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A GENETIC ALGORITHM FOR VLSI PHYSICAL DESIGN AUTOMATION

Volker Schnecke, Oliver Vornberger University of Osnabruck, Dept. of Math. /Computer Science D{49069 Osnabruck, Germany

>ABSTRACT

Solving discrete optimization problems with genetic algorithms is in many aspects di erent from the solution of continuous problems. The blindness of the algorithm during the search in the space of encodings must be abandoned, because this space is discrete and the search has to reach feasible points after the application of the gentic operators. This can be achieved by the use of a problem speci c genotype encoding, and hybrid, knowledge based techniques, which support the algorithm during the creation of the initial individuals and the following optimization process. In this paper a genetic algorithm for the layout generation of VLSI-chips is presented, which optimizes two, usually consecutively solved tasks simultaneously: together with the placement of the modules, the routes for the interconnection nets are optimized.

after crossing, which reduces the parent to o spring correlation. For solving discrete real-world optimization problems, there has to be an application speci c genotype encoding and `intelligent' operators, which only create admissible individuals. During the application of these operators, problem speci c knowledge can be used to generate high quality o spring. Here | in contrast to function optimization for example | bad genes, which would never be a building-block in a global optimal solution, could be recognized. The designer of a genetic algorithm can take care that these bad genes are not included in the population by hill-climbing strategies, which could be integrated in the construction of the initial individuals or during the application of the operators. In opposite to this, one could not know the good genes, i. e. the building-blocks which construct the optimal solution. Therefore, the designer has to support the genetic algorithm by presenting a pool of good genes. From this pool the algorithm can compose some good and (hopefully) eventually the optimal solution to the given optimization problem. In the following, after a short description of the physical design process of VLSI-chips, a problem speci c genetic algorithm for the layout generation is described. This approach takes into account the previous mentioned items by covering the following features: a problem speci c genotype representation a hybrid approach for the creation of the initial population problem speci c, `intelligent' operators multiple gene representation in a single individual

INTRODUCTION

One of the main feature of a genetic algorithm applied to an optimization problem is the fact, that it does not deal with the problem itself, but with encodings of solutions for this problem. Thus the genetic algorithm explores the space of these encodings rather than the solution space itself. For continuous parameter optimization problems, both spaces are identically. A straight-forward genotype encoding in this case is a string of genes, which are simple oats. Each gene represents an element of the vector de ning a point in the solution space. The standard mutation operator randomly modi es single genes, and crossover is done by direct merging of two gene strings, which results in two o spring. All o spring represent correct encodings and these encodings dene admissible solutions to the given optimization problem, because of the one-to-one (genotype to phenotype) mapping between both spaces. For discrete problems with string type genotype encoding, not every possible string represents a correct solution. It is even worse that simple crossing-over of two individuals does not necessarily lead to a correct o spring. There some repairment has to be done

This work is being supported in the BMBF-project `HYBRID{Applications of Parallel Genetic Algorithms for Combinatorial Optimization'.

LAYOUT GENERATION

Modern VLSI (very large scale integrated) microchips contain some million transistors. The design cycle for these chips consists of di erent serial steps (e. g. system speci cation, functional design, logic design, circuit design, physical design) 8]. The 1

physical design outlines the transformation of a cir-

Due to its complexity, the physical design is usually divided into various, consecutive sub-steps: The circuit has to be partitioned to get a number of modules (macro cells) which have to be placed on the chip ( oorplanning). During placement there has to be enough space reserved to ensure the completion of all interconnections later on. In the routing phase, pins on the border of the modules have to be connected. This is done in two steps: In the global routing the `loose' routes are determined, while in the detailed routing the exact routes for the nets in each channel between two modules are computed. The last step in the physical design is the compaction of the layout, where it is compressed in all dimensions so that the total area is reduced. The algorithm described in this paper combines the routing with the placement process during layout generation. For a more detailed description of the usual phases and possible solution methods in contrast to the approach described in this paper see 7].

height

cuit description (which is the result of the preceding circuit design process) into the layout of a chip, which is needed for the following fabrication step 3, 5]. The layout includes the geometric description of the circuit components and the information for the routes of the interconnections between them. It also has pads positioned on its borders for the I/Oconnections of the chip. The objectives in layout generation are to minimize the area of the circumscribing rectangle and to produce a routing with short wirelengths, especially for some critical nets.

