genetic algorithms by: jacob noyes 4/16/2013. traveling salesman problem given: a list of cities ...
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Traveling Salesman Problem
Given: A list of cities Distances between each city
Find: Shortest path to reach every city once
Traveling Salesman Problem
Brute force Exact Algorithms
Pro: Will find the right answer
Con: Does not scale well
Evolution
Change in inherited characteristics over timeNatural Selection:
A mechanism through which evolution happens Survival of the fittest Genes of the more suitable organisms get passed on more
oftenDeoxyribonucleic acid(DNA):
Genes encoded in amino acids Used to pass genes from parent to offspring
Genetic Algorithms: The Basics
Genetic algorithms are specialized searchheuristics which use the fundamental principles of evolution through natural selection to find the best solution to a problem.
Encoding
Changeable representation of individual's traits is created
Completed only at the startIts “string” is designed a series of bitsConcatenate multiple parameters
Examples: Max y-values?
Example 1: y = -x^2 + 255x
0 ≤ x ≤ 255
String: 00000000 ≤ x ≤ 11111111Example 2:
y = 2w + x + 3z
0 ≤ w ≤ 7, 0 ≤ x ≤ 7, 0 ≤ z ≤ 7 String: 000/000/000 – 111/111/111
Initialization
Beginning population is created
Each bit(gene) is randomly generated to create variety
Performed only for the first generation and not repeated
Example: Initialization
Given y = -x^2 + 255x 0 ≤ x ≤ 255
X Binary x188 1011110048 0011000075 01001011104 01101000249 1111100110 00001010134 10000110125 01111101
Selection
Assign a fitness measure of how close a solution is to fulfilling the problem Assigned to each individual
Select individuals Individuals with higher fitness will reproduce more often Non-selected individuals will “die off”
Example: Fitness
Given y = -x^2 + 255x 0 ≤ x ≤ 255
X Binary x Fitness188 10111100 1259648 00110000 9936 75 01001011 13500104 01101000 15704249 11111001 1494 10 00001010 2450 134 10000110 16214125 01111101 16250
Optimums
Local optimum: A point where small changes will lead to worse results
Overall optimum: The best solution
Selection: Categories
Proportionate Selection: Fitness relative to other individuals
Ranking Selection: Chance to reproduce based on order
Tournament Selection: Pits individuals against each other in smaller brackets
Gender Specific Selection: Splits Individuals into groups based on “sex”
Genetic Relatedness Based Selection: Individuals are selected based on their genetic distance from others in the population
Selection: Proportionate Selection
Roulette wheel selectionDeterministic SamplingStochastic Remainder Sampling with ReplacementStochastic Remainder Sampling without ReplacementStochastic Universal Selection
Roulette Wheel Selection
1. Find Pf: Population fitness = sum of all fitness factors2. Find Psel: Each individual's probability of selection
Psel = (fitness factor) / Pf3. Load each Psel into an array4. Generate random number between 0-1005. Start at beginning of array, subtract each Psel from number until number <= 0
Deterministic Sampling
1. Average fitness is found2. Individual fitnesses are divided by the average3. Whole number results = number of spots in the mating pool4. Extra slots filled starting by highest decimal5. Random numbers generated to select individuals from the mating pool
Stochastic Remainder with Replacement
Uses Deterministic Sampling to fill slots with whole number resultsLeft over slots are then filled using the remainders with the Roulette Wheel
Selection Method
Stochastic Remainder without Replacement
Uses Deterministic Sampling to fill slots with whole number resultsUses a “weighted-coin toss” to determine the rest
1. Each remainder multiplied by 100 2. Random number between 0-100 generated 3. If random number <= remainder, accept 4. Loop until all spots are filled
Ranking Selection
Chance to breed based on order of fitness, not proportionPro: Easy to implement and understandCon: Generally less accurate, less efficient, and
phase out diversity too quicklyDue to cons, not used oftenTypes:
Linear ranking selection Truncate Selection
Linear Ranking Selection
1. Probabilities are set up for each rank before fitnesses are even assessed
2. Individuals are ordered based on fitness level
3. The predefined probabilities are assigned to their rank
4. Individuals are selected based on the probabilities
Truncate Selection
1. Candidates are put in order based on fitness2. The top predefined percentage are chosen to reproduce
Tournament Selection
Individuals are pitted against each other in smaller bracketsThe winner(s) of each bracket reproducesBracket participants only need to know fitness levels of others in bracket
No need for total or average population fitness factors Good for situations when it is impossible or implausible to
calculate totals
Tournament Selection: Categories
Binary Tournament SelectionLarger Tournament SelectionBoltzmann Tournament SelectionCorrelative Tournament Selection
Binary Tournament Selection
1. Two candidates are randomly selected out of possible solutions
2. Candidate with best fitness factor is chosen to reproduce
Larger Tournament Selection
1. More than two candidates are randomly selected out of possible solutions
2. Candidate with best fitness factor is chosen to reproduce
Only difference from Binary Tournament Selection is number of candidates in each bracket
More candidates = higher selection pressure
Boltzmann Tournament Selection
N = temperature = variable describing number of differences in bit string between two individuals
1. First candidate is chosen randomly2. Second candidate is chosen as having exactly n differences in gene
string from first candidate3. Third candidate is chosen
Half of the time has exactly n differences in gene string from first AND second candidate (strict choice)
Other half of the time has exactly n difference in gene string from ONLYfirst candidate (relaxed choice)
4. Choose the winner of the three to reproduce
Correlative Tournament Selection
Not so much a separate selection method as much as an extension of othertournament selections
Once mating pool is selected, pairs are created based on how closely theyare related
Pairing similar individuals allows a better chance of passing on their (probably) good similar trait
Gender Specific Selection
Genetic Algorithm with Chromosome Differentiation(GACD)Restricted MatingCorrelative Family-based Selection
Genetic Algorithms with Chromosome Differentiation
Every individual has an extra 00 or 01 attached to their bit string 00 = female, 01 = male When a male and female mate each parent randomly selects a bit to pass
onto the child Females(00) can pass on 0 or 0 Males(01) can pass on 0 or 1
Hamming distance: the sum of the differences between each bit of two individuals Ex: 00011111 and 11111111 have a hamming distance of 3.
