Complexity Analysis Backtracking is an algorithmic-technique for solving problems recursively by trying to build a solution incrementally, one piece at a time, removing those solutions that fail to satisfy the constraints of the problem at any point of time by time, here, is referred to the time elapsed till reaching any level of the search tree. According to the wiki definition, Backtracking can be defined as a general algorithmic technique that considers searching every possible combination in order to solve a computational problem. There are three types of problems in backtracking — Decision Problem — In this, we search for a feasible solution. Optimization Problem — In this, we search for the best solution. Enumeration Problem — In this, we find all feasible solutions. How to determine if a problem can be solved using Backtracking?

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It uses a recursive approach to explain the problems. We can say that the backtracking is needed to find all possible combination to solve an optimization problem. Backtracking is a systematic way of trying out different sequences of decisions until we find one that "works. A tree is composed of nodes. Backtracking can understand of as searching a tree for a particular "goal" leaf node.

Backtracking is undoubtedly quite simple - we "explore" each node, as follows: To "explore" node N: 1. If N is a goal node, return "success" 2. If N is a leaf node, return "failure" 3. Return "failure" Backtracking algorithm determines the solution by systematically searching the solution space for the given problem.

Backtracking is a depth-first search with any bounding function. All solution using backtracking is needed to satisfy a complex set of constraints. The constraints may be explicit or implicit. Explicit Constraint is ruled, which restrict each vector element to be chosen from the given set. Implicit Constraint is ruled, which determine which each of the tuples in the solution space, actually satisfy the criterion function.

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## Introduction of Backtracking

The completion is done incrementally, by a sequence of candidate extension steps. Conceptually, the partial candidates are represented as the nodes of a tree structure , the potential search tree. Each partial candidate is the parent of the candidates that differ from it by a single extension step; the leaves of the tree are the partial candidates that cannot be extended any further. The backtracking algorithm traverses this search tree recursively , from the root down, in depth-first order. At each node c, the algorithm checks whether c can be completed to a valid solution. If it cannot, the whole sub-tree rooted at c is skipped pruned. Otherwise, the algorithm 1 checks whether c itself is a valid solution, and if so reports it to the user; and 2 recursively enumerates all sub-trees of c.

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## Recursion and Backtracking

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## Backtracking Algorithms

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## Backtracking

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