Backtracking Algorithms

Backtracking is a powerful algorithmic technique that provides solutions to complex computational problems by exploring all possible outcomes and building solution incrementally. At any stage, if a solution cannot be found, the algorithm 'backtracks' to change the previous decision and tries to find a different path. It is often used in computing scenarios where possible solutions can be represented as a tree or a graph such as in puzzles, artificial intelligence, and games.

The most characteristic feature of Backtracking Algorithm is its capability of utilizing user-defined conditions to decide which track to continue with or leave. The algorithm will continue down the path till it meets an endpoint. If that endpoint is a solution, it will return the solution. If not, it will discard that path, move back up the tree, and start down a new path.

Let's take a look at the basic structure of a backtracking algorithm:

def BacktrackingApproach(candidates):
    if found_solution(candidates):
        output_solution(candidates)
        return

    for next_candidate in list_of_candidates:
        if is_valid(next_candidate):
            # Try this partial candidate solution
            candidate.add(next_candidate)
            # Given the candidate, explore further.
            BacktrackingApproach(candidates)
            # Backtrack
            candidates.remove(next_candidate)

Through an example of generating all permutations of an array, we can further understand the working of this algorithm:

def permute(data, i, length): 
    if i==length: 
        print(''.join(data) )
    else: 
        for j in range(i,length): 
            # swap
            data[i], data[j] = data[j], data[i] 
            permute(data, i+1, length) 
            data[i], data[j] = data[j], data[i]  
 
# test the function 
string = "ABC"
n = len(string) 
data = list(string) 
permute(data, 0, n)

In the above code, 'ABC' are swapped with every index to form multiple permutations. When it finds no more candidates to proceed with, it backtracks and takes the next available candidate.

Another popular instance where backtracking is used is the N-Queens problem, where the task is to place N queens on an NxN chessboard such that no two queens threaten each other.

Backtracking is not always the most efficient algorithm, especially in problems where there are a large number of potential solutions. However, it's an effective method for problems where the solution space is not too large, or for problems where we want to find all solutions.

In summary, backtracking can be an extremely useful tool in a programmer's toolkit for solving complex problems. This technique is versatile and can be used across a broad swath of different problem types.