Genetic Algorithm With Chromosome Differentiation
1. Males generated first randomly2. Females created for each male with maximum hamming distance3. Select individuals to put into mating pool by either:
Using a separate selection method for each sex Or, lumping them together and using one selection method over all of them
4. Mate each individual in the mating pool twice5. If there are fewer of one sex in the mating pool, mate leftovers with the
highest fitness individual of the opposite sex
Restricted Mating
In nature, different species cannot or will not mateRestricted mating is based on species differentiationsCertain traits (predefined sections of the bit string) must be the same to
mate two candidatesKeeps several variations from converging to a local optimum
Correlative Family-based Selection
1. Two candidates are mated together twice2. Between the two candidates and the two children, the most fit
solution is chosen3. The hamming distance is calculated for each individual compared
to the other three4. The individual with the highest hamming distance is also chosen
to reproduce
Genetic Relatedness Based Selection
Purpose is to search unexplored areas of the search spaceGroups candidates based on similar fitness factorsDoes not try to find most fit candidatesIncludes:
Fitness Uniform Selection Scheme(FUSS) Reserve Selection
Fitness Uniform Selection Scheme
Candidates with similar fitness factors are grouped togetherRandom numbers are generated from the range of minimum fitness to
maximum fitnessCandidates with fitnesses closest to the random number are selectedThis gives a higher probability of selecting unexplored areasHelps avoid local optimums
Reserve Selection
Candidates split into two categories Non-reserved: Normal candidates with normal selection process applied Reserved: Specific less fit candidates that are carried over from generation
to generation to keep variety in the populationKeeps pool out of local maximums
Elitism
Automatically carry over most fit individual to next generationExtension of other selection methodsMakes sure best fit does not just get unlucky
Example: Selection
Given y = -x^2 + 255x 0 ≤ x ≤ 255 Top half truncate selection
Gene pool01111101100001100110100001001011
X Binary x Fitness188 10111100 12596 48 00110000 9936 75 01001011 13500 104 01101000 15704 249 11111001 1494 10 00001010 2450 134 10000110 16214 125 01111101 16250
Crossover
Genes(bit strings) are combined from both parents to create offspringLocus: the randomly generated point(s) at which each parent's bit string
is separated
Example: Crossover
Candidate Gene pool Locus1 01111101 32 10000110 1 3 01101000 1 4 01001011 6
Parents P1 String P2 String Offspring1, 2 011 00110 011001102, 1 100 11101 100111012, 3 1 1101000 111010003, 2 0 0000110 000001103, 4 0 1001011 010010114, 3 0 1101000 011010004, 1 010010 01 010010011, 4 011111 11 01111111
Mutation in The Natural World
Brings diversity to a populationWithout mutation, just different combinations of the same traitsMutations happen when DNA is not copied properlyIf the mutation has a benefit, or is just not a hindrance, it may be passed
on to new generations
Mutation in Genetic Algorithms
Purposely inject after crossover Rate of mutation is decided beforehand
Ex: 1/2000th chance of mutation per bitFor every bit in a population, a random number is generatedIf the probability hits, the bit is XOR'ed with 1
Given mutation rate: 1/64
Example: Mutation
Pre-mutated XOR Offspring01100110 00000000 0110011010011101 00000000 1001110111101000 00000010 1110101000000110 00000000 0000011001001011 00000000 0100101101101000 00000000 0110100001001001 00000000 0100100101111111 00000000 01111111
Genetic Algorithms: End
Fitness threshold based Each solution's fitness level is checked after each generation If a given minimum fitness level is achieved, the algorithm finishes
running and outputs the maximum fitness candidateGeneration threshold based
Genetic algorithm runs for a predefined number of generations Most fit solution over all generations is outputted
Uses of Genetic Algorithms
Optimal water network layoutsFacial recognitionRoboticsTrajectories for spacecraftFun with walkingMuch More