1

3. Algorithms for self-configuration and evolution

• General perspective on search, optimization and adaptation algorithms

• Essence of evolutionary algorithms• Details of operation of Genetic Algorithms • Multi-criteria optimization, Hybrid Search

2

Objectives: control self-configuration for desired functionality

• A control C that creates a structure / topology / architecture S, that has the function F. Specification in terms of S or F. F may include constraints, preferences, etc.

• Behavior/Function may change in time, in simple case it doesn’t

• Often C, even for a set of states which can be decomposed, but could be a sequence C1 C2 C3 if system has memory

• Digital or analog controls (analog signals often obtained by conversion from digital)

3

Search problems

• A search space consists of a set of objects for potential consideration during the search.

• One may search for the minimum of a function or a circuit that does the function. Points in the search space are candidate solutions (or simply solutions).

• The goal of search problems is finding solutions that respect requirements – desired properties, defined usually in respect to a function, called the objective function f.

• If the search goal is optimization the goal is to maximize f, in which case it is called utility or fitness function, or figure of merit, or minimize f, in which case it is often called cost function of energy.

• If the search goal is constraint satisfaction, f measures the degree to which a solution violates the constraints and is called penalty function, and the goal is to reduce it to zero.

N. Radcliffe in Handbook of evolutionary computation.

4

Search landscapes

• Find peaks or valleys in rugged landscape• IF looking for peaks – higher means higher degree of satisfying the objective in maximization

problems (Wright)• In an inverted perspective populations advance toward lower behavioral error. (Atmar)

• Searching for peaks depicts evolution as slow and fragile, and an optimize solution may quickly fall off the peak under slight perturbations. In the inverted perspective once a solution is reached stagnation sets in. In varying environments this never happens.

5

Problems — Landscapes

Landscape in Space• Linear Functions: contain one global maximum that

can be reached using gradient methods • Deceptive Functions: contain isolated optima, the

best points tends to be surrounded by the worst

Landspace in Time• Stationary Environment• Non-Stationary/Dynamic Environment

link to one example of deceptive function : http://www.cs.unr.edu/~humphrey/deceptive.html

6

Search/optimization algorithms and NFL Theorems

• Start with an initial "guess" at a solution, • The estimated solution is updated on an iteration-by-iteration basis with the

aim of improving the performance measure (objective function). • Multiple variables influence the function: a multivariable optimization

problem of minimizing or maximizing an objective function.• No free lunch Theorem: No search algorithm is uniformly better than all other

algorithms across all possible problems.• Cheaper lunches in certain places: Some algorithms may work better than

others on certain classes of problems as a consequence of being able to exploit the problem structure.

• E.g. traditional nonlinear programming methods (e.g., constrained conjugate gradient) are well suited to deterministic optimization problems with exact knowledge of the gradient of the objective function; more generally, stochastic gradient methods are effective if one has direct (unbiased) measurements of the gradient of the objective function.

7

Search Techniques

Complexity of the Search Space

Calculus Based Methods (Gradient Notion)

Probabilistic Methods (Evolutionary Algorithms, Hill Climbing, Simulated Annealing)

Random Search

Scope of Evolutionary Algorithms:Discontinuous, non-differentiable, Multimodal and noisy response surfaces.

Efficiency

0

1Robust Scheme

SpecializedScheme

Random Walk

Combinatorial unimodal multimodalProblem Type

8

“No-gradient” population-based searches

• Explicit modeling information describing the relationship between the parameters to be optimized and the objective function is often lacking.

• No direct measurements of gradient/derivative of objective function. • A class of recursive optimization algorithms that rely on measurements of

only the objective function to be optimized,• Population-based searches: -(with “generate and test” strategies)• Evolutionary Algorithms: search algorithms based on a loose analogy with

evolution in nature, and mechanisms of natural selection and genetics. Key characteristic features: survival of fittest and variation.

• Found to outperform other techniques on difficult problems including search on discontinuous, non-differentiable, multimodal, noisy response surfaces.

9

Principles of evolutionary processes

• Genetic program -> genotype, expressed behavioral traits -> phenotype

• Pleiotropy: a single gene may simultaneously affect several phenotypic traits.

• Polygeny: a single phenotypic characteristic may be determiend by the simultaneous interaction of many genes.

• Epistasis: expression of one gene masks the phenotypic effects of another• There are no one-gene, one-trait relationships in naturally evolved systems.

• Very different genetic structures may code for equivalent behaviors; various circuits that implement a function with electronic components.

