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# Backtracking vs Branch and Bound

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In this article, we have compared Backtracking vs Branch and Bound algorithms.

1. Introduction to Backtracking Algorithm
2. Introduction to Branch and Bound Algorithm
3. Comparison of Backtracking vs Branch and Bound Algorithms
4. Conclusion

## Introduction to Backtracking Algorithm

Backtracking is a depth-first search algorithm used to find all possible solutions to a problem by building a solution incrementally and then undoing (backtracking) the last step if the solution is invalid. This process is repeated until a valid solution is discovered, or all possibilities are exhausted. Backtracking is frequently used in obtaining the solution of problems such as the `N-Queens problem` and `Sudoku`.

## Introduction to Branch and Bound Algorithm

Branch and bound is a technique for pruning the search space of a problem by using bounds on the possible solutions.
The algorithm begins by breaking the problem down into smaller subsets of the problem (branching), and then employs bounds (such as upper and lower limits) on the solutions to eliminate subsets of the problem that cannot contain the optimal solution. This process is repeated until a solution is found, or all options are exhausted. Branch and bound is frequently used to solve problems like the `Traveling Salesman` and `Knapsack` problem.

Backtracking Branch and Bound
Problem type Used to find all or a subset of all possible solutions to a problem. Used to solve optimization problems and provide the best solution.
Search strategy Traverses the state space tree by Depth-first search. Depth-first search with pruning using bounds on solutions and can also use Breadth-first search
Examples of problems it can solved N-Queens, Sudoku, M-coloring problem, Combination Sum, Subset Sum. Traveling salesman problem, Knapsack problem, Linear programming.
Time complexity It is exponential and determined by the problem's branching factor. It depends on the problem and bounds and it can be faster than brute force search.
Space complexity It is linear and determined by the depth of the recursion tree. It is linear and determined by the number of explored nodes.
Optimality Provides all possible solutions but not necessarily the best ones Provides the best solution when used with appropriate bounds and the problem is convex.
Complexity Simple and straightforward to understand. Depending on the bounds used, the problem, and the strategy, it can be complex and difficult to understand.

## Conclusion

The time and space complexity of an algorithm can vary depending on the problem and how it is implemented. Backtracking can also be improved by employing techniques such as memoization and dynamic programming to reduce time and space complexity.

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