N Queens Solution with Genetic Algorithm

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N- Queens Solution with Genetic Algorithm

• By
• Mohammad A. Ismael

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N-Queens problem Definition

• Place N queens on an NN board so that no queen
  is attacking another queen.
• A queen can move horizontally, vertically, or
  diagonally.
• The problem can be solved with genetic algorithm
  for a n queens problem. (n is between 8 and 30)

Here N8
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8-Queens problem

• Simple solutions may lead to very high search
  costs
• 64 fields, 8 queens gt 648 possible sequences
• Genetic algorithm solution trim the search space.

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Problem formulation

• Initial State n queens placed randomly on the
  board, one per column.
• Successor function moving one queen to a new
  location.
• Cost The number of queens that hit each others.

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Genetic consists of

• Genes
• Chromosomes
• Populations

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In n-queens

• A gene is a number between 0 to n-1.
• Is a position of any queen in the board
• A chromosome is an array of these genes. It
  could be the solution.
• Population is a generated set of chromosomes.

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Chromosomes and genes
A chromosome (array of genes. It could be an
answer)

• 3,6,8,5,1,4,0,7,9,2
• 7,6,9,5,1,4,0,3,8,2
• 9,6,1,5,8,4,0,7,3,2
• 6,3,8,5,2,4,0,7,9,1
• 3,6,8,5,1,4,0,7,9,2
• .
• .
• .

Gene
Population
Here N10
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Create a random initial population

• An initial population is created from a random
  selection of chromosomes.
• The number of generations needed depends on the
  random initial population.

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Finding the cost

• To find the assigned cost for each chromosome a
  cost function is defined.
• The result of the cost function is called cost
  value.
• This value is used for chromosomes ranking
• The best (minimum value) is placed on top and the
  worst (maximum) is placed in the bottom.

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Producing next generation

• Those chromosomes with a higher fitness (lesser
  cost) value are used to produce the next
  generation.
• The offspring (or Child) is a product of the two
  parents, whose composition consists of a
  combination of genes from them (this process is
  known as "crossing over").
• If the new generation (Child) contains a
  chromosome that produces an output that is close
  enough or equal to the desired answer then the
  problem has been solved.
• If this is not the case, then the new generation
  will go through the same process as their parents
  did. This will continue until a solution is
  reached.

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Steps to solving the problem with GA

• Clear the board.
• Generate the initial population.
• This generation is a purely random generation.
• Fill the chess board with a chromosome.
• For example,
• Let Chromosome Matrix 3,6,8,5,1,4,0,
• 7,9,2,
• Here Chess Board Length 10 (n10).

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Steps to solving the problem with GA

• Determines the cost value for each chromosome
  matrix.
• For example,
• For chromosome 2,6,9,3,5,
• 0,4,1,7,8 , the cost value will be 2
• Because of there are two queens
• that hit each other

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Steps to solving the problem with GA

• Sort the new generation according to their cost
  value.
• The best (minimum) is placed on top and the
  worst (maximum) is placed in the bottom.
• Generate the cross over matrix. This matrix
  contains 0s and 1s.

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Steps to solving the problem with GA

• Generate children from parents using cross over
  matrix.
• Genes are drawn from P0 and P1.
• A gene is drawn from one parent and it is
  appended to the offspring (child) chromosome.
• The corresponding gene is deleted in the other
  parent
• This step is repeated until both parent
  chromosomes are empty and the offspring contains
  all genes involved.

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Steps to solving the problem with GA

• Apply mutation to the current generation.
• First of all, a random chromosome is selected
  but the first (best) one in the list.
• Then, two random genes of this chromosome are
  selected and replaced with each other.
• Increasing the number of mutations increases the
  algorithms freedom to search outside the current
  region of chromosome space.

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Example of such software
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• Questions?