Backtracking Algorithm with Example

Last Updated on June 27, 2023 by Prepbytes

Backtracking is a powerful algorithmic technique used to solve problems that involve searching for a solution among a large set of possibilities. It is especially useful for solving combinatorial problems, such as puzzles, optimization problems, and constraint satisfaction problems. In this article, we will explore the backtracking algorithm in detail, understand its working principles, and examine some real-world examples to illustrate its application. We will also introduce the concept of state space trees, which provide a visual representation of the search process in backtracking.

Understanding Backtracking Algorithm

Backtracking is a systematic approach that involves exploring all possible solutions to a problem by incrementally building a solution candidate and backtracking whenever the candidate fails to satisfy certain conditions. It is based on the idea of depth-first search, where the search space is traversed in a depth-first manner, exploring one branch of the search tree at a time.

Working Principle of Backtracking Algorithm

The general working principle of the backtracking algorithm can be summarized in the following steps:

1. Start with an empty solution candidate or an initial partial solution.
2. Make a choice to extend the current solution candidate.
3. Check if the choice violates any constraints or conditions.
   • If the choice is valid, continue to the next step.
   • If the choice is invalid, backtrack and undo the choice.
4. If the solution candidate satisfies the problem’s goal condition, a valid solution is found.
5. If the solution candidate cannot be extended further, backtrack to the previous decision point and try an alternative choice.
6. Repeat steps 2-5 until all possible solutions are explored or a valid solution is found.

State Space Trees

To visualize the search process in backtracking, we can use state space trees. A state space tree represents the search space of a problem by showing all possible states and the transitions between them. Each node in the tree represents a specific state, and the edges represent the choices made to transition from one state to another. By following the branches of the tree, we explore different paths in the search space.

Applications of Backtracking Algorithm

The backtracking algorithm has various practical applications, including:

1. Finding Hamiltonian Paths in a Graph:
   Backtracking can be used to find all possible Hamiltonian paths in a graph, where each vertex is visited exactly once. This is useful in optimizing travel routes or exploring graph connectivity.

   

2. Solving the N-Queens Problem:
   Backtracking is commonly employed to solve the N-Queens problem, which involves placing N queens on an NxN chessboard without any two queens attacking each other. It helps in finding all the distinct solutions or a single valid solution.

3. Maze Solving:
   Backtracking algorithms are applied to solve maze problems, where the objective is to find a path from the starting point to the destination. By exploring different paths and backtracking when reaching dead ends, the algorithm determines a valid solution.

4. Knight’s Tour Problem:
   The backtracking algorithm is utilized to solve the Knight’s Tour problem, which involves finding a sequence of moves for a knight on a chessboard to visit every square exactly once. It helps in identifying all possible tours or a single valid tour.

Conclusion
Backtracking is a powerful algorithmic technique that allows us to systematically search for solutions in a large search space. By incrementally building a solution candidate and backtracking when necessary, we can efficiently find valid solutions. State space trees provide a visual representation of the search process, aiding in understanding and analyzing the backtracking algorithm. The N-Queens problem example showcased how backtracking and state space trees work together to solve real-world puzzles.

Remember, backtracking algorithms can be resource-intensive, and optimization techniques, such as pruning or heuristics, might be required to handle larger problem sizes. Nevertheless, understanding the principles of backtracking and state space trees provides a valuable tool in a programmer’s problem-solving toolkit.

Frequently Asked Questions (FAQs)

Q1. How does the backtracking algorithm differ from other search algorithms?
The backtracking algorithm is different from other search algorithms in that it systematically explores all possible solutions by incrementally building a solution candidate and backtracking whenever necessary. It exhaustively searches the entire solution space, whereas other algorithms may use heuristics or pruning techniques to optimize the search process.

Q2. Can the backtracking algorithm handle problems with a large search space?
The backtracking algorithm explores all possible solutions, which can be time-consuming and resource-intensive for problems with large search spaces. In such cases, optimization techniques like pruning or heuristics can be applied to reduce the search space and improve the algorithm’s efficiency.

Q3. How do I determine the constraints or conditions for backtracking?
The constraints or conditions for backtracking depend on the specific problem you are trying to solve. They define the rules that must be satisfied at each step of the solution-building process. Understanding the problem domain and requirements is crucial for identifying the constraints and formulating the backtracking algorithm accordingly.

Q4. What happens if there is no valid solution in the search space?
If there is no valid solution in the search space, the backtracking algorithm will exhaustively explore all possibilities and eventually backtrack to the previous decision point. At that point, the algorithm might terminate without finding a solution, indicating that no valid solution exists for the given problem.

Q5. Can backtracking be applied to real-world problems beyond puzzles?
Absolutely! Backtracking is a versatile algorithmic technique applicable to a wide range of real-world problems. It can be used for optimization problems, scheduling, constraint satisfaction, and more. By formulating the problem as a search space, defining constraints, and applying the backtracking approach, complex real-world problems can be effectively tackled.