1 7 7 2 6

4 5

{ 1, 4 } {7} { 7, 2, 6 } {5}

width

Figure 1: The input for a xed cell (top)

and a exible cell with the shapefunction (bottom)

A GENETIC ALGORITHM FOR LAYOUT GENERATION

A layout is described by the positions of the modules, the chosen implementation for the exible modules and the routes of the interconnection nets on the layout-surface between the cells. Such a complex phenotype can not directly be represented by a gene-string of elementary data-types. A feasible way to characterize the placement of the modules is a binary slicing tree. This tree is the problem speci c genotype encoding for the layout optimization (cf. g. 2).

THE INPUT

The input to the layout generation process for macrocell layouts is a set of modules, which are rectangular blocks. Each module represents a functional unit which consists of hierarchically arranged subcells. There are two kinds of modules (cf. g. 1): A xed module has xed dimensions with exact terminal positions for the interconnection nets on its borders ( g. 1, top). A exible module can have various implementations with di erent aspect ratios, which are de ned by a shape-function. This is a stepfunction which is characterized by a set of minimal width/height combinations ( g. 1, bottom). For the interconnection nets of a exible module, only a list of terminals for each side is given but no exact terminal positions, because these vary with the di erent implementations. 2

Figure 2: The genotype representation

The structure of the tree xes the relative placement of the single blocks (modules) which are represented by its leaves. Each inner node represents a metablock, which de nes the arrangement for the set of blocks characterized by the leaves of the corresponding sub-tree and information about the routing inside this partial layout. All possible implementations for exible blocks are taken into account by storing shape-functions for all nodes in the tree so that a single individual represents di erent layouts, if some blocks are exible. When combining two blocks to a meta-block, the arrangement and the rotation of the composed blocks are xed. Hence, the shapefunctions of both blocks can be added which results in a shape-function for the meta-block. The number of implementations for the meta-blocks in the higher levels of the tree does not grow exponentially with their height in the tree because there are redundant implementations 7]. For the example shown in gure 3, two exible blocks with three and two implementations are positioned upon. The shape-function for the resulting meta-block has only two di erent (non redundant) implementations. An upper bound for the routing space inside the meta-block is computed and added to its shape-function.

to create building-blocks which de ne high quality partial layouts. For that, a complete graph is constructed: the nodes represent the blocks, and each edge is weighted with a value which de nes the quality of a meta-block consisting of the two blocks characterized by the adjacent nodes. A matching in this graph is a set of disjunct node pairs and the maximum weighted matching is the matching with the maximal sum of edge weights (cf. g. 4). The quality of a meta-block is marked out by the number of common nets of the combined blocks.

100

65 80 100 75

60

(a)

(b)

(c)

80 + 75 = 155

60 + 100 = 160

100 + 65 = 165

Figure 4: A matching graph for four

blocks, the three possible matchings and the maximum weighted matching (c)

Figure 3: The combination of two exible

blocks to a meta-block and the addition of routing space

HYBRID CREATION OF INDIVIDUALS

The slicing-tree for an individual of the initial population is composed in a bottom-up fashion. A special heuristic { the iterated matching 2] { is used 3

The matching process is iterated for each level of the tree until the root node is computed. In the second iteration, meta-blocks which consist of two blocks are paired, in the third iteration meta-blocks with four blocks are combined, and so on. This heuristic places highly connected blocks together and so reduces the overall wirelength and the total area of the layout. Because the iterated matching is a deterministic process, care has to be taken to create various individuals. For that, randomness is included in the computation of the edge weights for some of the used matching graphs.

