A novel bottom-left packing genetic algorithm for analog module placement

Advances in Radio Science (2003) 1: 191–196 c

Copernicus GmbH 2003

Advances in

Radio Science

A novel bottom-left packing genetic algorithm for analog module

placement

L. Zhang and U. Kleine

Otto-von-Guericke University of Magdeburg, IESK, PO Box 4120, D-39016 Magdeburg, Germany

Abstract. This paper presents a novel genetic algorithm for analog module placement. It is based on a generalization of the two-dimensional bin packing problem. The genetic encoding and operators assures that all constraints of the problem are always satisfied. Thus the potential problems of adding penalty terms to the cost function are eliminated, so that the search configuration space decreases drastically. The dedicated cost function covers the special requirements of analog integrated circuits. A fractional factorial experi-ment was conducted using an orthogonal array to study the algorithm parameters. A meta-GA was applied to determine the optimal parameter values. The algorithm has been tested with several local benchmark circuits. The experimental re-sults show this promising algorithm makes the better perfor-mance than simulated annealing approach with the satisfac-tory results comparable to manual placement.

1 Introduction

The significant tendency of system-on-chip (SoC) intensifies the booming market share of mixed-signal integrated circuits (ICs). Although most of the functions in such an integrated system are performed with digital circuitry, analog circuits are always needed as an interface to the external, continuous-valued world. The design of digital portion can be tackled with modern cell-based tools (Wang et al., 2000) for synthe-sis, mapping and physical design. The analog counterpart, however, is still routinely designed by hand. The layout of analog circuits is intrinsically more difficult than the digi-tal one. To address this problem, an automated layout tool called ALADIN (Automatic Layout Design Aid for Analog Integrated Circuits) (Zhang et al., 2000) is currently being developed for analog experts who can and must bring their specific knowledge into the synthesis process in order to cre-ate high quality layouts. The focus of this paper is on the

Correspondence to: L. Zhang

([email protected])

placement phase. Due to analog constraints such as match-ing requirements, it is usually preferred to build more or less complex clusters of devices, hereafter called modules or macro-cells, which are parameterized for the processed sub-circuits. The objective of placement is to position the modules appropriately so that the chip area and the total wire length of the interconnections are minimized under certain constraints.

Many heuristic strategies for module placement based on iterative improvement have been published so far, such as force-directed, min-cut, passive resistive optimization, simu-lated annealing (SA) (Su et al., 2001; Plas et al., 2001) and genetic algorithm (GA) (Shahookar and Mazumder, 1990; Esbensen and Mazumder, 1994). Among them, SA and GA are the latest and most promising techniques. SA is widely used in the domain of both digital and analog (Cohn et al., 1994; Plas et al., 2001) circuits. Although it yields good placement solutions, it is a very time-consuming pro-cess. In contrast, GA has been mainly applied for digital cir-cuits. This paper describes a GA application in analog circuit placement. It is organized as follows. Section 2 introduces the design flow with the use of DesignAssistant in which this placement approach is included. Section 3 describes the im-plementation of this adaptive strategy. Section 4 gives the parameter optimization with fractional factorial experiment and meta- GA. Section 5 shows experimental results and the conclusion is drawn finally.

2 DesignAssistant

DesignAssis-192 L. Zhang and U. Kleine: A novel bottom-left packing genetic algorithm

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3 Algorithm implementation

The GA is a search strategy based on the mechanics of natu-ral selection and natunatu-ral reproduction in a biological system. It differs from the other stochastic search techniques by being able to encode and exploit past information efficiently during a search.

