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Hindawi Publishing CorporationVLSI DesignVolume 2011, Article ID 896241, 9 pagesdoi:10.1155/2011/896241

Research Article

Wirelength Minimization in Partitioning and FloorplanningUsing Evolutionary Algorithms

I. Hameem Shanavas1 and Ramaswamy Kannan Gnanamurthy2

1 Department of Electronics and Communication, M. V. J. College of Engineering, Bangalore 560067, India2 Vivekanandha College of Engineering for Women, Trichengode 637205, Tamilnadu, India

Correspondence should be addressed to I. Hameem Shanavas, [email protected]

Received 16 December 2010; Revised 23 April 2011; Accepted 6 July 2011

Academic Editor: Zhuo Li

Copyright © 2011 I. H. Shanavas and R. K. Gnanamurthy. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Minimizing the wirelength plays an important role in physical design automation of very large-scale integration (VLSI) chips. Theobjective of wirelength minimization can be achieved by finding an optimal solution for VLSI physical design components likepartitioning and floorplanning. In VLSI circuit partitioning, the problem of obtaining a minimum delay has prime importance.In VLSI circuit floorplanning, the problem of minimizing silicon area is also a hot issue. Reducing the minimum delay inpartitioning and area in floorplanning helps to minimize the wirelength. The enhancements in partitioning and floorplanninghave influence on other criteria like power, cost, clock speed, and so forth. Memetic Algorithm (MA) is an Evolutionary Algorithmthat includes one or more local search phases within its evolutionary cycle to obtain the minimum wirelength by reducing delayin partitioning and by reducing area in floorplanning. MA applies some sort of local search for optimization of VLSI partitioningand floorplanning. The algorithm combines a hierarchical design technique like genetic algorithm and constructive technique likeSimulated Annealing for local search to solve VLSI partitioning and floorplanning problem. MA can quickly produce optimalsolutions for the popular benchmark.

1. Introduction

Partitioning and floorplanning (PF) has been an active areaof research for at least a quarter of a century. The mainreason that partitioning has become a central and criticaldesign task today is due to the enormous increase of systemcomplexity in the past and the expected further advancesof microelectronic system design and fabrication. Widelyaccepted powerful high-level synthesis tools allow the design-ers to automatically generate huge systems by just changinga few lines of code in a functional specification. Synthesisand simulation tools often cannot cope with the complexityof the entire system under development, and also designerswant to concentrate on critical parts of a system to speed upthe design cycle. Thus the present state of design technologyoften requires a partitioning of the system with fast andeffective optimization. Moreover, fabrication technologymakes increasingly smaller feature sizes and augmented diedimensions possible to allow a circuit for accommodating

several million transistors. However, circuits are restrictedin size and in the number of external connections. So thefabrication technology requires partitioning of a system intocomponents by arranging the blocks without wasting freespace. The direct implementation of large circuit will occupylarge area. Hence the large circuit is to be split into smallsubcircuit. This will minimize the area of the system andthe complexity of the system. When they are partitioned, theconnection between two modules should be minimum (orthe number net cut by the partition). This is known as cutsize and hence this plays a major role in partitioning. It isa design task by applying an algorithmic method to solvedifficult and complex combinatorial optimization problemslike breaking a large system into pieces [1].

The process of determining block shapes and positionswith area minimization objective is referred to as floor-planning. A common strategy for blocks floorplanning is todetermine in the first phase and then the relative location ofthe blocks to each other based on connection-cost criteria.

2 VLSI Design

In the second step, block sizing is performed with thegoal of minimizing the overall chip area and the locationof each block is finalized [2]. When the partitioning andfloorplanning are combined (PF), the criteria like power,cost, and clock speed of each module are the subobjectiveto be optimized. The main objective to be optimized is thewirelength that can be achieved by incorporating MemeticAlgorithm (MA) in both the partitioning and floorplanning[3, 4].

In early stages, many interchanging methods have beenused which resulted in local optimum solution. And latersome of the mathematical methods are followed. Someheuristics are also used which resulted in better result butit has its own advantage and disadvantage. Since there maybe many solutions possible for this problem, stochasticoptimization techniques are utilized and until now manytechniques have been known like Simulated AnnealingAlgorithm (SA) which combines the Local Search Algorithmwith the Metropolis algorithm.

