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Backtracking Algorithm with Example

Backtracking Algorithm with Example

In this blog, we will examine the foundations of backtracking, its uses, and the secret to its ability to methodically traverse decision spaces. Come along as we walk you through every aspect of this effective algorithm, illuminating its operation and presenting instances from real-world applications where it excels. This blog serves as your entryway to grasp the particulars of backtracking, regardless of your level of programming experience or curiosity.

Table of Contents:

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Introduction to Backtracking Algorithm

Backtracking is a systematic problem-solving technique employed in computer science and mathematics to find solutions to problems, especially those involving combinatorial choices, such as finding all possible paths or arrangements. The fundamental idea behind backtracking is to explore all possible options systematically and backtrack when a dead end is reached.

Imagine you’re in a maze and want to find the exit. Backtracking works like this: you explore a path until you reach a point where it doesn’t lead to the solution. At that point, you backtrack to the last decision point and try a different path. This process continues until you find the exit or exhaust all possibilities.

In programming, backtracking is often implemented using recursion. The algorithm explores a branch of the solution space, backtracking to the previous step if it fails and exploring another branch. This continues until a solution is found or all possibilities are explored.

For example, solving Sudoku or the N-Queens problem involves backtracking. In Sudoku, you fill in numbers, and if you reach a point where the puzzle becomes unsolvable, you backtrack to the last valid move. Backtracking is a powerful technique for systematically solving complex problems by exploring solution spaces.

Sudoko puzzle problem using backtracking

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State Space Tree

A state space tree in the context of a backtracking algorithm as a roadmap for problem-solving. Imagine being on a journey, facing multiple choices at each point. The tree represents all potential paths, with each node serving as a decision point. Backtracking involves exploring these paths; if encounter a dead end, you backtrack to the last decision point and explore a different route.

For instance, in a maze, nodes might represent various paths, and backtracking aids in exploration until the correct path is found. The state space tree guides this exploration systematically, facilitating a methodical approach to problem-solving by navigating through choices, trying options, and backtracking when necessary to find the optimal solution.

State Space Tree

Types of Backtracking Algorithm

Backtracking is a general algorithm for finding all (or some) solutions to computational problems, particularly constraint satisfaction problems. It incrementally builds candidates for solutions and abandons a candidate as soon as it determines that the candidate cannot possibly be completed to a valid solution. Here are some types of backtracking algorithms:

  • Recursive Backtracking
  • Non-Recursive Backtracking

Recursive Backtracking

In this basic form of backtracking, the algorithm explores each branch of the solution space recursively. If a solution is found, it is returned; otherwise, the algorithm backtracks to the previous decision point and explores alternative choices.

Here’s a simple example of a recursive backtracking algorithm for generating permutations of a set of elements:

def permute_recursive(nums, path, result):
    # Base case: if all elements are used, add the current permutation to the result
    if not nums:
    # Explore all possible choices
    for i in range(len(nums)):
        # Make a choice
        # Explore with the chosen element removed from the options
        permute_recursive(nums[:i] + nums[i+1:], path, result)
        # Backtrack by undoing the choice
# Example usage:
nums = [1, 2, 3]
result = []
permute_recursive(nums, [], result)

Output: [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

This recursive algorithm generates all permutations of the input list “nums”. The “permute_recursive” function takes three parameters: the remaining elements to permute (nums), the current partial permutation (“path”), and the list to store the final result (“result”).

Non-Recursive Backtracking

Backtracking can also be implemented using iteration and an explicit stack or queue to manage the state of the search. This can be useful in situations where recursive calls are not desirable due to factors like memory constraints.

