diversity, and the hope is that a potentially poorly scoring solution will enable the population to exit a local optimum in a future crossover operation. Results Unfortunately, it is hard to draw many conclusions from the results of this project. Both Genie and Timber Wolf (TW) are algorithms that have a lot of run-time parameters, and I am certain that both algorithms could do better with parameter tuning. However, working off of the two 50- cell circuits, I attempted to even out the algorithms run time. In this way, we can get a better idea of the possible effectiveness of each algorithm. It should be further noted that while the Genie algorithm is fully implemented, TW is a pared down version of the complete algorithm. Table 1. has a summary of my results. All scores were calculated using the TW cost function. For a more complete Idea of how this algorithm performs, Table 2. Shows the results presented by the authors in the original paper. chip Genie Genie Score Time TW Score TW Time 10_ _ _ _ Table 1. Genie and TW best-cost and run-time Table 2. Authors results My biggest disappointment with Genie was the run time. I felt that it delivered very good solutions, but often ran well over 10 times longer than TW. The recommended termination point is when the best solution has not improved in 10,000 generations. Through experimentation, I found that for my chips, I could reduce that number to 1,000 generations without experiencing a noticeable worsening of the final solution. The times listed in table 1. were achieved using the 1,000 generation cutoff. Despite this, the small chips still take 10-times longer to run than TW. One aspect of Genie that I was very impressed with was the consistency of the solution. Multiple runs consistently produced results that varied by less than 10% of the average cost. TW, on the other hand, would occasionally return a very poor result that was 50% worse than the average score. This actually makes the run time picture for Genie look better, as it suggests that multiple runs should be made with TW in order to get a good solution, while it will only take one run with Genie. 5

6 Discussion After completing this project, I believe that there is a lot of room for improvement upon Genie. One very big issue that I have not yet touched upon is that Genie requires same sized modules. While it is certainly possible to modify the algorithm to accommodate variable width modules, the reliance on a fixed grid structure severely limits the quality of results that could be achieved. Since only one module can be in a grid location at a time, any variation from the predefined shape would result in wasted space, and a sub-optimal solution. Other issues mentioned earlier include a mutation operator that I believe is too deterministic to achieve the desired diversifying effect, and a crossover operator that does not lend well to small placement problems. Unfortunately, due to time constraints, I have been unable to test most of these conjectures, so I can not say if they would hold up under experimentation. I believe that genetic algorithms have many advantages over simulated annealing. The biggest one stems from maintaining an active population. This provides a search-history, which can be used to provide a more guided search. For instance, new solutions could be limited to something that is not currently in the population. The population also allows you to keep inferior solutions in the hope that the may combine to form a better path in the future, without destroying the best ones. Another advantage comes from crossover. This potentially allows for the best parts of multiple solutions to form together into one. However, as stated, genetic algorithms do tend to be heavily dependant on the settings of initial parameters, which make them harder to port from one situation to another. Also, they have considerably higher memory requirements than SA, as they must store many solutions simultaneously. Despite these few problems, I believe that the advantages are strong enough to keep genetic algorithms as a viable placement problem technique. References Genetic Placement James P. Cohoon, member, IEEE, and William D. Paris IEEE Transactions on Computer Aided Design, Vol. Cad-6, No. 6, November 1987 VLSI Cell Placement Techniques K. SHAHOOKAR AND P. MAZUMDER ACM Computing Surveys, Vol. 23, No. 2, June