3.1 Genetic encoding

The conventional chromosomal representation of the GA is based on bit-string (Shahookar and Mazumder, 1990). A GA for the two-dimensional bin packing problem has been de-veloped by Kroeger et al. (1991). Esbensen and Mazumder (1994) used this representation for the digital circuit place-ment. In this paper a bottom-left GA (BLGA) for the analog module placement is developed based on comprehensive ex-tensions and modifications of the genetic encoding and op-erators found in the above work (Esbensen and Mazumder, 1994; Kroeger et al., 1991). In GA a distinction is made be-tween genotype and phenotype of an individual. Here the ge-netic encoding is inspired by the two-dimensional bin pack-ing problem, which is the problem of compactly packpack-ing a number of rectangular blocks into a bin with a fixed width and infinite height in such a way that the distance from the top edge of the highest placed block to the bottom edge of the bin is minimized. The standard algorithm for this prob-lem places each block at a time at the downmost and then at the leftmost position. The placement algorithm is based on a generalization of this scheme. The solution space consid-ered by the algorithm is restricted to the set of all possible BL-placements.

Assume that the given problem hasncellsc1, ..., cn. An

example genotype withn = 6 cells is shown in Fig. 2

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Fig. 3. Crossover operators.

descendantγ) is constructed as follows. From the cell tree ofα, a connected subset

Ts =(Vs, Es), Vs ⊂V , Es ⊂Eα (3)

is chosen.Tsis chosen at random but subject to the constraint

that decodingTs in the order defined byPα (i.e., priority of

individualα), using

c∈Vs|∀c0∈Vs/{c} :Pα(c) < Pα(c0) (4)

as root, causes no constraint violations. In Fig. 3, the chosen Ts is indicated by the dashed line. Initially Eγ is defined

to beEs. Hence,γ has inherited all cells inVs fromα. The

remaining cellsV−Vsare then inherited fromβby extension

of Eγ. The cell tree ofβ is traversed in ascending order according toβ. At any node it is checked if the corresponding cellcbelongs toVs, that is, whether it has been placed inγ

already. If so, the cell is skipped. Otherwise,cis added to the cell tree ofγ by extendingEγ randomly. The orientation of any cell is inherited unaltered together with the cell itself. That is,

tγ(c)=

tα(c) if c∈Vs

tβ(c) if c∈V −Vs

(5)

Pγ should correspond to the order in which the cells were placed when creatingEγ. SinceP is a bijection, the follow-ing constraints in Eq. (6) uniquely determinesPγ:

∀ci ∈Vs, ∀cj ∈V −Vs :Pγ(ci) < Pγ(cj)

∀ci, cj ∈Vs :Pα(ci) < Pα(cj)⇒Pγ(ci) < Pγ(cj)

∀ci, cj ∈V −Vs :Pβ(ci) < Pβ(cj)⇒

Pγ(ci) < Pγ(cj) (6)

The second proposal differs from the first one by the con-struction of the remaining cellsV −Vs, which inherits from

βby the ordered extension. In detail, the concatenation tries to follow the structure ofβ first of all. If impossible, ran-domly add to any free position ofγ. In Fig. 3d the nodeBis added to the left of the nodeE, instead of the left of the node Din Fig. 3c. As well the nodeAis added to the left of the nodeD, instead of the left of the nodeCin Fig. 3c.

Five different mutation operators are developed. Each per-forms some random change in the given genotype. When performing each of these mutations, a part of the genotype has to be decoded to check if the mutated individual satisfies all constraints. A mutation is only performed if it does not cause any constraint violations. The purpose of the inver-sion operator is to weaken the linkage among genes. Given a genotype, the inversion operator computes a new genotype by rearranging the components in such a way that their mu-tual distances changes, while at the same time assuring that the corresponding phenotype is still the same. The inversion operator selects a subtree at random and moves it to another free position in such a way that no constraints are violated and so that the corresponding phenotype is still the same. An example of this is shown is Fig. 2b and c. This genotype tree is generated by moving the subtree rooted at the nodeF. 3.3 Cost function

The cost function is the goodness criterion of searched con-figurations. It consists of four parts, which are given in Eq. (7).