Here’s an example of a non-recursive backtracking algorithm for generating permutations using an explicit stack:

def permute_non_recursive(nums):
    result = []
    stack = [(nums, [])]
    while stack:
        current_nums, current_path = stack.pop()
        if not current_nums:
        for i in range(len(current_nums)):
            stack.append((current_nums[:i] + current_nums[i+1:], current_path + [current_nums[I]]))
    return result
# Example usage:
nums = [1, 2, 3]
result = permute_non_recursive(nums)

Output: [[3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2], [1, 2, 3]]

In this non-recursive version, a stack is used to simulate the recursive calls. The stack contains tuples representing the state of the algorithm, and the while loop continues until the stack is empty. The algorithm explores choices iteratively, and the stack is manipulated to simulate the recursive calls and backtracking.

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Working of Backtracking Algorithm

Let us now break down how backtracking works, step by step. It builds solutions bit by bit, backtracking when needed, and cleverly explores different paths until it finds the right solution.

Step 1: Understanding the Problem

Backtracking is a technique used to systematically search for a solution to a problem, especially in cases where a brute-force approach would be impractical. Typically, backtracking is applied to problems with multiple possible solutions, such as puzzles, mazes, or optimization problems.

Step 2: Choose a Path

The algorithm begins by choosing a path to explore from the current state. It makes a decision and moves forward, marking the chosen path.

Step 3: Explore the Chosen Path

The algorithm explores the selected path and reaches a new state. At this point, it evaluates whether the current state is a solution to the problem. If it is, the algorithm stops; otherwise, it continues to explore further.

Step 4: Backtrack if Necessary

If the algorithm reaches a state where it cannot proceed or the chosen path does not lead to a solution, it backtracks. Backtracking involves undoing the last decision made, returning to the previous state, and exploring alternative paths.

Step 5: Explore Alternative Paths

Having backtracked, the algorithm continues to explore alternative paths from the previous state. It repeats the process of choosing a path, exploring, and backtracking until a solution is found or all possibilities are exhausted.

Let’s consider a scenario where you have three persons named A, B, and C. Now we have to create a seating arrangement for the three of them. There are multiple ways or combinations in which they can sit. The Backtracking algorithm helps in that.

Let us visualize the tree for this:

Working of backracking algorithm

In this example, we began by placing A at the first position, followed by B and then C. Subsequently, we backtracked, positioning C at the second spot and B at the third. This yielded two options: ABC and ACB.

Continuing, we placed B at the first position, followed by A at the second, and C at the third. Once again, backtracking occurred, with C in the second position and A in the third. This resulted in two additional options: BAC and BCA.

Lastly, we positioned C at the first spot, followed by A in the second, and B in the third. A final backtrack led to B at the second and A at the third position, concluding with the last two options: CAB and CBA.

The entire backtracking process led us to the final six combinations of their seating, which are:



Examples of Backtracking Algorithm

Consider the N-Queens problem, where you need to place N queens on an N✕N chessboard in such a way that no two queens threaten each other. The backtracking algorithm for this problem involves placing queens row by row and backtracking when conflicts arise.

  • Choose a Path: Start with the first row and place a queen in an empty column.
  • Explore the Chosen Path: Move to the next row and repeat the process, ensuring that the queens do not threaten each other.
  • Backtrack if Necessary: If a conflict is detected, backtrack to the previous row and explore alternative columns for placing the queen.
  • Explore Alternative Paths: Continue exploring and backtracking until a valid placement for all queens is found.
N-Queens problem

There are ten solutions to the problem after it is solved, so we must use the backtracking algorithm to get each of them.

When to use Backtracking Algorithm

The backtracking algorithm explores all possible candidates for a solution and backtracks when it determines that a candidate cannot lead to a valid solution. Here’s when you might use a backtracking algorithm:

1. Permutation or Combination Problems: One can use this algorithm when one needs to generate all possible permutations or combinations of a set of elements. For example, solving puzzles like Sudoku, and N-Queens, or generating all possible combinations of elements from a set.

2. Constraint Satisfaction Problems: When you have a problem that can be broken down into smaller decisions, each with constraints, this technique can be used. Backtracking can efficiently solve problems where choices made need to satisfy specific conditions or constraints, like scheduling or resource allocation problems.