10

Selected characteristics of neo-Darwinism and EA

1. The individual is the primary target of selection2. Genetic variation is largely a chance phenomenon: stochastic

process.3. Genotypic variation is largely a product of recombination and

“only ultimately of mutation” 4. “Gradual” evolution may incorporate phenotypic discontinuities5. Selection is probabilistic not deterministic

Mayr (1988) also in Handbook of Evolutionary Computation

11

Initialize a population of candidate solutions Evaluate

population

Select the best

individuals

Acceptable solution found?

Output solution

Use genetic operators tocreate a new

population based on old one

No

Principle of operation of evolutionary algorithms

1 0 1 11 0 1 0 0

0 1 1 0 0

Coding solutions as chromosomes. Operates on code not on solution.A string is a candidate solution.

•Switch states 11011 Bitstring •Program (+x(*x (- x 1)))•Vector (4.3 3.2 500)

12

Mechanisms that power the EA• Survival of the best - Selection pressure, chooses the basis for

new samples, in normal contexts tends to cluster solutions around the best point, or a few best points if some Pareto selection is made and solutions are ranked according to multiple criteria

• Diversification of search: mutation – trying to expand the search domain to new areas modifying the code at random

• Exploitation of search: cross-over – trying to combine known search areas by combining codes at random

• Selection clusters solutions, diversification spreads them – a fine balance needs to be obtained during the search to avoid falling into premature convergence into local optima, or, at the other extreme, with too much mutation, a derivate of random walk.

Types of evolutionary algorithms

Classes of EAs:

Evolutionary Strategies (ES);

Evolutionary Programming (EP);

Genetic Algorithms (GA);

Genetic Programming (GP);

Basic components of EAs:

Representation;

Fitness Evaluation Function.

Selection;

Recombination

• Crossover;

Mutation;

14

Evolutionary Programming/Evolutionary Strategies

• Evolutionary Programming (Lawrence J. Fogel -1962)– Representation: State machines;– Motivation: forecast nonstationary series;– Population of state machines is deterministically selected based on fitness value

assigned according to the performance in the particular task;– Selected machines are then mutated, creating then a new generation of state

machines, or a new population.

• Evolutionary strategies (Rechenberg and Schwefel – 1965) – Representation: tuple (X, ) Rn x R+ where X is a real vector representing an

individual and is a control parameter for the mutation operator;– Motivation: Mechanical shape optimization;– (+) Evolutionary Strategy: a total of individuals makes up the population.

new individuals are then selected among the original individuals, with a probability of 1/. The new pool of (+) individuals undergoes a process of mutation and recombination to generate a new population;

).exp(1 Ztt

ZXX ttt .11

Mutation in Evolutionary Strategies:New mutation parameter t+1 computed to generate new individual Xt+1. Z is a standard normal random vector and is a control parameter.

Genetic Programming

    

• Motivation for GP (Koza, early 90’s) evolutionary algorithms to evolve computer programs; • Representation: Trees composed of functional and terminal nodes;

• Genetic programming breeds a population of rooted, point-labeled trees (trees without cycles) with ordered branches opposed to the labeled cyclic graphs encountered in the world of electrical circuits;

• Developmental approach to map circuits to GP trees:- GP tree encodes a set of instructions to build a circuit from an initial one (embryonic). - Circuit structure encompasses a fixed and a modifiable part: the fixed part contains circuit source and load; the modifiable part consist of pieces of wires subject to the application of building instructions;

16

Constructing circuits with Genetic Programming

LIST

L

L GND

589u END

*

SERIESC L

15200nEND 218

uEND

LC

0.0367u END5490n

END

END

Rload   Rload

589u

0.0367u 5490n

15200n  218u

Rin

Embryonic

Rload   

Wr1, Wr2, Wr3 – Writing Heads

Vin

WR1 WR2 WR3

List

Rin

Development

Constructing tree

Final Circuit

17

• Claim: EA are blind (I.e. work well with no info about problem domain)• Fact: Seeing may helpIncorporation of domain knowledge helps the search• Representation (e.g contingency of physical cells)

• E.g. useful for not destroying useful blocks by crossover• Specific operators

• E.g. • Known good sub-circuits can get higher probability for

insertion• Constraints

• E.g. restrictions for interconnect (cells, terminals)• Increased probability for using certain connects (e.g. more

VDD points)

