When is backtracking the best problem solving approach for programming?

Backtracking is a problem-solving technique used to find all possible solutions to a problem. It involves systematically trying out different potential solutions and backtracking from choices that do not satisfy the problem's constraints. The algorithm explores various paths of a solutions, and when it realizes that the current path cannot lead to a valid solution, it retreats (backtracks) to the previous decision point and explores other choices. A classic example of backtracking is the N-Queens problem, where the objective is to place N chess queens on an N×N chessboard in a way that no two queens threaten each other. This means that no two queens can be in the same row, column, or diagonal.

Usually, backtracking is used to optimize the exhaustive approach. When a problem has many permutations, with CHOOSE and NOT CHOOSE options, backtracking can be efficient, to use the earlier solution, by just undoing the last action, and adding the current one. Simplest example - if you want to evaluate this expression -> 1+2+3, the answer is (1+2) =3; 3+3 -> 6! now if the ask is to find 1+2+5, just undo the last action in 1st example -> 6-3 =3, This is exactly what we do in backtracking. Now add 5 to (1+2); 3+5 =8!! This saves us time recalculating and saves space.

The process involves making choices, checking their validity, and backtracking when necessary. Choose: Make a choice to extend the current solution. This could involve selecting an option from a set of possibilities. Explore: Recursively explore all possible choices and their consequences. This often involves moving to the next decision point and making another choice. Backtrack/Unchoose: If the current choice does not lead to a valid solution, undo the choice (backtrack) and explore other alternatives. This involves reverting the changes made and going back to the previous decision point.

Backtracking serves as the best approach whenever there is a problem which can be broken down into smaller, overlapping subproblems, and the optimal solution to the overall problem can be constructed from optimal solutions to its subproblems. If there is a problem which involves constraints that limit the valid solutions, backtracking is particularly useful when constraints need to be satisfied, and solutions must adhere to specific rules. Classic examples of problems where backtracking is commonly applied include the N-Queens problem, the Sudoku puzzle, and the Knight's Tour problem.

Most often, a backtracking solution naturally arises when optimizing a recursive brute-force approach to a problem. The primary advantage of backtracking is its ability to optimize the time and space complexity of a brute-force solution by pruning ineffective "branches." Additionally, backtracking tends to be a more intuitive approach compared to other optimization methods, such as dynamic programming.