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УДК 681.324
V.V. Miagklkh, A.P. Topchy, S.A. Chertkov GENETIC ALGORITHMS: SOME NEW FEATURES FOR PREMATURE CONVERGENCE AVOIDANCE
One of the major difficulties with Genetic Algorithms (GAs) (and In facf with most search algorithms) is that sometimes premature convergence, i.e. convergence to a suboptimal solution, occurs. This paper describes some new features in GA with local optimization/preferences aimed to premature convergence avoidance. Described approach was successfully applied to genetic solution of the well known traveling Salesman Problem [I] (TSP) and the Graph Coloring Problem [5].
In case of TSP, standard Greedy Crossover [1] very easily kills all 'bad1 changes which are produced by randomizing genetic operators. It was experimentally noted, that recombination of the best ranking individual with not so good one almost always produces an offspring, which is the same as the better of the parents. If place such an individual into the population, then it doesn't introduce anything new. Such 'good'
Материалы Всероссийской конференции “Интеллектуальные САПР-95”
solutions will result in ineffective usage of the population pool. There are some ways to change the situation: change crossover or doesn't place the offspring. One of possible measures to be undertaken is acceptance of the more expensive path to the cheaper with definite probability or taking about fifty-fifty of the parental representations into the offspring. Random picking of a town in case of deadlock or choice of the closest may be optional. The fifty-fifty approach may result in the following crossover rule, which is oriented to application in TSP defined upon the undirected graphs: To find a random town in the first parent, that becomes the current town. Current parent is the first parent. Body of the crossover: decide, which of two adjacent towns (adjacent towns means locatefi in the position to the left and to the right from the current town) in the current parent is closer to the current town. -The closer, if not visited, becomes the current town\ If only one of the adjacent towns is not visited then it will become the current town. If both adjacent towns were visited, then choose the closest from the number of remaining unattended. Current parent is another parent. Repeat the body until all the towns are visited. Offspring is forming from the succession of the towns in order they were becoming current towns.
Decrease of the crossover greediness will have good effect against the lack of variety in the population if there is a good income of new changes, produced by random factors or additional heuristics. After reaching deep local minima, there is usually a lot of the same tours. Authors decided to construct special operators for striving against it directly. Hence, after determining of being in a suboptimal, the special algorithm, changing a number of individual m the population, is activated. The fact of being in a local (global) minimum was concluded in our case when the N-first individuals have the same length of tour. We called this N an elite number. We assume, that in TSPs with a large number of towns, the equal length indicates the same tour configuration. There are some other ways to determine this, like not improvement of the length after K trails. When suboptimal is determined one of the following operators may be performed: 1. Packing: Of all the tours, having the same length, only one remains unchanged. Under all the others the Warp Operators are performed. 2. JudgmentDay: Only the best remains unchanged. Under all the others the Warp Operators are performed. Warp Operators may be full or partial randomization.
The application was tested on a number of TSP benchmarks - Oliver's 30 [2], bilon's 50 and Eilon's 75 (both taken from [3]). Thirty towns problem was solved the most easily. Being good tuned (Population Size=20, mutation rate = 0.3, elite number=10), the GA reached the optimum every time run. In average (100 runs) GA reaches optimum solution after about 5000 genetic operators performed. 50 and 75 towns TSPs seem to be harder for this GA. There are still many local minima which it strives with difficulties. For more complicated TSPs the combination of proposed modification with other measures against premature convergence like r-opt operators may be useful. It is also very interesting to try this features in multi-populational (parallel) GAs, which populations are formed with help of different approaches using various migration schemes.
One more good feature of described TSP solving technique is that this GA is very effective with small population sizes (about 20) even with TSPs of large number of towns. Authors consider it to be a consequence of very efficient usage of the population pool, due to the regular 'purges' produced by proposed operators. We also concluded, that these best tours are the result of both crossover modification and operators, working with whole population, because one without another works not so well. TSP benchmark results proved good effectiveness of the proposed approach.
Results in TSP encourages to try this approach in genetic solution of other problems in Computer Aided Design [4] and in different domains of Artificial Intelligence. Nowadays we applied it to the solution of the Graph Coloring Problem. It was built on the similar two principles: 1. Greedy crossover with moderate greediness. 2. Randomizing operators working with whole population, which are two generalized points of proposed approach called social disasters technique. We tested the method on
Davis's 100 nodes graph benchmark [5]. The approach yielded 5.3"» the best solution fitness improvement under the rest conditions Remained unchanged. In both cases greedy (local improvement) crossovers were used.
REFERENCES
1 Grefenstette J et al, 1985, 'Genetic algorithms for the traveling salesman problem', Proc. Intern. Conf. of Genetic Algorithms and their applications. Pp: 160-165.
2 Oliver I et al, 1987, 'A study of permutation crossover operators on the traveling salesman problem', Proc. of the Second Int. Conf.'on Genetic Algorithms, Pp: 224-230.
3 Whitley et al, 1989, ’Scheduling Problems and Traveling Salesman: The Genetic Edge Recombination Operator', Proc. of the Third Int. Conf. on Genetic Algorithms. Pp: 133-140.
4 Goodman E et al, 1994, 'A Genetic Algorithm Approach to Compaction, Bin Packing, and Nesting Problems', Tech. report #940702. Case center for computer-aided engineering and manufacturing. Michigan State University.
5 Davis L, 1991, 'Handbook of genetic algorithms', Van Nostrand Reinhold. New York.
УДК 658.512
С.Н. Щеглов Применение генетических алгоритмов для решения задачи назначения слоев в базовых матричных кристаллах
Введение
В последнее время возникает все больше новых высокоэффективных методов, позволяющих разрабатывать СБИС большой степени интеграции, требующих высококачественного топологического решения. В связи с этим происходит объединение задач новых концепций с казалось бы глубоко исследованными и изученными подходами автоматизации проектирования ИС. К такому типу можно отнести задачу распределения соединений по слоям.
Данная задача возникает при разработке топологии многослойных схем для распределения "конфликтующих" соединений по отдельным слоям схемы. Самой общей целью при ее решении является наиболее эффективное использование площади КП при одновременной оптимизации таких конструктивных параметров схемы, как число слоев, количество межслойных переходов, процент реализованных соединений.
I. Постановка задачи
Известно, что базовый матричный кристалл (БМК) - это компактный модуль с высокой степенью интеграции, служащий для расположения нескольких сотен кристаллов и их соединения несколькими тысячами цепей. Верхний слой, содержащий кристаллы, называется слоем кристаллов. Под слоем кристаллов находится группа слоев, на которых распределены контакты. Ниже слоя распределения контактов находятся слои расположения цепей. Обычно это парные сдои, и каждая пара называется Х-У пара-плоскость. Х-плоскость имеет проводящие каналы только в Х-направлении, а У-плоскость имеет проводящие каналы только в У-направлении. Проводники используются для межсоединения распределенных контактов кристаллов и размещаются соответственно на паре Х-У
