Examples of Backtracking Techniques

Explore practical examples of backtracking techniques in problem solving, including puzzles and optimization problems.
By Jamie

Introduction to Backtracking Techniques

Backtracking is a powerful algorithmic technique used to solve problems incrementally by trying partial solutions and eliminating those that fail to satisfy the conditions of the problem. It is particularly useful for solving constraint satisfaction problems, such as puzzles, combinatorial optimization problems, and pathfinding issues. Below are three diverse, practical examples of backtracking techniques.

1. Solving the N-Queens Problem

The N-Queens problem is a classic example in computer science and mathematics where the goal is to place N queens on an N×N chessboard such that no two queens threaten each other. This is a great use case for backtracking as it requires exploring multiple configurations to find a valid arrangement.

To solve the problem, we can start placing queens in columns, one at a time, and check for conflicts with already placed queens. If a conflict arises, we backtrack by removing the last placed queen and try the next row in the current column.

For example, let’s say we want to solve the 4-Queens problem:

  • Start placing the first queen in the first column, first row.
  • Move to the second column, and place the queen in the first row. This causes a conflict, so backtrack to the second column and place the queen in the second row.
  • Continue this process until all queens are placed or all rows in a column are exhausted.

This approach ensures that all possible configurations are checked, and only valid ones are recorded.

Notes:

  • The backtracking algorithm can be optimized further by avoiding checking already attacked rows and diagonals.
  • Variations include different board sizes and constraints on the queens’ movements.

2. Finding All Subsets of a Set

Another practical example of backtracking is generating all possible subsets (the power set) of a given set. This technique is beneficial in various applications, such as database query optimization and combinatorial problems.

For instance, given a set of numbers {1, 2, 3}, we want to find all possible subsets:

  • Start with an empty subset and recursively add elements from the original set.
  • At each step, you can either include the current element or exclude it, leading to two branches.
  • Continue this process until you reach the end of the set, at which point you will have formed a complete subset.

The recursive function might look like this:

  1. Start with an empty subset: [].
  2. Include 1: [1]. Then include 2: [1, 2]. Include 3: [1, 2, 3]. Backtrack to [1, 2]. Exclude 3: [1, 2]. Backtrack to [1]. Exclude 2; include 3: [1, 3].
  3. Backtrack to []. Exclude 1; include 2: [2], and so on.

This method effectively explores all combinations of included/excluded elements.

Notes:

  • The total number of subsets for a set of size n is 2^n.
  • The algorithm can be adapted to generate subsets of a specific size or satisfy certain conditions.

3. Solving Sudoku Puzzles

Sudoku is a popular puzzle game that requires placing numbers in a 9x9 grid according to specific rules. Backtracking is a widely used technique to solve Sudoku puzzles efficiently.

In Sudoku, we need to place numbers 1-9 in such a way that each row, each column, and each of the nine 3x3 grids contain all of the numbers without repetition. The backtracking approach involves:

  • Scanning the grid to find an empty cell (denoted by 0).
  • Attempting to place a number (1-9) in that cell while checking if it violates Sudoku rules.
  • If a number can be placed without conflict, move to the next empty cell and repeat.
  • If no number fits, backtrack by removing the last placed number and trying the next possible number in the previous cell.

For example, start with a partially filled grid:

  • Locate an empty cell, say (0, 2), where you can try placing a number like 5.
  • Check whether placing 5 violates any Sudoku rules. If not, move to the next empty cell.
  • If you reach a state where no numbers can be placed, backtrack to (0, 2) and try the next number (6, 7, etc.).

This systematic exploration guarantees finding a solution if one exists.

Notes:

  • Sudoku solvers can be optimized by using constraint propagation techniques to reduce the search space.
  • Variations of Sudoku with different sizes or additional constraints can also be tackled using backtracking.