Use of domain knowledge

18

Representations

• Binary; Real-values vectors; Trees; State Machines.• Representation has a critical role in determining performance;• Some: Choose the representation that is most suitable for the

search algorithm; • My take: choose the representation most suitable to the problem,

then choose the search algorithm• A good representation should be:

• Simple and compact (small chromosomes);• Flexible to map solutions of various sizes and shapes;• can be of fixed or variable size, respectively processing

chromosomes of fixed or variable length. Improved flexibility of variable length representation may be accompanied by drawbacks, such as bloating problem

19

Representations

Epistasis: expression of one gene masks the phenotypic effects of another

Problems with low epistasy are too easy for GAs; • Problems with medium epistasy are the most adequate for GAs;• Problems with strong epistasy are too difficult for GAs, being random search problems;

-Use of less epistatic representation!

20

Common Representations in EHW

• Binary representations when evolving circuits on reconfigurable devices (JPL);

• Tree representation of analog circuits with developmental rules to grow analog circuits (Koza);

• Linear representation of analog circuits with developmental rules to grow analog circuits (Lohn);

• Linear representation with correction for invalid circuits and increasing length chromosomes (JPL);

21

From representation to manifestation

• A1 A2 A3 where each Block Ai has the structure• B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

B11 B12• Block Function, Block Interconnect, Analog Signal,

Parameter of passive component• 101                        2 input Amp with gain g1, out of 8 choices• 0101 1000 0011 NESW connections for 2 inputs

and output E W connect to In 1, N connects to In 2, Out goes to S and W

• 101 An analog bias of 6/8 on bias node• 1111 selection of 10K R

22

Measurement and evaluation of individuals

• Testbench, or in-system measurement:– Stimulation signals– Load on the output signals

• What do we measure in the testbench:– Time response, taking samples– Often A/D conversion for processing in digital– Frequency response, directly (Spectrum Analyzer) indirect (FFT)– Other measures, such as current– Effects: derived effects (if electrical device controls something else)

• How we assess quality of individual: an overall fitness value is determined based on individual fitness function associated to the testbench, and their weighting

23

Objective Functions

Objective function evaluates how well each individual performs.Goal: maximize the objective functionStandard Method: compute a distance to a target

Fitness F is computed over n samples;Ri – Individual Response;Ti – Target Response;Wi – Weights reflecting some knowledge of the problem

The design of a good fitness evaluation function is critical for evolution.

F = - (Wi . |Ri – Ti|)i=0

n-1

24

In evolving electronic circuits in simulations three kinds of circuit analysis are used: transient; DC transfer; and small signal analysis.

F = wi. eik

w - weight vector;ei - error of the output sample I to the desired response;k - power index applied to the error, commonly k = 2;i - index related to the time domain, DC transfer domain or frequency domain.To improve the performance:- Set weight vector components according to the problem;-Consider shape/significant point descriptors for the analysis domain: e.g. “peaks/valleys” for DC analysis; cut-off frequency for AC analysis; particular time intervals for transient analysis; - Probe internal points; - Co-evolve the weights.

Components of fitness function

25

Improvement in individual and population

• Often we care only of best individual

• Sometimes we care of a population:• For monitoring purposes

to understand better what is going on

• For fault-tolerance we may want several good “mutants” – a fault gives a mutant which still has high fitness

Example of improvement in fitness of the best individual over the generations (and improvement of average)

Generations

Fitn

ess

0 10 20 30 40 50 60 70 80 90 100-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0Fitness of best individual

Average Fitness across pop

26

Selection

• Based on the principle of survival of the fittest;• Better candidate solutions get more offspring with same/close genetic code •Deterministic in ES and EP;• Probabilistic in GA and GP

Spin theroulette

Slice in roulette and fitness of an individual are proportional

•Selection Techniques:•Proportional Selection; •Rank based selection;•Exponential Selection;•Tournament Selection;

Proportional SelectionRoulette wheel selection

Those who have higher fitness have higher probability to be selected for mating

27

Steady-state/Generational, Elitism

• Stationary environment. Steady-state we keep part of the old population

• Non-stationary/Dynamic Environment: Generational – regenerate all the population

• Elitist GA - Keep the best from old population• Keep best and some not as good

28

Keeping search diversified

• Islands, regions of search• Avoids clustering in regions that temporarily look most

promising Gold Rush• Works well for dynamic environments when best solution at

some moment may not be good at next

29

Exploitation by Recombination

• Combine good parents to exploit current solutions• Replace parents (some or all) with offspring

30

Crossover

• Genetic algorithms

• Genetic ProgrammingParents

Offspring

•Recombination of genetic material that contributes to the variability in the population;•Harmful effects: destroying potentially useful building blocks

•Automatically Defined Functions (ADFs):protection against disruptive effect of crossover.

