Genetic AlgorithmsCSCI-2300 Introduction to Algorithms
David Goldschmidt, Ph.D.Rensselaer Polytechnic InstituteApril 28, 2014
Evolutionary Computing Evolutionary computing produces
high-quality partial solutions to problems throughnatural selection andsurvival of the fittest – Compare to natural
biological systems thatadapt and learn over time
Integer Binary code Integer Binary code Integer Binary code1 112 7 123 8 134 9 145 10 15
6 1 0 1 11 1 0 01 1 0 11 1 1 01 1 1 1
0 1 1 00 1 1 11 0 0 01 0 0 11 0 1 0
0 0 0 10 0 1 00 0 1 10 1 0 00 1 0 1
Genetic Algorithm Example Find the maximum value of function
f(x) = –x2 + 15x
– Represent problem using chromosomes built from four genes:
http://www
.webgraphi
ng.com
Genetic Algorithm Example Initial random population of size N = 6:
Chromosomelabel
Chromosomestring
Decodedinteger
Chromosomefitness
Fitnessratio, %
X1 1 1 0 0 12 36 16.5X2 0 1 0 0 4 44 20.2X3 0 0 0 1 1 14 6.4X4 1 1 1 0 14 14 6.4X5 0 1 1 1 7 56 25.7X6 1 0 0 1 9 54 24.8
x
50
40
30
20
60
10
00 5 10 15
f(x)
(a) Chromosome initial locations.x
50
40
30
20
60
10
00 5 10 15
(b) Chromosome final locations.
Genetic Algorithm Example Determine chromosome fitness for
each chromosome:
218 100.0
Chromosomelabel
Chromosomestring
Decodedinteger
Chromosomefitness
Fitnessratio, %
X1 1 1 0 0 12 36 16.5X2 0 1 0 0 4 44 20.2X3 0 0 0 1 1 14 6.4X4 1 1 1 0 14 14 6.4X5 0 1 1 1 7 56 25.7X6 1 0 0 1 9 54 24.8
x
50
40
30
20
60
10
00 5 10 15
f(x)
(a) Chromosome initial locations.x
50
40
30
20
60
10
00 5 10 15
(b) Chromosome final locations.
fitness function here issimply the original function
f(x) = –x2 + 15x
100 0
36.743.149.5
75.2
X1: 16.5%X2: 20.2%X3: 6.4%X4: 6.4%X5: 25.3%X6: 24.8%
Genetic Algorithm Example Use fitness ratios to determine which
chromosomes are selected for crossoverand mutation operations:
Genetic Algorithm Example Converge on a near-optimal solution:
Chromosomelabel
Chromosomestring
Decodedinteger
Chromosomefitness
Fitnessratio, %
X1 1 1 0 0 12 36 16.5X2 0 1 0 0 4 44 20.2X3 0 0 0 1 1 14 6.4X4 1 1 1 0 14 14 6.4X5 0 1 1 1 7 56 25.7X6 1 0 0 1 9 54 24.8
x
50
40
30
20
60
10
00 5 10 15
f(x)
(a) Chromosome initial locations.x
50
40
30
20
60
10
00 5 10 15
(b) Chromosome final locations.
Convergence Example
local maximum
global maximum
Genetic Algorithms – Step 1 Represent the problem domain as
a chromosome of fixed length– Use a fixed number of genes to represent a
solution– Use individual bits or characters for
efficientmemory use and speed
– e.g. Traveling Salesman Problem (TSP) http://www.lalena.com/AI/Tsp/
1 10 1 0 1 0 0 0 0 0 1 0 1 10
Genetic Algorithms – Step 2 Define a fitness function f(x) to
measurethe quality of individual chromosomes
The fitness function determines– which chromosomes carry over to the next
generation – which chromosomes are crossed over with
one another– which chromosomes are individually
mutated
Genetic Algorithms – Step 3 Establish our genetic algorithm
parameters:– Choose the size of the population, N – Set the crossover probability, pc – Set the mutation probability, pm
Randomly generate an initial populationof chromosomes:– x1, x2, ..., xN
1 10 1 0 1 0 0 0 0 0 1 0 1 101 10 1 0 1 0 0 0 0 0 1 0 1 10
1 10 1 0 1 0 0 0 0 0 1 0 1 10
1 10 1 0 1 0 0 0 0 0 1 0 1 10
1 10 1 0 1 0 0 0 0 0 1 0 1 10
1 10 1 0 1 0 0 0 0 0 1 0 1 10
1 10 1 0 1 0 0 0 0 0 1 0 1 10
. . .
