CS211 The n queens problem By the “n queens problem” we mean the problem of plac
ID: 3758347 • Letter: C
Question
CS211 The n queens problem By the “n queens problem” we mean the problem of placing n queens on an nXn “chessboard” in such a way that no queen can capture any other on the next move. In class we solved the “8 queens” problem. Write a function that inputs an integer n and returns the number of solutions to the “n queens” problem. Your function should use the one dimensional representation for the board, the algorithm we discussed in class, and no gotos. Test your function with a main program that prompts the user for an integer n. The main program then calls the function n times, once for each number from 1 – n, and then prints the number of solutions to each of these problems, one on a line. For example, if you enter n=5 your program should output: 1. There are _____ solutions to the 1 queens problem. 2. There are _____ solutions to the 2 queens problem. 3. There are _____ solutions to the 3 queens problem. 4. There are _____ solutions to the 4 queens problem. 5. There are _____ solutions to the 5 queens problem. Now, since each time through the loop you will need an array q of a different length, you will need to allocate the array off of the heap (which we mentioned in class) and not the run-time stack. To do this you use the “new operator” to request the heap to dynamically allocate the memory for you. For example, to get a one dimensional array of integers of size n and called q, we use the following syntax: int* q = new int[n]; This allocates the array for us dynamically, at run-time. After this we can use the array q just as if it had been declared “normally” and it has n elements denoted q[0] through q[n-1]. When we no longer need the memory that was allocated to the array, we write: delete [ ] q; For this problem, you pass the required length for the array q to your function, which then allocates q dynamically (using “new”) and uses it to hold the solutions for the given size. Each time the function exits, you must deallocate q by calling delete [ ] q. For this problem, test your program for n=8. For some more details on how to use new and delete, go to the video section of the course web site and see the video called “Dynamically allocated arrays.”
Explanation / Answer
/* Java program to solve N Queen Problem using
backtracking */
public class NQueenProblem
{
final int N = 4;
/* A utility function to print solution */
void printSolution(int board[][])
{
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
System.out.print(" " + board[i][j]
+ " ");
System.out.println();
}
}
/* A utility function to check if a queen can
be placed on board[row][col]. Note that this
function is called when "col" queens are already
placeed in columns from 0 to col -1. So we need
to check only left side for attacking queens */
boolean isSafe(int board[][], int row, int col)
{
int i, j;
/* Check this row on left side */
for (i = 0; i < col; i++)
if (board[row][i] == 1)
return false;
/* Check upper diagonal on left side */
for (i=row, j=col; i>=0 && j>=0; i--, j--)
if (board[i][j] == 1)
return false;
/* Check lower diagonal on left side */
for (i=row, j=col; j>=0 && i<N; i++, j--)
if (board[i][j] == 1)
return false;
return true;
}
/* A recursive utility function to solve N
Queen problem */
boolean solveNQUtil(int board[][], int col)
{
/* base case: If all queens are placed
then return true */
if (col >= N)
return true;
/* Consider this column and try placing
this queen in all rows one by one */
for (int i = 0; i < N; i++)
{
/* Check if queen can be placed on
board[i][col] */
if (isSafe(board, i, col))
{
/* Place this queen in board[i][col] */
board[i][col] = 1;
/* recur to place rest of the queens */
if (solveNQUtil(board, col + 1) == true)
return true;
/* If placing queen in board[i][col]
doesn't lead to a solution then
remove queen from board[i][col] */
board[i][col] = 0; // BACKTRACK
}
}
/* If queen can not be place in any row in
this colum col, then return false */
return false;
}
/* This function solves the N Queen problem using
Backtracking. It mainly uses solveNQUtil() to
solve the problem. It returns false if queens
cannot be placed, otherwise return true and
prints placement of queens in the form of 1s.
Please note that there may be more than one
solutions, this function prints one of the
feasible solutions.*/
boolean solveNQ()
{
int board[][] = {{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}
};
if (solveNQUtil(board, 0) == false)
{
System.out.print("Solution does not exist");
return false;
}
printSolution(board);
return true;
}
// driver program to test above function
public static void main(String args[])
{
NQueenProblem Queen = new NQueenProblem();
Queen.solveNQ();
}
}
/* Java program to solve N Queen Problem using
backtracking */
public class NQueenProblem
{
final int N = 4;
/* A utility function to print solution */
void printSolution(int board[][])
{
for (int i = 0; i < N; i++)
{
for (int j = 0; j < N; j++)
System.out.print(" " + board[i][j]
+ " ");
System.out.println();
}
}
/* A utility function to check if a queen can
be placed on board[row][col]. Note that this
function is called when "col" queens are already
placeed in columns from 0 to col -1. So we need
to check only left side for attacking queens */
boolean isSafe(int board[][], int row, int col)
{
int i, j;
/* Check this row on left side */
for (i = 0; i < col; i++)
if (board[row][i] == 1)
return false;
/* Check upper diagonal on left side */
for (i=row, j=col; i>=0 && j>=0; i--, j--)
if (board[i][j] == 1)
return false;
/* Check lower diagonal on left side */
for (i=row, j=col; j>=0 && i<N; i++, j--)
if (board[i][j] == 1)
return false;
return true;
}
/* A recursive utility function to solve N
Queen problem */
boolean solveNQUtil(int board[][], int col)
{
/* base case: If all queens are placed
then return true */
if (col >= N)
return true;
/* Consider this column and try placing
this queen in all rows one by one */
for (int i = 0; i < N; i++)
{
/* Check if queen can be placed on
board[i][col] */
if (isSafe(board, i, col))
{
/* Place this queen in board[i][col] */
board[i][col] = 1;
/* recur to place rest of the queens */
if (solveNQUtil(board, col + 1) == true)
return true;
/* If placing queen in board[i][col]
doesn't lead to a solution then
remove queen from board[i][col] */
board[i][col] = 0; // BACKTRACK
}
}
/* If queen can not be place in any row in
this colum col, then return false */
return false;
}
/* This function solves the N Queen problem using
Backtracking. It mainly uses solveNQUtil() to
solve the problem. It returns false if queens
cannot be placed, otherwise return true and
prints placement of queens in the form of 1s.
Please note that there may be more than one
solutions, this function prints one of the
feasible solutions.*/
boolean solveNQ()
{
int board[][] = {{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}
};
if (solveNQUtil(board, 0) == false)
{
System.out.print("Solution does not exist");
return false;
}
printSolution(board);
return true;
}
// driver program to test above function
public static void main(String args[])
{
NQueenProblem Queen = new NQueenProblem();
Queen.solveNQ();
}
}