2-point Crossover

Parents Offspring

31

Various crossovers, mutation as crossover with random string

1 point Crossover

Parents

Offspring

n point Crossover

Parents

Offspring

uniform Crossover

Parents

Offspring

32

Schemata, building blocks, ADFs

• GA/GP (EAs using crossover) use the building block theory – useful components of what makes a solution (chunks of chromosomes) can be efficiently manipulated and used to lead to the solution.

• A problem decomposition• Looking for similarities patterns in chromosomes of similarly performing

solutions• 1100 10• 0010 3• 0101 4• 1101 20 11xx may be a good building block – • schemata set of all combinations based on same pattern

Goldberg: Ensure BB supply, growth, understand BB spead, ensure good BB decisions, know BB challenges, ensure good BB mixing

• Crossover probability – rules of thumb

33

Mutation

• Each bit of a new string can be changed (mutated) with a probability given by the mutation rate;

• Low values for the mutation rate are often used;• Traditional interpretation: only support for crossover;• More recent voices: driving force of GAs: (something other

EA camps have always stated) (or it depends on problem/rep)– GAs performance largely affected by the mutation rate.

Crossover with random string

Parent 1

Offspring

Parent 2 random string

34

Adaptive operators

• Adapting the probability associated to evolutionary operators improves convergence

• Crossover probability• Mutation probability• Change representation• Change selection probability and method – all is permitted

35

Specific Crossover and Mutation

MUTATION(RETIME)

Node (Gene )Selected For

Mutation

D

Initial Solution

D

Node (Gene)'Mutated' To

Another'Location'

Mutated Solution

Example of Mutation Operation

Examples from T. Arslan’s work

36

Specific Genetic Algorithm Implementation

D D

D

Critical Path=2 Pipeline

D

Critical Path=2

Retime

D

Pipeline

Parent 1

Parent 2

D D

D

Critical Path=1Pipeline

Retime

D D

D

Critical Path=1 Pipeline

Retime

D

Pipeline

Child 1

Child 2

Problem Specific Crossover

• Pipeline Identified In Parent 1• Retime Identified In Parent 2• Transformations Crossed-Over To Produce 2 Children

Examples from T. Arslan’s work

37

Population size, Generations, Stopping, multiple runs

• Fixed or variable• Small populations – more generations, vice-versa, ballanced• ~ 100 individuals very common • Usually GP asks for more: eg 640,000 in some of Koza’s

experiments• Hundreds of generations• Sampling a small % of space • See if it is still improving tracing amount of changes in last

generations• Stop: nr of gen, time, lack of improvement• Re-start, change initial population, seed with solutions

38

PopulationInitialization(Randomly)

Evaluation• Simulators (SPICE)• Hardware (PTAs)

New Population(Pop. Size 512,

24 bits)

Evaluations(30 samples)

Fitness(MSE)

0.11

Population

01001110111001

0

0,5

1

1,5

2

2,5

3

3,5

0,22

0,36

0,51

0,66

0,82

0,97

1,12

1,27

1,43

1,58

1,73

1,89

2,04

2,19

2,34 2,5

2,65 2,8

2,95

3,11

3,26

3,41

3,57

3,72

3,87

4,03

4,18

4,35 4,5

4,46

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

0,22 0,51 0,82 1,12 1,43 1,73 2,04 2,34 2,65 2,95 3,26 3,57 3,87 4,18 4,5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

4

0,22 0,51 0,82 1,12 1,43 1,73 2,04 2,34 2,65 2,95 3,26 3,57 3,87 4,18 4,5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0,22

0,36

0,51

0,66

0,82

0,97

1,12

1,27

1,43

1,58

1,73

1,89

2,04

2,19

2,34 2,5

2,65 2,8

2,95

3,11

3,26

3,41

3,57

3,72

3,87

4,03

4,18

4,35 4,5

4,46

Vin(V)

Vin(

V)

0.34

0.10

0.53

0.10

01001010110000

10110111110110

01011110111001

0.15

0.29

01001110110000

11001110111001

01100010100001

01000111101000

0.11

0.27

0.34

0.530.39

Sort

01001010110000

01000111101000 01001010110000

Two Points Crossover(Prob. 70%)

Elite(10%)

Uniform Mutation(Prob. 4%)

Elite(10%)

Binary Tournament Selection (size 2)