Genetic Algorithms – Step 4 Calculate the fitness of each
individual chromosome using f(x):– f(x1), f(x2), ..., f(xN)
Order the population based on fitness values
100 0
36.743.149.5
75.2
X1: 16.5%X2: 20.2%X3: 6.4%X4: 6.4%X5: 25.3%X6: 24.8%
Genetic Algorithms – Step 5 Using pc, select pairs of chromosomes
for crossover Using pm, select chromosomes for
mutation Chromosomes are selected
based on their fitnessvalues using aroulette wheel approach:
X6i 1 00 0 01 0 X2i
0 01 0X2i 0 11 1 X5i
0X1i 0 11 1 X5i1 01 0
0 10 0
11 101 0
Genetic Algorithms – Step 6 Create a pair of offspring chromosomes
by applying a crossover operation:
0 11 1X5'i 01 0
X6'i 1 00
0 01 0X2'i 0 1
0 0
0 1 111X5i
1 1 1 X1"i1 1
X2"i0 1 0
0X1'i 1 1 1
0 1 0X2i
Genetic Algorithms – Step 6 Mutate an offspring chromosome by
applyinga mutation operation:
Genetic Algorithms – Steps 7 & 8 Step 7:
– Place all generated offspringchromosomes in a new population
Step 8:– Go back to Step 5 until the size of the new
population is equal to the size of the initial population, N
1 01 0X1i
Generation i
0 01 0X2i
0 00 1X3i
1 11 0X4i
0 11 1X5i f = 56
1 00 1X6i f = 54
f = 36
f = 44
f = 14
f = 14
1 00 0X1i+1
Generation (i + 1)
0 01 1X2i+1
1 10 1X3i+1
0 01 0X4i+1
0 11 0X5i+1 f = 54
0 11 1X6i+1 f = 56
f = 56
f = 50
f = 44
f = 44
Crossover
X6i 1 00 0 01 0 X2i
0 01 0X2i 0 11 1 X5i
0X1i 0 11 1 X5i1 01 0
0 10 0
11 101 0
Mutation
0 11 1X5'i 01 0
X6'i 1 00
0 01 0X2'i 0 1
0 0
0 1 111X5i
1 1 1 X1"i1 1
X2"i0 1 0
0X1'i 1 1 1
0 1 0X2i
Genetic Algorithms – Steps 9 & 10 Step 9:
– Replace the initial population withthe new population
Step 10:– Go back to Step 4 and repeat the process
until termination criteria are satisfied– Typically repeat this process for 50-5000+
generations
1 01 0X1i
Generation i
0 01 0X2i
0 00 1X3i
1 11 0X4i
0 11 1X5i f = 56
1 00 1X6i f = 54
f = 36
f = 44
f = 14
f = 14
1 00 0X1i+1
Generation (i + 1)
0 01 1X2i+1
1 10 1X3i+1
0 01 0X4i+1
0 11 0X5i+1 f = 54
0 11 1X6i+1 f = 56
f = 56
f = 50
f = 44
f = 44
Crossover
X6i 1 00 0 01 0 X2i
0 01 0X2i 0 11 1 X5i
0X1i 0 11 1 X5i1 01 0
0 10 0
11 101 0
Mutation
0 11 1X5'i 01 0
X6'i 1 00
0 01 0X2'i 0 1
0 0
0 1 111X5i
1 1 1 X1"i1 1
X2"i0 1 0
0X1'i 1 1 1
0 1 0X2i
Iteration
Crossword Puzzle Construction Given:
– Dictionary of valid wordsand phrases
– Empty crossword grid
Problem:– Fill the crossword grid such
that all words both acrossand down are valid
(assign clues later)
1 01 0X1i
Generation i
0 01 0X2i
0 00 1X3i
1 11 0X4i
0 11 1X5i f = 56
1 00 1X6i f = 54
f = 36
f = 44
f = 14
f = 14
1 00 0X1i+1
Generation (i + 1)
0 01 1X2i+1
1 10 1X3i+1
0 01 0X4i+1
0 11 0X5i+1 f = 54
0 11 1X6i+1 f = 56
f = 56
f = 50
f = 44
f = 44
Crossover
X6i 1 00 0 01 0 X2i
0 01 0X2i 0 11 1 X5i
0X1i 0 11 1 X5i1 01 0
0 10 0
11 101 0
Mutation
0 11 1X5'i 01 0
X6'i 1 00
0 01 0X2'i 0 1
0 0
0 1 111X5i
1 1 1 X1"i1 1
X2"i0 1 0
0X1'i 1 1 1
0 1 0X2i
Crossword Puzzle Construction Genetic Algorithm (GA)
– Evolve a solution by crossovers andmutations through many generations
– Initial population of crossword grids: Random letters? Random letters based on Scrabble® frequencies? Random words from dictionary?
