8 QUEENS PROBLEM SOLUTION USING GENETIC ALGORITHM

There are optimization problems that couldn't be solved using classical bit-string chromosome approach. The 8 queens problem gives us the simple and clear example of such kind problems. Problem is how to place 8 queens on an ordinary chess board so that none of them can hit any other in one move.

ENCODING

Since classical recombination and mutation operators don't preserve the number of bits we need to make their analogues suitable for this particular case. I think that the most simple 'chromosome' for 8 queens problem is the array holding queens positions (integer numbers from 0 to 63 for 8x8 chess board). For example:  MEASURING FITNESS

Naturally here, fittest 'chromosomes' are those with smaller total number of queens under attack. To evaluate fitness of 'chromosome' we execute this simple algorithm: For each queen in chromosome we calculate number of queens sharing the same row, column and diagonal. Since relation 'being attacked' is symmetric and total number of queens being attacked always will be even we may divide it by 2. And at last we need to negate fitness value, because our algorithm is suited for maximization. For example, above chromosome {2,0,55,48,16,28,62,38} has fitness: -10.

MUTATION

To mutate 'queen' chromosome at nth position (locus) we simply move nth queen to randomly selected unoccupied position. For example, let mutate our sample chromosome at 4th position (counting from 0): CROSSOVER

Crossover is more sophisticated procedure. We take two parent chromosomes and put all genes (avoiding duplicates) to one combined chromosome (1st gene comes from 1st parent, 2nd - from 2nd parent, 3rd again from 1st parent and so on). the length of combined chromosomes ranges from 8 to 16 (parents have no common genes) . After combined chromosome is created we copy the first 8 genes from the beginning to first offspring. Other offspring receives the last 8 genes of combined chromosome:
The case where parents has no common genes: The case when parents has some common genes: And situation when combined chromosome offsprings are identical: CROSSOVER EXAMPLE Results of modeling:   VB.NET chromosome class implementation for 8 queens problem:

 ```Imports System.Collections.Generic Imports BaseGA5.GeneticAlgorithm Public Class QueenChromosome : Implements IChromosome #Region "Local Variables" Private Shared _symbols() As String = {"A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P"} Private Const _Undefined As Integer = -1 Private _NQ As Integer = 8 Private _MaxCol As Integer = 8 Private _MaxRow As Integer = 8 Private _N As Integer Public _queens As List(Of Integer) Public _histogram As List(Of Integer) Private Shared _random As Random = New Random((DateTime.Now.Hour * 3600 + DateTime.Now.Minute * 60 + DateTime.Now.Second) * 1000 + DateTime.Now.Millisecond) #End Region Public Sub New(ByVal maxCol As Integer, ByVal maxRow As Integer, ByVal NQ As Integer) If maxCol > 16 Then Throw New Exception(String.Format("Numebr of columns is out of range. Must be < 17")) End If _MaxCol = maxCol _MaxRow = maxRow _N = _MaxRow * _MaxCol _NQ = NQ _queens = CreateList(_NQ, _Undefined) _histogram = CreateList(_N, 0) End Sub Private Function CreateList(ByVal N As Integer, ByVal value As Integer) Dim o As New List(Of Integer)(N) While N > 0 o.Add(value) N -= 1 End While Return o End Function Public Sub RandomInit() Implements IChromosome.RandomInit Dim N As Integer = _NQ While N > 0 Dim pos As Integer = _random.Next(0, _N) If Not _queens.Contains(pos) Then _queens(N - 1) = pos N -= 1 End If End While End Sub Private Sub doCopy(ByVal dest As QueenChromosome, ByVal src As QueenChromosome) For j As Integer = 0 To _NQ - 1 dest._queens(j) = src._queens(j) Next End Sub Public Sub Copy(ByVal dest As IChromosome) Implements IChromosome.Copy doCopy(dest, Me) End Sub Public Function ValidPosition(ByVal r As Integer, ByVal c As Integer) As Boolean Return (r >= 0 And r < _MaxRow) AndAlso (c >= 0 And c < _MaxCol) End Function Private Function ChangePosition(ByVal oldPos As Integer) As Integer Dim newPos As Integer = 0 While True newPos = _random.Next(0, _N) If Not _queens.Contains(newPos) Then Return newPos End If End While Return _Undefined End Function Public Sub MutateAt(ByVal locus As Integer) Implements IChromosome.MutateAt _queens(locus) = ChangePosition(_queens(locus)) End Sub Public Function ValidLocus(ByVal locus As Integer) As Boolean Implements IChromosome.ValidLocus Return locus >= 0 AndAlso locus < _NQ End Function Public Sub Validate() For j As Integer = 0 To _N - 1 _histogram(j) = 0 Next For Each pos As Integer In _queens If (pos < 0) Or (pos >= _N) Then Throw New Exception(String.Format("Invalid queen position {0}.", pos)) End If If _histogram(pos) <> 0 Then Throw New Exception(String.Format("Duplicate queen position {0}.", pos)) End If Next End Sub Private Sub doCrossOver(ByVal dad As QueenChromosome, ByVal mom As QueenChromosome, ByVal offspring1 As QueenChromosome, ByVal offspring2 As QueenChromosome) Dim qjoined As New List(Of Integer) For j As Integer = 0 To _NQ - 1 If Not qjoined.Contains(dad._queens(j)) Then qjoined.Add(dad._queens(j)) End If If Not qjoined.Contains(mom._queens(j)) Then qjoined.Add(mom._queens(j)) End If Next Dim N As Integer = qjoined.Count If N < _NQ Or N > _NQ * 2 Then Throw New Exception(String.Format("Invalid number of queens ({0}) in joined list.", N)) End If For j As Integer = 0 To _NQ - 1 Dim pos As Integer = qjoined(j) offspring1._queens(j) = pos pos = qjoined(j + N - _NQ) offspring2._queens(j) = pos Next End Sub Public Sub CrossOver(ByVal mom As IChromosome, ByVal offspring1 As IChromosome, ByVal offspring2 As IChromosome) Implements IChromosome.CrossOver doCrossOver(Me, mom, offspring1, offspring2) End Sub Public Sub GetRowCol(ByVal pos As Integer, ByRef row As Integer, ByRef column As Integer) row = pos \ _MaxCol column = pos Mod _MaxCol End Sub Public Function GetPosition(ByVal r As Integer, ByVal c As Integer) As Integer Return r * _MaxCol + c End Function Public Function GetRow(ByVal pos As Integer) As Integer Return pos \ _MaxCol End Function Public Function GetCol(ByVal pos As Integer) As Integer Return pos Mod _MaxCol End Function Public Function GetSymbolic(ByVal pos As Integer) As String Return String.Format("{0}{1}", _symbols(pos Mod _MaxCol), (pos \ _MaxCol) + 1) End Function Public Function EvalFitness() As Double Dim sum As Double = 0 For Each pos As Integer In _queens 'Debug.WriteLine(String.Format("{0} --> {1};{2}", _queens(j), row, col)) sum += OnSameColumn(pos) sum += OnSameRow(pos) sum += OnSameDiagonal(pos) Next 'Debug.WriteLine(String.Format("Sum={0}", sum)) Return sum End Function Private Function OnSameRow(ByVal pos As Integer) As Integer Dim sum As Integer = 0 Dim row As Integer = GetRow(pos) For Each q As Integer In _queens sum += IIf((q \ _MaxCol) = row, 1, 0) Next Return sum - 1 End Function Private Function OnSameColumn(ByVal pos As Integer) As Integer Dim sum As Integer = 0 Dim col As Integer = GetCol(pos) For Each q As Integer In _queens sum += IIf((q Mod _MaxCol) = col, 1, 0) Next Return sum - 1 End Function Private Function OnSameDiagonal(ByVal pos As Integer) As Integer Dim sum As Integer = 0 Dim row As Integer = GetRow(pos) Dim col As Integer = GetCol(pos) For Each q As Integer In _queens sum += IIf(Math.Abs(GetRow(q) - row) = Math.Abs(GetCol(q) - col), 1, 0) Next Return sum - 1 End Function End Class ```

Genetic algorithm specialization:

