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Genetic Algorithms CSCI-2300 Introduction to Algorithms

Genetic Algorithms CSCI-2300 Introduction to Algorithms. David Goldschmidt, Ph.D. Rensselaer Polytechnic Institute April 28, 2014. Evolutionary Computing. Evolutionary computing produces high-quality partial solutions to problems through natural selection and survival of the fittest

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Genetic Algorithms CSCI-2300 Introduction to Algorithms

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  1. Genetic AlgorithmsCSCI-2300 Introduction to Algorithms David Goldschmidt, Ph.D. Rensselaer Polytechnic Institute April 28, 2014

  2. Evolutionary Computing • Evolutionary computing produceshigh-quality partial solutionsto problems throughnatural selection andsurvival of the fittest • Compare to naturalbiological systems thatadapt and learn over time

  3. Genetic Algorithm Example • Find the maximum value of function f(x) = –x2 + 15x • Represent problem using chromosomes built from four genes: http://www.webgraphing.com

  4. Genetic Algorithm Example • Initial random population of size N = 6:

  5. fitness function here issimply the original function f(x) = –x2 + 15x Genetic Algorithm Example • Determine chromosome fitness foreach chromosome: 218 100.0

  6. Genetic Algorithm Example • Use fitness ratios to determine which chromosomes are selected for crossoverand mutation operations:

  7. Genetic Algorithm Example • Converge on a near-optimal solution:

  8. Convergence Example global maximum local maximum

  9. Genetic Algorithms – Step 1 • Represent the problem domain asa 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/

  10. Genetic Algorithms – Step 2 • Define a fitness functionf(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

  11. . . . 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

  12. Genetic Algorithms – Step 4 • Calculate the fitness of eachindividual chromosome using f(x): • f(x1), f(x2), ..., f(xN) • Order the population based on fitness values

  13. Genetic Algorithms – Step 5 • Using pc, select pairs of chromosomesfor crossover • Using pm, select chromosomes for mutation • Chromosomes are selectedbased on their fitnessvalues using aroulette wheel approach:

  14. Genetic Algorithms – Step 6 • Create a pair of offspring chromosomes by applying a crossover operation:

  15. Genetic Algorithms – Step 6 • Mutate an offspring chromosome by applyinga mutation operation:

  16. 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

  17. Genetic Algorithms – Steps 9 & 10 • Step 9: • Replace the initial population withthe new population • Step 10: • Go back to Step 4 and repeat the processuntil termination criteria are satisfied • Typically repeat this process for 50-5000+ generations

  18. Iteration

  19. Crossword Puzzle Construction • Given: • Dictionary of valid wordsand phrases • Empty crossword grid • Problem: • Fill the crossword grid suchthat all words both acrossand down are valid • (assign clues later)

  20. 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

  21. 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 ?

  22. Computational Complexity • How long does it take for an algorithm toproduce a solution? • Depends on the size of the input andthe complexity of the algorithm • The size of the input is n • The complexity of the algorithm is classifiedbased on its expected run time

  23. 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(cn) • Factorial time: O(n!) P NP

  24. Genetic Algorithms • Genetic algorithms are often well-suited to producing reasonable solutions to intractable problems • Intractable problems are problems withexcessive 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

  25. yikes! Genetic Algorithms Example • 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 seehttp://bio.math.berkeley.edu/classes/195/2000/lec14/index.html

  26. Genetic Algorithms Example • 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 membersof the population • e.g. combining ABCDEF and BDAFCE may result in ABCFCE

  27. 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...

  28. 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

  29. Genetic Algorithms Example • Crossing ACEDB with ABCED... Crossover Operation

  30. another approach: http://www.dna-evolutions.com/dnaappletsample.html Genetic Algorithms Example • Combining ACEDB with ABCED... ...yields ACBED from A.K. Dewdney’s The (New) Turing Omnibus, Computer Science Press, New York, 1993

  31. 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

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