We have discussed Knight’s tour and Rat in a Maze problems in Set 1 and Set 2 respectively. Let us discuss N Queen as another example problem that can be solved using Backtracking. The N Queen is the problem of placing N chess queens on an N×N chessboard so that no two queens attack each other. For example, following is a solution for 4 Queen problem.

The expected output is a binary matrix which has 1s for the blocks where queens are placed. For example, following is the output matrix for above 4 queen solution.

{ 0, 1, 0, 0}

{ 0, 0, 0, 1}

{ 1, 0, 0, 0}

{ 0, 0, 1, 0}

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:  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. 

• 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.

Algorithm:

1. Place the queens column wise, start from the left most column

2. If all queens are placed.

   1. return true and print the solution matrix.

3. Else

   1. Try all the rows in the current column.

   2. 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.

   3. If placing the queen in above step leads to the solution return true.

   4. 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.

4. 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(N 2 ) space. We can reduce it to O(N). We will solve it by taking one dimensional array and consider solution[1] = 2 as “Queen at 1st row is placed at 2nd column