C++ Program to Solve N-Queen Problem

This problem is to find an arrangement of N queens on a chess board, such that no queen can attack any other queens on the board.

The chess queens can attack in any direction as horizontal, vertical, horizontal and diagonal way.

A binary matrix is used to display the positions of N Queens, where no queens can attack other queens. Here, we solve 8 queens problem.

Input

The size of a chess board. it is 8 here as (8 x 8 is the size of a normal chess board).

Output

The matrix that represents in which row and column the N Queens can be placed.

If the solution does not exist, it will return false.

1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0

In this output, the value 1 indicates the correct place for the queens.

The 0 denotes the blank spaces on the chess board.

Algorithms

isValid(board, row, col)

Begin
   if there is a queen at the left of current col, then
      return false
   if there is a queen at the left upper diagonal, then
      return false
   if there is a queen at the left lower diagonal, then
      return false;
   return true //otherwise it is valid place
End

solveNQueen(board, col)

Begin
   if all columns are filled, then
      return true
   for each row of the board, do
      if isValid(board, i, col), then
         set queen at place (i, col) in the board
         if solveNQueen(board, col+1) = true, then
            return true
         otherwise remove queen from place (i, col) from board.
      done
   return false
End

Example

#include<iostream>
using namespace std;
#define N 4
void printBoard(int board[N][N]) {
   for (int i = 0; i < N; i++) {
      for (int j = 0; j < N; j++)
         cout << board[i][j] << " ";
         cout << endl;
   }
}
bool isValid(int board[N][N], int row, int col) {
   for (int i = 0; i < col; i++) //check whether there is queen in the left or not
      if (board[row][i])
         return false;
   for (int i=row, j=col; i>=0 && j>=0; i--, j--)
      if (board[i][j]) //check whether there is queen in the left upper diagonal or not
         return false;
   for (int i=row, j=col; j>=0 && i<N; i++, j--)
      if (board[i][j]) //check whether there is queen in the left lower diagonal or not
         return false;
   return true;
}
bool solveNQueen(int board[N][N], int col) {
   if (col >= N) //when N queens are placed successfully
      return true;
   for (int i = 0; i < N; i++) { //for each row, check placing of queen is possible or not
      if (isValid(board, i, col) ) {
         board[i][col] = 1; //if validate, place the queen at place (i, col)
         if ( solveNQueen(board, col + 1)) //Go for the other columns recursively
            return true;
         board[i][col] = 0; //When no place is vacant remove that queen
      }
   }
   return false; //when no possible order is found
}
bool checkSolution() {
   int board[N][N];
   for(int i = 0; i<N; i++)
   for(int j = 0; j<N; j++)
   board[i][j] = 0; //set all elements to 0
   if ( solveNQueen(board, 0) == false ) { //starting from 0th column
      cout << "Solution does not exist";
      return false;
   }
   printBoard(board);
   return true;
}
int main() {
   checkSolution();
}

Output

1 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 1
0 1 0 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0