Elite(10 %)

0.390.15

01011110111001

10110111110110 0.53

0.29 01011110111001

01001110110000

01011010111001

Best

01101111110000

11011010011011

RecombinedIndividuals

RecombinedIndividuals

0

0,5

1

1,5

2

2,5

3

3,5

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

Vin(V)

Vout

(V)

0

0,5

1

1,5

2

2,5

3

3,5

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

Vin(V)

Vout

(V)

Vout

Vin

Vout

Vin

Vout

Vin

Vout

Vin

Compare to

Target

2*2 rand

Evolutionary algorithms visualized

39

Guiding evolution

• Shaping – increase difficulty• Guide in certain directions of search space rewarding certain

types of solutions• Pressure can guide away from certain regions – e.g. avoid

patented solutions, avoid areas with faults• Island based GA – search with multiple populations

interacting during evolution

40

Multi-criteria optimization, trade-offs, Pareto optimality

• The simultaneous optimization of multiple, possibly competing, objective functions deviates from the single-function optimization in that it seldom admits a perfect (or Utopian) solution;

• Instead, multi-objective optimization problems tend to be characterized by a family of alternatives that must be considered equivalent in the absence of information concerning the relevance of each objective relative to the others;

• Two different methods: Plain aggregating approaches and Pareto-based approaches;

• Plain aggregating approaches perform the scalarization of the objective vectors:

• each objective, fi(x), multiplied by the weight wi.

n

iii xfwxf

1

)()(

41

Multi-criteria optimization, trade-offs, Pareto optimality

• Given a population of GA individuals, one particular individual v dominates another individual u if and only if :

• Where vi and ui represents the fitness value referring to objective i for the individuals u and v respectively. In this particular example, there are n objectives, that have to be minimized.

• Different implementations:• all the non-dominated individuals are removed from the

population and given the same probability of reproduction, higher than the one for dominated individuals (Goldberg);

• individual’s rank (using rank selection) corresponds to the number of individuals in the current population by which it is dominated (Fonseca and Flemming) .

• Non-dominated individuals are always selected for crossover/mutation (Schnier and Yao)

iiii uvniuvni },,...,1{},,...,1{

42

• Analog circuit design is intrinsically multi-objective;• Conventional design usually decomposes synthesis tasks into two sub-tasks:

• general performance requirements (Ex: frequency response) • specific circuit requirements (Ex: noise and fault-tolerance);

• The designer may choose among a number of solutions provided by the Genetic Algorithm.

General Fitness Expression

Fitness = (wi. fi)

wi Weight Vector component for objective i;fi Fitness of the objective i.

How to find “optimal” weight vector? Co-evolution of circuits and weights (Lohn,1998) (Zebulum, 1998)

Multi-Objective Circuit Optimization

43

Enhancements in EAs

• Adaptive mutation rate:– Escape local optima by increasing rate of mutation;

• Speciation:– Keep diversity by creating sub-populations;– Multimodal problems: subpopulations sampling

different and interesting solutions to a particular problem.

• Variable Length Representations:– Map solutions of different sizes;– Evolution of electronic circuits of different sizes.

44

Co-evolution idea

• Two populations that evolve together• Powerful evolutionary dynamics: arms’ races• Predator’s and prey’s performance improve together to maintain

equilibrium. If predator gets faster so does pray in order to survive. If predator gets too fast they eat too much pray and later they have not enough left to survive. The system self-regulates.

45

In coevolution a second population mimics dynamic fitness functions.

po p ulation o fhard w are de signs

fitn e ssfu nction

re pre se n te d asge ne tic

prog rams

f = f(g1 , g2 , ... gn)

• Static fitness function• Difficult to weight objectives

p op u lation o ftarg e t v e cto rs

re pre se nte d asge ne tic

p ro grams

th re sh o ldv alu e s

<g1 , g2 , ... gn>

<g1 , g2 , ... gn>

<g1 , g2 , ... gn>

po pu lation o fhard ware d e sign s

Std Evolution Coevolution Dynamic, multiple fitness functions Objectives/weights are evolved

Lohn in evolving antennas

46

Coevolutionary Algorithm• Concept: the goals start out easy, then increase in difficulty as the

hardware designs improve (ZPD: zone of proximal development)• Hardware designs are rewarded for solving goal vectors, with extra points

for solving difficult goals:

• Goals that are too difficult (s.t. no design can solve it) or too easy are given low fitness. Difficulty is defined:

igvj

i

i

gv

s

solvethatdesigns#

1

10.1

)1(1

00.1)(

1jj

j

j xxM

xgvF

leastdifficult

m ostdifficult

1 M 1

go al v e ctord ifficu lty

# d e sign sth at so lv e ag o al v e ctor

47

Other techniques: SPSA

• Simultaneous perturbation stochastic approximation (SPSA) method.