SA is a simple algorithm and does not need muchmemory, but it takes a long time to reach the desiredsolution. Kernighan and Lin [5] proposed a two-way graphpartitioning algorithm which has become the basis formost of the subsequent partitioning algorithms. Fiducciaand Mattheyses [6] modified the K-L algorithm to a moreefficient algorithm by suggesting moving one cell at a timeand by designing a new data structure separately. As akind of global optimization technique Genetic Algorithm(GA) which borrows the concept of generation from bio-logical system had been used for physical design problemslike circuit partitioning, floorplanning, and so forth. Thistechnique has been applied to several problems, most ofwhich are graph related because the genetic metaphor canbe most easily applied to these types of problems. GArequires more memory but it takes less time than SA [7].Lots of researchers have proposed their theories to partitioncircuit using GA. Work of [8] proposed hardware geneticalgorithm by developing GA and local search processor thatuses some external memory to overcome the problem of localmaxima/minima. Authors of [8] expressed the probability ofselection of chromosome as function of both the best andworst chromosome while [4] proposed different cost func-tions in order to achieve multiple objectives of minimizingdelay, cutsize area, and power. The authors of [9] proposedtwo GA one based on 0-1 encoding and other based oninteger encoding. Work done in [10] developed an adaptivestrategy for partitioning of circuits in which populationsize, crossover rate, and mutation rate are modified duringthe execution in order to enhance performance. Number ofenhancements like crossover operator mutation or choosingdifferent fitness functions can still be made to achieve opti-mal solutions. This means that theory of GA still providesopportunities for new inventions that can help in inventingnew solutions for physical design problems. This paperproposes memetic algorithm to solve the graph partitioningand floorplanning problem. The algorithm incorporatesseveral genetic algorithm features, namely, selecting a pop-ulation and crossover of the selected chromosomes to getbetter stable solutions. The algorithm starts by incorporating

Evolutionary algorithm

Local search (SA)

Figure 1: Memetic Algorithm.

the circuit as unweighted connected graph in which eachvertex represents gate and edge represents an interconnectionbetween two gates and thereafter applying the GA metaphorto generate a partition that is highly interconnected withinbut disconnected from other subpartitions while trying tominimize the number of cuts and time consumed. Thiswork tried to hybrid two algorithms like Genetic Algorithmand Simulated Annealing for overcoming the disadvantageof one another. Such type of algorithms is called memeticalgorithm, (see Figure 1).

This work addresses the problem of VLSI netlist parti-tioning with the objective of reducing delay and then floor-planning with the objective of reducing area to minimize thewirelength.

2. Partitioning

A better circuit partition will reduce connection among sub-circuits and result in a better routing area of the layout. Thechallenge is that the circuit partitioning problem belongsto the class of well-known NP-hard optimization problems[11]. The problem can be viewed as a graph partitioningproblem where each module (gates etc.) is taken as verticesand the connection between them represents the edgesbetween the nodes [12, 13].

To start the algorithm, n gates are placed on the graphas n vertex, and an initial population is chosen as thedifferent permutations of various vertices of the given graph.The problem reduces to associate each chromosome andeach partition of the graph. An algorithm based on Geneticalgorithm is proposed that can be used to partition a numberof nodes. The example for a circuit and graph representationis given (Figure 2).

Given an unweighted connected graph G = (V ,E) on setof vertices V and edges E. Let k ≥ 2 be a given integer, find apartition V1, V2, V3, . . . ,Vk set of vertices V such that

(i) the induced graph Gi = (Vi,Ei), for all values i =1, 2, 3, . . . , k, is connected;

(ii) the following values are minimized min{|V1|, |V2|,|V3|, . . . , |VK |}.

