Backtracking – N Queens Problem
Objective : In chess, a queen can move as far as she pleases, horizontally, vertically, or diagonally. A chess board has 8 rows and 8 columns. The standard 8 by 8 Queen’s problem asks how to place 8 queens on an ordinary chess board so that none of them can hit any other in one move.(Source: http://www.math.utah.edu/~alfeld/queens/queens.html)
Here we are solving it for N queens in NxN chess board.
Approach: In this approach we will see the basic solution with O(N^2) extra space we will improve it further to O(N) space. Click here to see the solution.
- Create a solution matrix of the same structure as chess board.
- Whenever place a queen in the chess board, mark that particular cell in solution matrix.
- At the end print the solution matrix, the marked cells will show the positions of the queens in the chess board.
- Place the queens column wise, start from the left most column
- If all queens are placed.
- return true and print the solution matrix.
- Try all the rows in the current column.
- Check if queen can be placed here safely if yes mark the current cell in solution matrix as 1 and try to solve the rest of the problem recursively.
- If placing the queen in above step leads to the solution return true.
- If placing the queen in above step does not lead to the solution , BACKTRACK, mark the current cell in solution matrix as 0 and return false.
- If all the rows are tried and nothing worked, return false and print NO SOLUTION.
Better Solution: If you notice in solution matrix, at every row we have only one entry as 1 and rest of the entries are 0. Solution matrix takes O(N2) space. We can reduce it to O(N). We will solve it by taking one dimensional array and consider solution = 2 as “Queen at 1st row is placed at 2nd column. Click here to see the Better Solution.
Output: 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0
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If you find anything incorrect or you feel that there is any better approach to solve the above problem, please write comment.