THE INTEGRATED ROUTING

Because of the hierarchical construction of the in-

90

Figure 6: Fixing a global route by a topdown traversal (left) for the pseudo-optimal way (right)

Figure 5: The construction of the detailed

routing when combining a metablock

dividuals and the binarity of the tree, the detailed routing on the layout can be computed during the placement of the blocks. Note that a meta-block is considered to be a xed unit in the higher levels of the tree. When combining two blocks, all routing inside the resulting meta-block is done (cf. g. 5). Terminals on the outer borders are passed on as terminals to the outer border of the meta-block. Terminals in the channel and on the outer sides of the blocks are connected, if they are shared by a common net. Nets inside the channel which could not be connected or have to be connected to more terminals than only those contained in this meta-block are passed on as terminals to one of the borders which are adjacent to the channel. For these nets, the direction of the way out of the channel is determined by a top-down traversal of the tree following the bottomup construction phase. During this phase there is a check for each net, whether it is included inside of one of the partial layouts which are joined in an inner node. In this case, the net has to cross the channel and therefore a pseudo-optimal way for this net out of the channels in both sons of this node can be xed (cf. g. 6). This method achieves that the nets follow the hierarchy of the cuts de ned by the slicing tree structure. This way is often not optimal, but during the optimization process, the optimal structure of the slicing tree | regarding layout area and routing | is computed. In comparison to former experiments, where 4

the global routes were chosen randomly when combining the meta-blocks, here the total wirelength is considerably reduced. When choosing random routes for all nets out of the channels, many nets are routed to the outer border of the layout and have to be connected after composing the root node. Of course it could be possible to do the optimization of these routes by the genetic algorithm later on, but if one is able to compute (nearly) optimal routes `by hand', there is no need for passing this work to the genetic algorithm.

MULTIPLE GENES

As mentioned before, all resulting implementations for the meta-blocks containing exible blocks are stored. Storing all alternatives is useful because one could not decide in the lower levels of the tree, which implementation of a meta-block would be the best to minimize the overall area of the layout. Due to the simple adding of shape-functions and the binarity of the tree, the optimal sizing of the exible modules to reach an optimal layout (optimal for the special placement) can directly be determined by a top-down traversal of the tree after xing the best implementation for the root node 7]. This technique can be characterized as storing multiple genes in some locations of the genotype. It is a very good example for an opportunity the designer of a gentic algorithm has to increase its performance without directly guiding the search: when two blocks are combined, good combinations can be determined and bad combinations can be eliminated. But which of these good combinations are the best to be composed to a global optimal solution? Just let the genetic algorithm decide by making available a pool of good building-blocks!

THE OPTIMIZATION PROCESS

After the construction of the initial individuals, which already contain a lot of good components, the genetic algorithm starts the optimization by modifying individuals (mutation), and by combining building-blocks (crossover). Beside a mutation operator for changing the arrangement of two blocks inside a meta-block, the main mutation operators modify the slicing tree (cf. g. 7). One operator exchanges blocks or meta-blocks, which corresponds to exchanging cells or partial layouts on the layout surface. The other important mutation operator changes the structure of the tree by randomly picking out a subtree and inserting it into the tree at a di erent position, which corresponds to moving cells or partial layouts on the layout surface. Here storing all important implementations for the metablocks once more enhances the performance of the genetic algorithm, because for a moved partial layout a di erent implementation might be better in its new environment.

genes from the parents) is smaller than for problems where crossing-over directly leads to a correct individual. During the adding of the missing blocks into the tree of the o spring, the iterated matching is used once more to construct new (good) buildingblocks. It has turned out to be not helpful to do some `intelligent' crossover, i. e. looking for large disjunct subtrees, for example. Research is carried out to include a recombination operator for gene pool recombination. Here an o spring is constructed by combining building-blocks out of a pool of good partial solutions. When designing a genetic algorithm for a speci c problem, it is very important that a global optimum can be reached starting from any set of individuals by the application of the genetic operators. For an algorithm with tree-structured genotype encoding this means, if recombination only combines real subtrees with at least two leaves, then each pair of leaves contained in the encoding of an optimal solution must already exist in one of the initial individuals, or must be able to be constructed by mutation. Otherwise the genetic algorithm will never reach the optimum.