This k-Connected Problem is a particular instance ofgraph partitioning problem [14]. Because exact and approx-imation algorithms that run in polynomial time do not existfor graph partitioning problems in general, it is necessaryto solve the problem using heuristic algorithms. GeneticAlgorithm is a heuristic technique that seeks to imitatethe behavior of biological reproduction and their abilityto collectively solve a problem. The GA starts with several

VLSI Design 3

e2

e4

e1

e3

e7

e8

e5

e6

P1

P2

P6P5

P3

P4

P7 O1

O2

(a)

e2

e4

e1

e3

e7

e8

e5

e6

P1

P2

P6

P5

P3

P4

P7 O1

O2

(b)

Figure 2: Representation of a circuit and Directed acyclic graph.

alternative solutions to the optimization problem, which areconsidered as individuals in a population. These solutionsare coded as binary strings, called chromosomes. The initialpopulation is constructed randomly. These individuals areevaluated using partitioning by specific fitness function. GAthen uses these individuals to produce a new generation ofhopefully better solutions. In each generation, two of theindividuals are selected probabilistically as parents, with theselection probability proportional to their fitness. Crossoveris performed on these individuals to generate two newindividuals, called offspring, by exchanging parts of theirstructure. Thus each offspring inherits a combination offeatures from both parents. The next step is mutation.An incremental change is made to each member of thepopulation, with a small probability. This ensures that GAcan explore new features that may not be in the populationyet. It makes the entire search space reachable, despite thefinite population size. The basic foundation of the algorithmis to represent each vertex in the graph as a location that canrepresent a logic gate, and a connection is represented by anedge [15, 16].

3. Floorplanning

A module B is a rectangle of height HB, width WB, and areaAB. A supermodule consists of several modules, also called asubfloorplan, (see Figure 3). A floorplan for n modules con-sists of an enveloping rectangle R subdivided by horizontallines and vertical lines into n non-overlapping rectangles.Each rectangle must be large enough to accommodate themodule assigned to it. In the given problem, a set of hardmodules and outline-constraints are provided. The modulesinside the given outline have freedom to move. A feasiblepacking is in the first quadrant such that all the modulesinside the outline should not duplicate and overlap eachother. The objective is to construct a feasible floorplan Rsuch that the total area of the floorplan R is minimized andsimultaneously satisfy floorplanning constraint. Given a set

1

7 65

2

3

4

AB

HB

WB

Figure 3: Floorplan module.

of modules to a specified number of cluster satisfying thepredescribed properties. To solve the floorplanning problem,first construct a network graph, and then run the givenalgorithm to get the solution. The graph consists of twokinds of vertices horizontal and vertical. The network graphG = (V ,E) has to be constructed. “∗” represents verticalslicing of blocks. “+” represents horizontal slicing of blocks(see Figure 4).

4. Memetic Algorithm

The memetic algorithm [17, 18] is a combination of anEvolutionary Algorithm (EA) and Local Search (LS). TheEAs are used for finding the global optimum. The LS usedhere will aid the EA to convergence speed. The evolutionaryalgorithm used in this work is genetic algorithm. There aretwo types of memetic algorithm.

Exhaustive MA. Few solutions are selected from the finalgeneration and improved using local search.

Intermediate MA. After a predetermined number of iterationby GA, local search is applied to few random individuals.This is done to have a better solution than the local maxima.This work deals with intermediate memetic algorithm.

4 VLSI Design

4.1. Populations and Chromosomes. In GA-based optimiza-tions, a set of trial solutions are assembled as a population.The parameter set representing each trial solution or indi-vidual is coded to form a string or chromosome and eachindividual is assigned a fitness value by evaluation of theobjective function. The objective function is to only linkbetween the GA optimizer and the physical problem.

4.2. Parents. Following this initialization process, pairs ofindividuals are selected (with replacement) from the pop-ulation in a probabilistic manner weighted by their relativefitness and designated as parents.

4.3. Children. A pair of offspring, or children, are thengenerated from the selected pair of parents by the applicationof simple stochastic operators. The principle operators arecrossover and mutation. Crossover occurs with a probabilityof pcross (typ. 0.6–0.8) and involves the random selectionof a crossover site and the combining the two parent’sgenetic information. The two children produced share thecharacteristics of the parents as a result of this recombinationoperators. Other recombination operators are sometimesused, but crossover is the most important. Recombination(e.g., crossover) and selection are the principle way thatevolution occurs in a GA optimization.