– Fitness of each grid is number of valid words
Termination Criteria When do we stop?
– Pause a genetic algorithm after agiven number of generations, thencheck the fittest chromosomes
If the fittest chromosomes are fitbeyond a given threshold,terminate the genetic algorithm
– Also consider stopping when the highest fitness value does not change for a large number of generations
?
How long does it take for an algorithm toproduce a solution?– Depends on the size of the input and
the complexity of the algorithm
– The size of the input is n
– The complexity of the algorithm is classifiedbased on its expected run time
Computational Complexity
Big-O notation measures the expected run timeof an algorithm (i.e. its computational complexity)– Constant time: O(1) – Logarithmic time: O(log n) – Linear time: O(n) – Linearithmic time: O(n log n) – Quadratic time: O(n2) – Exponential time: O(c n) – Factorial time: O(n!)
Computational Complexity
P
NP
Genetic algorithms are often well-suited to producing reasonable solutions to intractable problems– Intractable problems are problems with
excessive computational complexity i.e. in the Nondeterministic Polynomial (NP) class
of problems
– A reasonable solution is a partial or inexact solution that adequately solves the problem in polynomial time
Genetic Algorithms
Consider the Traveling Salesman Problem (TSP) in which a salesman aims to visit n cities exactly once covering the least distance http://mathworld.wolfram.com/TravelingSalesmanProblem.html http://www.tsp.gatech.edu/games/index.html
– Starting at any given node, choose from n–1 remaining nodes, then choose from n–2 remaining nodes, etc.
– Testing every possible route takes (n–1)! steps
see http://bio.math.berkeley.edu/classes/195/2000/lec14/index.html
Genetic Algorithms Example
yikes!
Use a genetic algorithm to evolve a near-optimal solution to the TSP– Label cities A, B, C, D, E, F, etc.– Example circuits: ABCDEF, BDAFCE,
FBECAD
– How do we perform crossover operations? Basic crossovers might result in invalid members
of the population e.g. combining ABCDEF and BDAFCE may result
in ABCFCE
Genetic Algorithms Example
Key challenge of developing a genetic algorithm is often the representation of the problem– For TSP, consider a standard ordering
ABCDEF, assigning the code 123456– All other sequences encoded
based on the removal of letters
– Basic crossover works...
Genetic Algorithms Example
All other sequences encoded based on the removal of letters from standard ordering– Sequence BDAFCE has code 231311
B is 2 in ABCDEF D is 3 in ACDEF A is 1 in ACEF F is 3 in CEF C is 1 in CE E is 1 in E
Genetic Algorithms Example
Crossing ACEDB with ABCED...
Crossover Operation
Genetic Algorithms Example
Combining ACEDB with ABCED......yields ACBED
from A.K. Dewdney’s The (New) Turing Omnibus, Computer Science Press, New York, 1993
Genetic Algorithms Example
another approach: http://www.dna-evolutions.com/dnaappletsample.html
Genetic Algorithms Advantages of genetic algorithms:
– Often outperform “brute force” approaches by randomly jumping around the search space
– Ideal for problem domains in which near-optimal (as opposed to exact) solutions are adequate
Disadvantages of genetic algorithms:– Might not find any satisfactory partial
solutions– Tuning can be a challenge