• The essential feature of SPSA, which provides its power and relative ease of use in difficult multivariate optimization problems, is the underlying gradient approximation that requires only two objective function measurements per iteration regardless of the dimension of the optimization problem. These two measurements are made by simultaneously varying in a "proper" random fashion all of the variables in the problem (the "simultaneous perturbation"). This contrasts with the classical ("finite-difference") method where the variables are varied one at a time. If the number of terms being optimized is p, then the finite-difference method takes 2p measurements of the objective function at each iteration (to form one gradient approximation) while SPSA takes only two measurements. A fundamental result on relative efficiency then follows:

• Under reasonably general conditions, SPSA and the standard finite-difference SA method achieve the same level of statistical accuracy for a given number of iterations even though SPSA uses p times fewer measurements of the objective function at each iteration (since each gradient approximation uses only 1/p the number of function measurements). This indicates that SPSA will converge to the optimal solution within a given level of accuracy with p times fewer measurements of the objective function than the standard method.

• http://www.jhuapl.edu/SPSA/  

48

Hybrid search: efficient-global followed by efficient-local. Combine evolution with learning

• “Standard” GA are efficient global search algorithms but no as good for local search

• Solutions: Hybrid GA• Combine with local search:• Genetic local search• Lamarkian GA• Imanishian GAs

In general, methods involving a population of candidate solutions, such as evolutionary algorithms, may be useful for a broad search over the domain of the parameters being optimized and subsequent initialization of more powerful local search algorithms.

(the local search is not as often implemented as it should)

49

• First Stage: GA-based Evolution of the circuit topology;• Second Stage: GA-based Optimization of the transistor sizes for the best

topology resulted in the first stage. Initialization is made with the best topology and random parameters.

Second Stage (40 generations)

First Stage (200 generations)

1.2/4.8 3.6/1.2

1.2/2.4 8.4/2.4

6.0/4.8 9.6/9.6

4.8/4.8 4.8/3.6

7.2/1.2 8.4/7.2

8.4/7.21.2/7.2

6.0/2.4 4.8/3.6

4.8/6.0 8.4/3.6

Multiplier Evolved Through Multi-stage search W/L in um

Multi-stage Search: Search for topology followed by parameter optimization

50

Example of Software Structure

Initialize

Randomize

Main

App_init Initdata

Initmalloc Initpop

App_malloc Warmup_random Flip Objfunc

Generation

Randomperc

Rnd

StatisticsSelectCrossoverPreselectMutation

Advance_random

Flip

T. Arslan’s work

51

Name ARM Thumb Power Code SizeCurrent Time Code Size Current Time Code Size Ratio Ratio(mA) (CLK) (Byte) (mA) (CLK) (Byte) (%) (%)

App malloc 3.3 146 1835 3.1 253 1563 61 85Advance random 3.8 8051 3763 3.6 10751 3335 79 88Objfun 3.9 38910 11739 3.6 46619 10023 90 85Statistic 4.0 14875 4259 3.7 19134 4187 84 98Initdata 4.7 1789 20 3.5 2555 12 94 60App init 3.4 403 488 3.4 559 324 72 66Preselect 4.3 6334 3691 4.0 7826 3567 87 96Initmalloc 3.6 14008 1987 3.1 21996 1667 74 83Warmup random 4.0 44572 4431 3.6 59904 4063 83 91Randomperc 3.8 8281 4631 3.6 12159 4195 7290Randomize 4.0 43020 4519 3.6 57608 4219 83 93Flip 3.7 8332 4759 3.6 11061 4556 77 95Rnd 3.8 8375 6143 3.2 11064 5415 90 88Initpop 3.7 948736 22508 3.6 1355264 15815 72 70Mutation 3.9 427520 4859 3.6 627712 4591 74 94Crossover 3.8 8704 6403 3.6 12544 6011 73 93Select 3.8 17018 8219 3.6 24752 7871 72 95Initialize 4.0 1001472 7439 3.5 1268480 6583 90 88Generation 3.8 1014272 16343 3.6 1281792 15207 91 80

Currents and Execution Times for GA subroutines Using ARM and Thumb

T. Arslan’s work