4.4. Mutation. Mutation introduces new and unexploredpoints into the GA optimizer’s search domain. Mutationrandomly changes the genetic makeup of the population.Mutation is much less important than recombination andoccurs with a probability pmutation (typ. 0.05) which ismuch less than pcross.

4.5. New Generation. Reproduction consisting of selection,recombination and mutation continues until a new gener-ation is created to replace the original generation. Highlyfit individuals, or more precisely highly fit characteristicsproduce more copies of themselves in subsequent generationresulting in a general drift of the population as a wholetowards an optimal solution point. The process can beterminated in several ways: threshold on the best individual(i.e., the process stops when an individual has an errorless than some amount E), number of generations exceedsa preselected value, or some other appropriate criteria. Asimple Genetic Algorithm must be able to perform fivebasic tasks: encode the solution parameters in the form ofchromosomes, initialize a starting point population, evaluateand assign fitness values to individuals in the population,perform reproduction through the fitness weighted selectionof individuals from the population, and perform recom-bination and mutation to produce members of the nextgeneration.

Consider the graph G = (V ,E) with vertex |v| = uand in integer 1 < k < n/4. Initialize a randomly generatedpopulation P of k elements. Population P has 1 to k elements.Assume each parent p1 to pk belongs to the population P.Perform two point crossover for pa and pb from populationP using the fitness function ( f ) = k ·M(p)/n, where M(p) is

the number of node of partition with maximum cardinalityamong n partitions. Assume Pa(I) and Pb(I) are the childrenfrom pa and pb, respectively. If Pa(I) has not satisfied thefitness (Pa(I) is not in I), then choose pa randomly fromj > i. Swap Pa(i) and Pa( j). Copy the first element k elementsof pa in q1, q3. If Pb(I) has not satisfied the fitness (Pb(I)is not in I), then choose pb randomly from h > k. SwapPb(h) and Pb(i). Copy the first element k elements of pb inq2, q4. Create two vectors L, L′ with 2(n − k) elements. Ifj · mod2 = 1, then L(i) = Pa[k + ( j + 1)/2] and L′(i) =Pb[k + ( j + 1)/2], else L( j) = Pa[k + ( j + 1)/2] and L′( j) =Pb[k + ( j + 1)/2]. Check the fitness of L(i), L′(i), L( j), andL′( j). If L(i) is not in q1, then copy L′(i) in q1 and L(i) inq2. If L( j) is not in q3, then copy L′( j) in q3 and L( j) in q4.Repeat the process again with L(i) and L( j), L′(i) and L′( j) toget new offspring. The new offsprings can have more fitnessvalue or less fitness value depending upon the parents. Lessfitness offspring can be discarded then to reduce the numberof cycles.

The fitness of the individual in partitioning is basedon the delay of the module.The fitness of the individualis given by weighted evaluations of maximum delay (D).Da identifies a particular subgraph, Dm is a predeterminedmaximum value, and Df is the weighting factor. The sumof the weighting factors equals one. The complete fitnessfunction is

Gp ={Da

Dm> 1,

Da

Dm≤ 1,

Da

Dm∗Df

}(1)

Assuming an individual is fully feasible and meetsconstraints, the value of Gp ≤ 1, with smaller values beingbetter.

At the beginning, a set of Polish expressions is given forfloorplannig. It is denoted as P, randomly generated expres-sion to compose a population. The fitness for floorplanningof the individual is based on the area of the module. Area ofa block can be calculated by the general formula A = LW ,where L stands for length of the module and W stands forwidth of the module.

Total Area =∑

An. (2)

The fitness of the individual is given by weightedevaluations of maximum area (A). Aa identifies a particularsubgraph, Am is a predetermined maximum value, and Af isthe weighting factor. The sum of the weighting factors equalsone. The complete fitness function is

Gf ={Aa

Am> 1,

Aa

Am≤ 1,

Aa

Am∗ Af

}. (3)

5. Local Search

After a prescribed number of iterations by evolutionaryalgorithm, local search algorithm is applied to few randomindividual, to have better solution. Local search methodsare iterative algorithms that tend to enhance solutionby stepwise improvement and make an attempt to reachoptimum solutions, which more often results in suboptimal

VLSI Design 5

solutions by trapping themselves in local minima/maxima.The simplest form of local search is repetitively flippingelements in a solution resulting again in the objectivefunction. Eventually local minima will be reached wherebyflipping any element in the solution will result in loss ofobject. Although these algorithms are simple, there have beenmany complex improvements for CAD tools which involvelarge dynamic memory and linked list usages. For refiningthe solution obtained by GA, the local search (LS) is applied.This can be used before crossover or after crossaover, it canalso be used for parents selection, and used before or aftermutation to increase the number of fitness variables (seeAlgorithm 1).

