Backtracking Problems

Backtracking is an algorithmic technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, and removing those solutions that fail to satisfy the constraints of the problem at any point in time. It is a form of depth-first search (DFS) on the state-space tree of the problem. A state-space tree is a conceptual tree where the root represents the initial state, and the children of a node represent the next possible choices. A path from the root to a leaf represents a complete candidate solution. The general template for a backtracking function looks like this: `solve(state)`: 1. Check if the current `state` is a valid solution. If so, process it (e.g., print it or add it to a list of solutions). 2. Iterate through all possible `choices` that can be made from the current `state`. 3. For each `choice`: a. Make the choice (modify the state). b. Check if the new state is promising (i.e., it could potentially lead to a solution). c. If it is, recursively call `solve(new_state)`. d. **Backtrack**: Un-make the choice (revert the state to how it was before). This last step is the most crucial part of backtracking. It's what allows us to explore other branches of the state-space tree after we're done with the current one. This technique is used for problems like generating all permutations of a set, solving puzzles like Sudoku or N-Queens, and pathfinding in mazes.