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recursive backtracking tutorial in python

what is recursive backtracking?

recursive backtracking is a problem-solving technique used for solving constraint satisfaction problems, such as puzzles, combinatorial problems, and optimization problems. the method involves exploring all possible solutions by building a solution incrementally, and abandoning solutions (backtracking) as soon as it is determined that they cannot lead to a valid solution.

key concepts

1. **state space tree**: represents all possible configurations that can be reached during the search for a solution.
2. **decision variables**: the variables that can change in order to find a solution.
3. **base case**: condition that stops the recursion.
4. **constraints**: conditions that must be met for a solution to be valid.

when to use recursive backtracking

puzzles (e.g., sudoku, n-queens)
pathfinding problems (e.g., maze solving)
combinatorial problems (e.g., generating permutations, combinations)

basic structure of recursive backtracking

1. **choose**: make a choice from the available options.
2. **explore**: recursively explore the next decision.
3. **un-choose (backtrack)**: remove the last choice and try the next option.

example: n-queens problem

one classic problem that can be solved using recursive backtracking is the n-queens problem. the goal is to place n queens on an n×n chessboard so that no two queens threaten each other.

steps to solve the n-queens problem

1. place a queen in the first row.
2. move to the next row and try placing a queen in each column.
3. if placing a queen leads to a solution, proceed; if not, backtrack and try the next column.
4. continue until all queens are placed or all options are exhausted.

code example

here's a python implementation of the n-queens problem using recursive backtracking:



explanation of the code

1. **`print_solution(board)`**: prints the chessboard configuration where 'q' represents a queen and '.' represents an empty ...

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