5.1. Optimization by Simulated Annealing. Simulated An-nealing algorithm is applied for Local Search process since SAis not a Local Search Algorithm. Here simulated annealingmethod is performed on finally generated offspring toimprove the fitness. This method is called as intermediateMA.

Simulated annealing is a stochastic computationalmethod for finding global extrema to large optimizationproblems. It was first proposed as an optimization techniqueby Kirkpatrick et al. in 1983 [19] and Cerny in 1984 [20].The optimization problem can be formulated by describinga discrete set of configurations (i.e., parameter values) andthe objective function to be optimized. The problem isthen to find a vector that is optimal. The optimizationalgorithm is based on a physical annealing analogy. Physicalannealing is a process in which a solid is first heateduntil all particles are randomly arranged in a liquid state,followed by a slow cooling process. At each (cooling)temperature, enough time is spent for the solid to reachthermal equilibrium, where energy levels follow a Boltzmanndistribution. As temperature decreases the probability tendsto concentrate on low energy states. Care must be taken toreach thermal equilibrium prior to decrease the temperature.At thermal equilibrium, the probability that a system isin a macroscopic configuration with energy is given bythe Boltzmann distribution. The behavior of a system ofparticles can be simulated using a stochastic relaxationtechnique developed by Metropolis et al. [21]. The candidateconfiguration for the time is generated randomly. The newcandidate is accepted or rejected based on the differencebetween the energies associated with states. The condition tobe accepted is determined by

p = prpq=

exp(−Et − Eq

)

Kt> 1. (4)

One feature of the Metropolis way of Simulated Anneal-ing algorithm is that a transition out of a local minimumis always possible at nonzero temperature. Another evenlyinteresting property of the algorithm is that it performs akind of adaptive divide and conquer approach. Gross featuresof the system appear at higher temperatures, fine featuresdevelop at lower temperatures. For this application, it usedthe implementation by Ingber [22].

BeginInitialize population P;For i := 1 To size of(P) Do

Individual := Pi;Individual := Local Search (Individual);

Repeat Until (terminate=True) DoFor i := 1 To #recombination Do

Select two parents ia, ibε P randomly;ic := Recombination(ia,ib);ic := Local Search(ic);

Add individual ic to P;P := Select(P);If (P converged Then

For i := 1 To size of(P), i != index(Best) DoIndividual := Pi

Individual := Local Search(Mutate(Individual));End

Algorithm 1

5.2. Partitioning Based on Simulated Annealing. The basicprocedure in simulated annealing is to start with an initialpartitioning and accept all perturbations or moves whichresult in a reduction in cost. Moves that result in a costincrease are accepted. The probability of accepting such amove decreasing with the increase in cost and also decreasingin later stages of the algorithm are given in (4). A parameterT, called the temperature, is used to control the accep-tance probability of the cost-increasing moves. SimulatedAnnealing algorithm for partitioning the modules will bedescribed here. The cells are partitioned using simulatedannealing so as to minimize the estimated wire length. Aformal description of the simulated annealing algorithm wasgiven in Section 5. There are two methods for generatingnew configurations from the current configuration. Eithera cell is chosen randomly and placed in a random locationon the chip, or two cells are selected randomly and inter-changed. The performance of the algorithm was observed todepend upon r, the ratio of displacements to interchanges.Experimentally, r is chosen between 3 and 8. A temperature-dependent range limiter is used to limit the distance overwhich a cell can move. Initially, the span of the range limiteris twice the span of the chip. In other words, there is noeffective range limiter for the high temperature range. Thespan decreases logarithmically with the temperature.