C=(αall areaCall area+αN areaCN area+αP areaCP area

+αA areaCA area+αD areaCD area)+αnet sCnet s

+αasp ratCasp rat (7)

α∗ is a weight factor for the corresponding costC∗, which balances the importance of all the possible considerations according to different design requirements. The first is the area cost which is made up of the whole area, NWELL and PWELL region areas, analog and digital region areas. They will help to decrease the whole area. Moreover they could make NWELL and PWELL regions relatively concentrated in order to ease the fabrication. And analog and digital re-gions are separated from each other so that the constraints for mixed signal circuits can be imposed. The second is the net-length cost Cnet s, in which priority coefficient can

194 L. Zhang and U. Kleine: A novel bottom-left packing genetic algorithm

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Fig. 4. Outline of BLGA.

The third is the aspect-ratio costCasp rat, which is used to

control the shape of the final layout. Designers can define the ideal width-length ratio or exact width value. The more the real layout shape differs from the ideal definition, the more penalty is brought about. The graphic input window of the cost function is provided in the DesignAssistant.

3.4 Algorithm outline

The algorithm outline is depicted in Fig. 4.st opCrit erion() makes the evolution process terminated if no improvement has been observed for a predefined number of consecutive generations or a fixed number of generations is over. A fit-ness is the reciprocal of the corresponding cost.

4 Parameter optimization

Since the operators are of paramount importance to the over-all performance of the algorithm, their parameters, including the population size, crossover, mutation and inversion rates, have to be investigated with care. The study of the correla-tion and sensitivity helps to shrink the regarded ranges and set up the exact optimal parameters in the problem of analog module placement.

4.1 Parameter analysis

The fractional factorial experiment is an important technique in Robust Design (Park, 1996). A Taguchi orthogonal ar-rayL27(313)is employed to construct the fractional factorial experiment. The population size, crossover, mutation and inversion rates are taken as the factors. The reason why to choose such an array with 13 columns is to give a deliberate

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Fig. 5. Outline of the meta-GA.

study about inter-correlation between two factors. The exper-iment design is depicted in Table 1,crmeans crossover rate, mr means mutation rate,ir means inversion rate,Mmeans population size and∗means the combination of two factors. Three columns (4, 7 and 11) are left for error estimation.

For the crossover and inversion rates, three levels are cho-sen as (0.2, 0.55, 0.85). (0.05, 0.1, 0.2) are chocho-sen for the levels of the mutation rate. Two groups of experiments were performed in order to cover a wide range for the population size, one with (10, 35, 60) and the other with (40, 70, 100). The cost and execution time are taken as the search target, where the cost is the primary consideration and execution time is secondary. As a result, the correlation between pa-rameters is found quite weak. The population size becomes insignificant to the cost if it is more than 60. The crossover and mutation rates are sensitive for the search. Even though the inversion rate is insensitive to the cost, it is 0.25 optimally with the consideration of its effect on the execution time. So the optimal value or ranges for the four parameters are set as follows. The population size was set as 60, the mutation range was from 0 to 0.1, the crossover rate from 0.55 to 1 and the inversion rate from 0.20 to 0.55.

4.2 Parameter determination

represent-L. Zhang and U. Kleine: A novel bottom-left packing genetic algorithm 195

Table 1. Design of the fractional factorial experiment

Exp. No. 1 2 3 4 5 6 7 8 9 10 11 12 13

cr mr cr∗mr ir cr∗ir mr∗ir M cr∗M mr∗M ir∗M

1 1 1 1 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 2 2 2 2 2 2 2 2 2

... ... ... ... ... ... ... ... ... ... ... ... ... ...