Figure 7: Mutation by exchaning blocks

(top) and by changing the structure of slicing tree (bottom)

The implementation of the crossover operator is straight-forward: Two individuals are randomly chosen to produce one o spring. Two disjunct subtrees are searched in the parents which are composed to a subtree in the o spring. Because these subtrees usually do not build a complete layout, a third part has to be added to the layout of the resulting o spring. Due to this, the heritability (number of transmitted 5

Figure 8: A layout with 33 macro-cells

RESULTS

The algorithm has been tested on real-world circuits with 10 to 49 modules and up to 500 nets. Figure 8 presents a layout for a circuit with 33 xed modules and 123 nets. For a direct comparison to commercial placement and routing tools, an e cient channel routing algorithm has to be implemented. In the current version, a very simple routing strategy is applied when combining two blocks: for each net in the channel, one special track is added which results in an excessive routing space. Apart from the quantitative comparison it can be said that this is the only known approach to the layout generation process, which concurrently optimizes the placement together with the detailed routing.

combine both steps. The described application is a good example for the tasks which research on genetic algorithms should deal with: Because of the nondeterministic behaviour and the long runtimes, genetic algorithms will never succeed against other optimization methods for low complexity problems that allow fast greedy solutions. But for high complexity problems without any known sophisticated solution techniques, a genetic approach is well suited. By solving such a hard and complex problem, critics can easily be convinced of the power and the advantages of genetic algorithms. When designing a genetic algorithm for such a special application, it is important to withdraw from usual solution methods in one part, but using hybrid approaches and problem speci c knowledge for hill-climbing in another. By doing so, there always is the trade-o between directed search and random search. The designer has to nd out, which part of the optimization can be done `by hand', how the genetic algorithm can be supported during the search process, and which tasks must be left for the genetic algorithm to work on.

FUTURE RESEARCH

A parallelization of the genetic algorithm is planned and along with this the implementation of a strategy adaptation 6]. There are many mutation operators, which are applied with di erent frequencies. It might be ingenious to exchange or move large parts of the layout during the early stage of optimization, and doing only minor changes when the population converges to an optimum. This can become possible by adapting the frequencies of the di erent mutation operators during the optimization. Further a gene-pool recombination operator 1, 4] will be implemented which might replace the current crossover operator. For a combinatorial optimization problem like the layout generation, the biologically motivated crossing of two parent chromosomes is likely to be less e cient. The construction of an o spring out of a pool of good building-blocks seems to be more suitable.

REFERENCES

1. A. E. Eiben, P.-E. Raue, Z. Ruttkay, Genetic algorithms with multi-parent recombination, 3rd Conf. on Parallel Problem Solving from Nature, Springer Lecture Notes in Computer Science 866, 1994, 78{87 2. A. Fritsch, O. Vornberger, Cutting Stock by Iterated Matching, Operations Research Proceedings, Selected Papers of the Int. Conf. on OR 94, U. Derigs, A. Bachem, A. Drexl (eds), Springer Verlag, 1995, 92{ 97 3. T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, John Wiley & Sons, 1990 4. H. Muhlenbein, H.-M. Voigt, Gene Pool Recombination in Genetic Algorithms, Procs. of the Metaheuristics Int. Conf., I. H. Osman, J. P. Kelly (eds.), Kluwer Academic Publishers, Norwell, 1995 5. S. M. Sait, H. Youssef, VLSI Physical Design Automation: Theory and Practice, McGraw-Hill (1995) 6. D. Schlierkamp-Voosen, H. Muhlenbein, Strategy Adaptation by Competing Subpopulations, 3rd Conf. on Parallel Problem Solving from Nature, Springer Lecture Notes in Computer Science 866, 1994, 199{ 208 7. V. Schnecke, O. Vornberger, Genetic Design of VLSILayouts, Procs. First IEE/IEEE Int. Conf. on GAs in Engineering Systems: Innovations and Applications, GALESIA '95, IEE Conference Publication No. 414, 1995, 430{435 8. N. Sherwani, Algorithms for VLSI Physical Design Automation, Kluwer Academic Publishers, 1993

CONCLUSIONS

For the application of genetic algorithms to optimize combinatorial problems, the focal point is to nd a proper genotype encoding and all genetic operators, which are necessary to enable the algorithm to reach a global optimum. Solutions to real-world optimization problems are too complex for being represented as a simple gene-string. In the layout optimization process, due to the problem speci c genotype encoding as a binary tree, the genetic algorithm is able to compute and optimize the routing on a chip concurrently with the placement of the modules. Usually in VLSI-CAD tools this is done in consecutive steps because of the complexity of the single optimization problems. But according to the strong interdependencies between the arrangement of the modules and the routing of the interconnection nets, it is wise to 6

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