1.1) What is Backtracking?

Backtracking is a problem-solving technique that involves exploring all possible solutions to find the correct one. It is often used when dealing with problems that have multiple potential solutions, and the goal is to find one or more solutions that satisfy certain constraints or conditions. Backtracking explores a solution space systematically by trying out different options at each step and undoing choices that lead to dead ends.

How Backtracking Works:

1. Recursive Approach: Backtracking uses a recursive approach to explore all possible choices at each step of the problem-solving process.
2. Make a Choice: At each step, the algorithm makes a choice based on the available options.
3. Explore: The algorithm explores the consequences of that choice by further applying the same process to the remaining part of the problem.
4. Backtrack: If the current choice leads to an invalid solution or a dead-end, the algorithm backtracks by undoing the last choice and trying a different option.
5. Continue Exploration: The process continues until all possible choices have been explored or until a valid solution is found.

Backtracking Example – N-Queens Problem:

The N-Queens problem is a classic backtracking problem where the goal is to place N queens on an NxN chessboard in such a way that no two queens threaten each other.

N-Queens Solution:

#include <iostream>
using namespace std;

const int N = 4;
bool board[N][N] = {false};

bool isSafe(int row, int col) {
    // Check if no queen is present in the same column or diagonals
    for (int i = 0; i < row; i++) {
        if (board[i][col] || board[i][col - row + i] || board[i][col + row - i])
            return false;
    }
    return true;
}

bool solveNQueens(int row) {
    if (row == N) {
        // Base case: All queens are placed, a solution is found
        return true;
    }

    for (int col = 0; col < N; col++) {
        if (isSafe(row, col)) {
            board[row][col] = true;  // Place the queen

            if (solveNQueens(row + 1)) {
                return true;  // Found a solution, exit the loop and return true
            }

            board[row][col] = false;  // Backtrack, remove the queen and try the next column
        }
    }

    return false;  // No solution found for this row, backtrack to the previous row
}

int main() {
    if (solveNQueens(0)) {
        cout << "Solution Found:\n";
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < N; j++) {
                cout << (board[i][j] ? "Q " : ". ");
            }
            cout << "\n";
        }
    } else {
        cout << "No solution exists for N = " << N << "\n";
    }

    return 0;
}

Explanation:
In this example, we are solving the N-Queens problem for N = 4. The solveNQueens function uses backtracking to find a valid solution. It recursively places queens on the chessboard row by row, trying out all possible positions. If a valid solution is found, the function returns true; otherwise, it backtracks and tries different options.

Pros of Backtracking:

1. Exhaustive Search: Backtracking guarantees that all possible solutions are explored, ensuring that no valid solution is overlooked.
2. Versatility: Backtracking can be applied to a wide range of problems, including combinatorial optimization, puzzles, and constraint satisfaction problems.
3. Simplicity: The basic idea of backtracking is simple and easy to implement.

Cons of Backtracking:

1. Exponential Time Complexity: In many cases, backtracking explores an exponential number of potential solutions, leading to slow execution for large problem sizes.
2. Memory Usage: The recursive nature of backtracking may lead to high memory consumption, especially for deep recursion levels.
3. Performance: Backtracking may not be the most efficient approach for some problems, especially when more specialized algorithms exist.

Despite its drawbacks, backtracking remains a powerful and essential technique in problem-solving, especially when it comes to finding optimal or satisfying solutions for complex problems with multiple constraints.