Added Leetcode-0050 N-Queen Solution

dsa-solutions/lc-solutions/0000-0099/0051-N-Queens.md

@@ -0,0 +1,388 @@
+---
+id: n-queens
+title: N-Queens (LeetCode)
+sidebar_label: 0051-N-Queens
+tags:
+    - Array
+    - Backtracking
+description: "The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other."
+---
+
+## Problem Description
+
+| Problem Statement                                                                                           | Solution Link                                                                                                                               | LeetCode Profile                                   |
+| :----------------------------------------------------------------------------------------------------------- | :------------------------------------------------------------------------------------------------------------------------------------------ | :------------------------------------------------- |
+| [N-Queen](https://leetcode.com/problems/n-queens/)                                         | [N-Queen](https://leetcode.com/problems/n-queens/) | [DaminiChachane](https://leetcode.com/u/divcxl15/) |
+
+### Problem Description
+
+The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other.
+
+Given an integer n, return all distinct solutions to the n-queens puzzle. You may return the answer in any order.
+
+Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' both indicate a queen and an empty space, respectively.
+
+
+### Examples
+
+#### Example 1
+
+- **Input:** `n = 4`
+- **Output:** `[[".Q..","...Q","Q...","..Q."],["..Q.","Q...","...Q",".Q.."]]`
+- **Explanation:** There exist two distinct solutions to the 4-queens puzzle as shown above.
+
+#### Example 2
+
+- **Input:** `n = 1`
+- **Output:** `[["Q"]]`
+- **Explanation:** Here the number of rows and column is one thatswhy queen is placed in row 1 and column 1.
+
+
+
+### Constraints
+
+- `1 <= n <= 9`
+
+### Approach
+
+To solve the problem, we can use the Backtracking technique technique. Here is the step-by-step approach:
+
+1. **Initialize :**
+    -Start in the leftmost column
+    -If all queens are placed return true
+    -Try all rows in the current column. Do the following for every row.
+2. **Iterate Through the Array:**
+   - Try all rows in the current column. Do the following for every row.
+    -If the queen can be placed safely in this row
+    -Then mark this [row, column] as part of the solution and recursively check if placing queen here leads to a solution.
+    -If placing the queen in [row, column] leads to a solution then return true.
+    -If placing queen doesn’t lead to a solution then unmark this [row, column] then backtrack and try other rows.
+
+3. **Return Result:**
+    -If all rows have been tried and valid solution is not found return false to trigger backtracking.
+
+### Solution Code
+
+#### Python
+
+```
+# Python3 program to solve N Queen
+# Problem using backtracking
+
+global N
+N = 4
+
+
+def printSolution(board):
+    for i in range(N):
+        for j in range(N):
+            if board[i][j] == 1:
+                print("Q",end=" ")
+            else:
+                print(".",end=" ")
+        print()
+
+
+# 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 placed in columns from 0 to col -1.
+# So we need to check only left side for
+# attacking queens
+def isSafe(board, row, col):
+
+    # Check this row on left side
+    for i in range(col):
+        if board[row][i] == 1:
+            return False
+
+    # Check upper diagonal on left side
+    for i, j in zip(range(row, -1, -1),
+                    range(col, -1, -1)):
+        if board[i][j] == 1:
+            return False
+
+    # Check lower diagonal on left side
+    for i, j in zip(range(row, N, 1),
+                    range(col, -1, -1)):
+        if board[i][j] == 1:
+            return False
+
+    return True
+
+
+def solveNQUtil(board, 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 i in range(N):
+
+        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
+            # queen from board[i][col]
+            board[i][col] = 0
+
+    # If the queen can not be placed in any row in
+    # this column 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
+# placement of queens in the form of 1s.
+# note that there may be more than one
+# solutions, this function prints one of the
+# feasible solutions.
+def solveNQ():
+    board = [[0, 0, 0, 0],
+             [0, 0, 0, 0],
+             [0, 0, 0, 0],
+             [0, 0, 0, 0]]
+
+    if solveNQUtil(board, 0) == False:
+        print("Solution does not exist")
+        return False
+
+    printSolution(board)
+    return True
+
+
+```
+
+#### Java
+```
+
+public class Solution {
+	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++) {
+				if (board[i][j] == 1)
+					System.out.print("Q ");
+				else
+					System.out.print(". ");
+			}
+			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 the 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 the queen can not be placed in any row in
+		// this column 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 Solution()
+	{
+		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;
+	}
+}
+
+```
+
+#### C++
+```
+
+#include <bits/stdc++.h>
+#define N 4
+using namespace std;
+
+// A utility function to print solution
+void printSolution(int board[N][N])
+{
+	for (int i = 0; i < N; i++) {
+		for (int j = 0; j < N; j++)
+		if(board[i][j])
+			cout << "Q ";
+		else cout<<". ";
+		printf("\n");
+	}
+}
+
+// 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 placed in columns from 0 to col -1.
+// So we need to check only left side for
+// attacking queens
+bool isSafe(int board[N][N], int row, int col)
+{
+	int i, j;
+
+	// Check this row on left side
+	for (i = 0; i < col; i++)
+		if (board[row][i])
+			return false;
+
+	// Check upper diagonal on left side
+	for (i = row, j = col; i >= 0 && j >= 0; i--, j--)
+		if (board[i][j])
+			return false;
+
+	// Check lower diagonal on left side
+	for (i = row, j = col; j >= 0 && i < N; i++, j--)
+		if (board[i][j])
+			return false;
+
+	return true;
+}
+
+// A recursive utility function to solve N
+// Queen problem
+bool solveNQUtil(int board[N][N], 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 the 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))
+				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 the queen cannot be placed in any row in
+	// this column 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.
+bool Solution()
+{
+	int board[N][N] = { { 0, 0, 0, 0 },
+						{ 0, 0, 0, 0 },
+						{ 0, 0, 0, 0 },
+						{ 0, 0, 0, 0 } };
+
+	if (solveNQUtil(board, 0) == false) {
+		cout << "Solution does not exist";
+		return false;
+	}
+
+	printSolution(board);
+	return true;
+}
+```
+
+### Conclusion
+
+The N-Queens problem is a classic example of a combinatorial optimization problem that has been extensively studied in the field of computer science, particularly within the realms of algorithms and artificial intelligence. The problem is defined as placing N queens on an N×N chessboard in such a way that no two queens can attack each other. This means that no two queens can be placed in the same row, column, or diagonal.