LWV (T) = LWV (Ti)

[logTlogTi

],

LWH(T) = LWH(Ti)

[logTlogTi

],

(5)

where T is the current temperature, Ti is the initial tem-perature, LWV (Ti) and LWH(Ti) are the initial values of thevertical and horizontal window spans LWV (T) and LWH(T),respectively.

The wirelength cost C is estimated using the semiperime-ter method, with weighting of critical nets and independent

6 VLSI Design

weighting of horizontal and vertical wiring spans for eachnet:

C =∑

nets i

[x(i)WH(i) + y(i)WV (i)

], (6)

where x(i) and y(i) are the vertical and horizontal spansof the net i’s bounding rectangle and WH(i) and WV (i)are the weights of the horizontal and vertical wiring spans.When critical nets are assigned a higher weight, the annealingalgorithm will try to place the cells interconnected by criticalnets close to each other. Independent horizontal and verticalweights give the user the flexibility to prefer connections inone direction over the other.

The acceptance probability is given by exp(−ΔC/T),where ΔC (i.e., Et − Eq) is the cost increase and T is thecurrent temperature. When the cost increases, or when thetemperature decreases, the acceptance probability (4) getscloser to zero. Thus, the acceptance probability exp(−ΔC/T)less than random (0, 1) (a random number between 0and 1) is high when ΔC is small and when T is large.At each temperature, a fixed number of moves per cell isallowed. This number is specified by the user. The higher themaximum number of moves, the better the results obtained.However, the computation time increases rapidly. There isa recommended number of moves per cell as a function ofthe problem size in. For example, for a 200-cell and 3000-cell circuit, 100 and 700 moves per cell are recommended,respectively. The annealing process starts at a very hightemperature, for example, Ti = 4, 000, 000, to accept mostof the moves. The cooling schedule is represented by Ti +1 = a(T), where a(T) is the cooling rate parameter and isdetermined experimentally. In the high and low temperatureranges, the temperature is reduced rapidly (e.g., a(T) ≈ 0.8).However, in the medium temperature range, the temperatureis reduced slowly (e.g., a(T) ≈ 0.95). The algorithm isterminated when T is very small, for example, when T < 0.1.

5.3. Floorplannning Based on Simulated Annealing. Thissection describes an optimal floorplanning on simulatedannealing algorithm. Assume that a set of modules are givenand each module can be implemented in a finite numberof ways, characterized by its width and height. Some ofthe important issues in the design of a simulated annealingoptimization problem are as follows.

(1) the solution space,

(2) the movement from one solution to another,

(3) the cost evaluation function.

The branch cells correspond to the operands and theinternal nodes correspond to the operators of the Polishexpression. A binary tree can also be constructed from aPolish expression by using a stack as shown in Figure 4.The simulated annealing algorithm moves from one Polishexpression to another. A floorplan may have different slicingtree representations. For example, the trees in Figure 4represent the given floorplan. There is a one-to-one corre-spondence between a floorplan and its normalized Polish

7 6

1

3

5

2

4

+

+

+

∗

∗ ∗

Figure 4: Graph representation of the given floorplan module.Polish expression: 76∗1+43+2∗5+∗.

1

765

2

3

4

Figure 5: Condition 1 (67∗1+34+2∗5+∗).

expression. But this leads to a larger solution space andsome bias towards floorplans with multitree representations,since they have more chances to be visited in the annealingprocess. Assume three types of movement are defined tomove from one floorplan to another. They operate on thePolish expression representation of the floorplan.

Condition 1. Exchange two operands when there are no otheroperands in between (67∗1+34+2∗5+∗) (see Figure 5).

Condition 2. Complement a series of operators between twooperands (67+1∗34∗2+5∗+) (see Figure 6).

Condition 3. Exchange adjacent operand and operator ifthe resulting expression is a normalized Polish expression(67+43∗+25∗1+∗) (see Figure 7).