 ```Imports System.Collections.Generic Imports BaseGA5.GeneticAlgorithm Public Class QueenGA : Inherits BaseGeneticAlgorithm(Of QueenChromosome) Private _crossoverP As Double = Double.NaN Private _mutateP As Double = Double.NaN Private _maxiter As Integer = 100 Private _sideSize As Integer = 8 Private _NQ As Integer = 8 Public Event BestObject(ByVal sender As Object, ByVal obj As QueenChromosome) Public Sub New(ByVal SideSize As Integer, ByVal NQ As Integer, ByVal maxiter As Integer, ByVal crossoverP As Double, ByVal mutateP As Double) _sideSize = SideSize _NQ = NQ _maxiter = maxiter _crossoverP = crossoverP _mutateP = mutateP End Sub Public Function PickBestObject(ByVal population As IEnumerable(Of QueenChromosome)) As QueenChromosome Dim maxf As Double = Double.MinValue Dim o As QueenChromosome = Nothing For Each obj As QueenChromosome In population Dim value As Double = GetFitness(obj) If value > maxf Then maxf = value o = obj End If Next Return o End Function Protected Overrides Function CanStop(ByVal iteration As Integer, ByVal population As System.Collections.Generic.List(Of QueenChromosome)) As Boolean Debug.WriteLine(String.Format("Iteration={0}", iteration)) Dim o As QueenChromosome = PickBestObject(population) If o IsNot Nothing Then Dim ftn As Integer = o.EvalFitness() Debug.WriteLine(String.Format("Sum={0}", ftn)) RaiseEvent BestObject(Me, o) Return (iteration >= _maxiter) Or (ftn = 0) End If Return True End Function Public Overrides Function CreateChromosome() As QueenChromosome Return New QueenChromosome(_sideSize, _sideSize, _NQ) End Function Public Overrides Function EvalFitness(ByVal obj As QueenChromosome) As Double Return obj.EvalFitness() End Function Protected Overrides Function PerformCrossover() As Boolean Return _random.NextDouble <= _crossoverP End Function Protected Overrides Function PerformMutate() As Boolean Return _random.NextDouble() <= _mutateP End Function End Class```