27 3 3 2 1 3 2 1 2 1 3 1 3 2

Table 2. Comparison among distinct algorithms

SA BLGA1 BLGA2

Cmean 11799 11823 11698

Circuit 1 Cσ 176 46 39

Tmean(s) 846 213 222

Tσ(s) 210 40 69

Cmean 86653 83487 82122

Circuit 2 Cσ 3087 2925 2165

Tmean(s) 5068 3897 2848

Tσ(s) 783 163 1031

Cmean 215720 220740 216350

Circuit 3 Cσ 13554 4224 3932

Tmean(s) 1143 2586 3222

Tσ(s) 4 280 461

ing the crossover rate, inversion rate, and mutation rate of BLGA. The fitness of an individual (a BLGA with a certain parameter combination) is taken to be the fitness of the best placement that the meta-GA can find in the entire run, using these parameters. In meta-GA, the population size was 20, and the algorithm was run for 100 generations. The crossover probability was 1.

5 Experimental results

Because so far the analog benchmark circuits are unavailable for the synthesis purpose, three local circuits are used to eval-uate the above algorithms. Each algorithm is executed for ten times so that the mean and standard deviation are used for evaluation. The cost value is superior to the execution time. As the focus here is on the comparison of algorithms, the simple half-perimeter estimation is applied for all the trials. In order to demonstrate the efficiency of GA, one optimiza-tion with SA was also performed. The results are depicted in Table 2 including SA, BLGA1 means with the first crossover proposal and BLGA2 means with the second crossover pro-posal.

The result shows SA is generally poorer than BLGA while need more execution time. The BLGA2 works marginally better than BLGA1. So finally the second proposal, i.e., the close inheritance crossover scheme is applied. In order to

Table 3. Comparison among distinct algorithms

circuit 1 circuit 2 circuit 3 Average

cr 0.775 0.82 0.865 0.82

mr 0.01 0.02 0.01 0.013

ir 0.235 0.41 0.41 0.352

initCost 12453 82276 220330 endCost 11731 77992 203704

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Fig. 6. Convergence schedules of different algorithms.

keep the diversity during the evolution, the selection based on fitness rank is applied instead of the fitness-based such as roulette wheel selection (Kroeger et al., 1991). The con-vergence schedule is depicted in Fig. 6. Since the costs of BLGA1 and BLGA2 are the best cost in each generation, the variance amplitude is smoother than SA in the whole view.

The representation of BLGA improves the searching effi-ciency so that the search need not cover a wide scope as SA but with more accuracy. The results with the Meta-GA for the three circuits are given in Table 3. Finally the best pa-rameter set is the crossover rate of 0.82, the mutation rate of 0.023 and the inversion rate of 0.235. The program is writ-ten inC+ +running under Solaris-UNIX in a Sun-Ultra60 workstation. The weight factors in the cost function are set as follows: αall area = 2, αNarea = 0.2, αParea = 0.2,

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> @6XQJ + 3DUN 5REXVW 'HVLJQ DQG $QDO\VLV IRU 4XDOLW\ (QJLQHHULQJ &KDSPDQ +DOO /RQGRQ Fig. 7. Schematic (a) and placement layout (b) with BLGA2 of a

common mode feedback optional amplifier.

Figure 7a gives the schematic of the third circuit, a com-mon mode feedback optional amplifier, in which the parti-tioning is indicated by rectangles. The corresponding place-ment result with the BLGA2 is shown in Fig. 7b, which is comparable to the manual placement. In DesignAssistant the relative position of modules are recorded and passed to the router as the input. The router then compacts all the modules with the detailed routing.

6 Summary

In this paper a new technique with genetic algorithm to solve the analog module placement has been introduced. By using the notion of bin-packing, a genetic encoding has been devel-oped in which most constraints of the problem are implicitly represented. As a consequence, each individual always sat-isfies constraints. The advantage of the proposed strategy is that it allows a more accurate estimation of the layout quality with the configuration space shrunk dramatically, since the use of penalty terms have been avoided. Special constraints for analog integrated circuits are included in the cost func-tion. The fractional factorial experiment with an orthogonal array is employed in order to study the algorithm parameters.

A meta-GA is applied to determine the exact parameter val-ues. The experimental results show that this approach with the optimized parameters contributes high design efficiency with the satisfactory result, which is comparable to the man-ual counterpart.

References

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