It is obvious that the movements will generate onlynormalized Polish expressions. Thus in effect, the algorithmmoves from one floorplan to another. Starting from anarbitrary floorplan, it is possible to visit all the floorplansusing the movement. If some floorplans cannot be reached,there is a danger of losing some valid floorplans in thesolution. Starting from any floorplan, the modules can moveto the floorplan based on the given conditions. The costfunction is a function of the floorplan, or equivalently the

VLSI Design 7

1

7

6

5

2

3 4

Figure 6: Condition 2 (67+1∗34∗2+5∗+).

1

7

6

52

34

Figure 7: Condition 3 (67+43∗+25∗1+∗).

Table 1: Delay comparison of MA with GA for ISCAS89 bench-mark.

CircuitGA MA

Delay(ps) T (s) Best (s) Delay (ps) T (s) Best (s)

S1196 396 375 373 301 184 134

S1238 475 397 365 408 187 160

S1494 614 1228 1040 585 616 427

S2091 302 94 32 225 616 16

S3330 571 2096 2074 533 47 994

S5378 587 2687 2686 590 1078 1100

Polish expression. There are two components for the costfunction, area, and wire length. The area of a sliceablefloorplan can be computed easily using the floorplan sizingalgorithm. The wire-length cost can be estimated from theperimeter of the bounding box of each net, assuming thatall terminals are located on the center of their module. Ingeneral, there is no universally agreed upon method of costestimation. For simulated annealing, the cost function is bestevaluated easily because thousands of solutions need to beexamined. Figures 5, 6, and 7 shows a series of movementswhich lead to a solution.

6. Experimental Results

The memetic algorithm has been implemented in C++on a Linux installed personal computer. In this work, itcompares the performance of a pure genetic algorithmto a memetic technique that combines GA with simplelocal search techniques like Simulated Annealing. In circuitpartitioning, MA can increase the speed of execution timeon an average of 45% when compared to simple geneticalgorithm see Table 1 and Figure 8. Delay (ps) is the delayof the most critical path. T (s) is the total run time, andBest (s) is the execution time in seconds for reaching the

3000

2500

2000

1500

1000

500

0

Cir

cuit

S119

6

S123

8

S149

4

S209

1

S333

0

S537

8

GA MA

Figure 8: Comparison of MA with GA.

Table 2: Wirelength comparison of MCNC benchmark for n100,n200, & n300 WITH ASPECT RATIO R = 1, 2, 3, 4.

CircuitAspectratio R∗

Fast-SA MA

Wire(mm)

Time(sec)

Wire(mm)

Time(sec)

n100

1 33.56 30 32.06 26

2 35.44 30 34.39 24

3 35.48 30 34.23 27

4 36.89 29 32.74 27

n200

1 63.55 175 58.33 150

2 62.76 173 59.84 156

3 63.70 180 61.55 156

4 66.31 176 63.72 171

n300

1 76.05 399 71.00 363

2 77.60 386 74.22 358

3 81.67 391 79.56 371

4 88.58 382 82.18 370

Comparison 1.06 1.11 1.00 1.0

best solution. Thus the objective of wirelength minimizationcan be achieved by reducing the delay in circuit partitioning.In Floorplanning, wirelength and CPU time are comparedwith simulated annealing see Table 2. MA can reduce thewirelength on an average of 0.5 mm when compared withFast Simulated annealing see (Figures 9 and 10).

7. Conclusion

By reducing the wirelength, cost of very large-scale integratedchip manufacturing can be reduced. This paper explores theadvantage of memetic algorithm which can be proven to 45%

8 VLSI Design

500

400

300

200

100

0

Fast-SA wire (mm)Fast-SA time (sec)

1 5 9 13

Figure 9: Performance of Fast Simulated Annealing.

400

300

3

200

100

0

MA wireMA time (sec)

(sec)

1 5 9 11 137

Figure 10: Performance of Memetic Algorithm.

faster than the simple software genetic algorithm by reducingthe delay and area in partitioning and floorplanning, respec-tively, that would indefinitely reduce the wirelength. Theimplementation of multiobjective in the algorithm enablesto get the near optimal solution. After a predeterminednumber of iteration by GA, local search is applied to fewrandom individual to get the optimal solution by SimulatedAnnealing. In the future, the performance of the algorithmhas to be tested on different benchmarks.

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VLSI Design 9

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