Day 48: Introduction to Backtracking #

Welcome to Day 48 of our 60 Days of Coding Algorithm Challenge! Today, we’ll explore backtracking, a powerful algorithmic technique used for solving problems recursively by trying out different possibilities until finding a solution.

What is Backtracking? #

Backtracking is an algorithmic technique that considers searching every possible combination in order to solve a computational problem. It builds candidates to the solution incrementally and abandons a candidate (“backtracks”) as soon as it determines that the candidate cannot lead to a valid solution.

Key Concepts of Backtracking #

1. Choice: At each step, we make a choice from a set of available options.
2. Constraints: The choices we make must satisfy certain problem constraints.
3. Goal: We have a specific goal or set of conditions that define a solution.

General Structure of a Backtracking Algorithm #

def backtrack(candidate):
    if find_solution(candidate):
        output(candidate)
        return
    
    for next_candidate in list_of_candidates:
        if is_valid(next_candidate):
            place(next_candidate)
            backtrack(next_candidate)
            remove(next_candidate)

Example: N-Queens Problem #

Let’s implement a solution to the N-Queens problem using backtracking. The goal is to place N chess queens on an N×N chessboard so that no two queens threaten each other.

def solve_n_queens(n):
    def is_safe(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 solve(board, col):
        if col >= n:
            return True
        
        for i in range(n):
            if is_safe(board, i, col):
                board[i][col] = 1
                if solve(board, col + 1):
                    return True
                board[i][col] = 0
        
        return False

    board = [[0 for _ in range(n)] for _ in range(n)]
    
    if not solve(board, 0):
        print("Solution does not exist")
        return False
    
    for row in board:
        print(row)
    return True

# Example usage
solve_n_queens(4)

Time Complexity #

The time complexity of backtracking algorithms can be exponential in the worst case, as they potentially explore all possible combinations. However, the actual time taken is often much less due to the pruning of invalid paths.

Applications of Backtracking #

1. Puzzle solving (e.g., Sudoku, crosswords)
2. Combinatorial optimization problems
3. Parsing languages
4. Finding all or some solutions to a computational problem

Exercise #

1. Implement a backtracking solution for the Sudoku puzzle.
2. Solve the “Subset Sum” problem using backtracking: Given a set of integers and a target sum, find all subsets that add up to the target.
3. Implement a backtracking solution for the “Graph Coloring” problem.

Summary #

Today, we introduced backtracking, a powerful algorithmic technique for solving complex problems by incrementally building candidates to the solution. We implemented a solution to the N-Queens problem as an example of backtracking in action.

Backtracking is crucial for solving many types of problems, especially those involving combinatorial optimization or constraint satisfaction. As we progress through this challenge, you’ll see backtracking being used to solve various complex problems efficiently.

Tomorrow, we’ll explore more advanced backtracking problems and techniques. Stay tuned!