3. Decision Problems: This algorithm is employed when you need to find a feasible solution or not, rather than generating all possible solutions. Backtracking can be used to efficiently solve problems like graph coloring or finding a Hamiltonian cycle in a graph.

Applications of Backtracking Algorithm

Backtracking finds applications across various domains due to its ability to systematically explore and solve problems. Some common applications include:

Applications of Backtracking Algorithm

1. Puzzle Solving: Backtracking is extensively used in puzzles like Sudoku, crosswords, word search, and maze solving. It efficiently explores the solution space to find a valid solution.

2. Game Playing: Games like chess, tic-tac-toe, and other board games often use backtracking to search through possible moves and find the best move or determine if a winning strategy exists.

3. Combinatorial Problems: Problems involving permutations, combinations, and subsets, such as generating all possible combinations of a set of elements or finding permutations of a sequence, utilize backtracking for efficient exploration.

4. Constraint Satisfaction Problems: Backtracking helps solve problems where constraints need to be satisfied, such as scheduling tasks, resource allocation, or job sequencing.

5. Graph Problems: It’s used in graph-related problems like finding a Hamiltonian cycle, solving the traveling salesman problem, graph coloring, and finding paths in a graph.

6. Optimization Problems: In some cases, backtracking can be used as part of an optimization strategy, exploring possible solutions and improving them incrementally to reach an optimal solution.

7. Robotics and Pathfinding: Backtracking is employed in robotics for pathfinding algorithms, helping robots find the shortest path from one point to another while avoiding obstacles.


In conclusion, the backtracking algorithm stands as a powerful tool in the domain of problem-solving, offering an approach to exploring solution spaces while navigating through constraints and conditions. Through its recursive nature and depth-first search strategy, it efficiently seeks solutions to various combinatorial problems, puzzle-solving challenges, and constraint satisfaction scenarios.

Its adaptability across diverse domains, from puzzles and games to optimization and graph-related conundrums, showcases its versatility and applicability. Despite its effectiveness in exploring solution spaces, its efficiency may be affected by the size and complexity of the problem, warranting potential optimizations or alternative approaches for larger-scale scenarios.

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What is backtracking?

Backtracking is a systematic algorithmic technique used to find solutions to problems by incrementally exploring potential candidates and backtracking from paths that do not satisfy the problem constraints.

How does backtracking work?

Backtracking works by exploring the solution space incrementally. It makes a series of decisions, moving forward until it reaches a dead end or finds a solution. Upon encountering an invalid solution, it backtracks to the last valid decision point and explores other paths.

What are the key steps involved in implementing a backtracking algorithm?

The main steps in implementing a backtracking algorithm include defining the problem, identifying decision points, establishing constraints, recursively exploring solutions, making decisions, backtracking, and terminating when a solution is found or all possibilities are exhausted.

Are there any limitations or drawbacks of using backtracking?

Backtracking’s efficiency can decrease significantly with large problem space, leading to excessive exploration. Additionally, not all problems can be efficiently solved using backtracking, especially those with extremely large or complex solution spaces.

How can I optimize a backtracking algorithm?

Optimizations for backtracking may include pruning branches of the search tree early if they are determined to lead to invalid solutions, employing heuristics to guide the search, or using additional data structures to enhance efficiency.

Are there alternatives to backtracking for solving similar problems?

Yes, other techniques like dynamic programming, greedy algorithms, or even specialized algorithms tailored to specific problems can sometimes provide more efficient solutions compared to backtracking, especially for certain types of problems.

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About the Author

Senior Consultant Analytics & Data Science

Presenting Sahil Mattoo, a Senior Consultant Analytics & Data Science at Eli Lilly and Company is an accomplished professional with 14 years of experience across data science, analytics, and technical leadership domains, demonstrates a remarkable ability to drive business insights. Sahil holds a Post Graduate Program in Business Analytics and Business Intelligence from Great Lakes Institute of Management.