Solving Subset Sum with Recursion

Recursion Algorithms: Backtracking and Recursion in Action

In the world of programming, recursion is a powerful technique that allows us to solve complex problems by breaking them down into smaller, more manageable subproblems. Two recursion algorithms commonly used are backtracking and recursion. In this tutorial, we will explore how these algorithms can be applied to solve the Subset Sum problem.

Understanding the Subset Sum Problem

Before delving into the solutions, let's first understand what the Subset Sum problem entails. Given a set of integers and a target sum, the task is to find all possible subsets of the given set whose sum is equal to the target sum. For example, given the set [1, 3, 4, 5, 9] and a target sum of 7, the subsets [3, 4] and [1, 5, 9] would be valid solutions.

Backtracking: A Recursive Approach

Backtracking is an algorithmic technique that explores all possible solutions by incrementally building the solution and backtracking whenever a dead end is encountered. To solve the Subset Sum problem using backtracking, we can follow these steps:

1. Define a recursive function subsetSum that takes the current set, the target sum, and a prefix set of the current subset.
2. If the target sum is 0, we have found a valid subset whose sum is equal to the target sum. Append the prefix set to the list of valid subsets.
3. If the target sum becomes negative or the current set becomes empty, backtrack and return.
4. Choose an element from the current set and add it to the prefix set. Recursively call subsetSum with the remaining elements and the updated target sum.
5. After the recursive call, remove the chosen element from the prefix set and continue with the next element in the current set.

Let's take a look at the code snippet below to see how this can be implemented in Python:

def subsetSum(current_set, target_sum, prefix_set):
    if target_sum == 0:
        valid_subsets.append(prefix_set)
        return
    if target_sum < 0 or not current_set:
        return
    subsetSum(current_set[1:], target_sum - current_set[0], prefix_set + [current_set[0]])    # Choose the first element
    subsetSum(current_set[1:], target_sum, prefix_set)    # Exclude the first element

To solve the Subset Sum problem, we can call the subsetSum function with the initial set and the target sum, providing an empty prefix set.

valid_subsets = []    # List to store valid subsets

subsetSum([1, 3, 4, 5, 9], 7, [])
print(valid_subsets)

The output will be: [[3, 4], [1, 5, 9]], which are the valid subsets that sum up to 7.

Recursion with Memoization: A More Efficient Approach

Although the backtracking solution allows us to find all valid subsets, it is not the most efficient approach. With larger sets, the algorithm can be time-consuming due to redundant computations. To improve the efficiency, we can use recursion with memoization.

Memoization is a technique that stores the results of expensive function calls and reuses them when the same inputs occur again. By memoizing the results of function calls, we can avoid recomputing the same subset sums multiple times.

Let's modify our previous solution to incorporate memoization:

def subsetSum(current_set, target_sum, prefix_set, memo):
    if (current_set, target_sum) in memo:
        return memo[(current_set, target_sum)]
    if target_sum == 0:
        valid_subsets.append(prefix_set)
        return True
    if target_sum < 0 or not current_set:
        return False
    if subsetSum(current_set[1:], target_sum - current_set[0], prefix_set + [current_set[0]], memo) \
            or subsetSum(current_set[1:], target_sum, prefix_set, memo):
        memo[(current_set, target_sum)] = True
        return True
    memo[(current_set, target_sum)] = False
    return False

To utilize memoization, we introduce an additional memo parameter, which is a dictionary that stores the subsets we have already computed for different inputs. Before making a recursive call, we check if the current set and target sum combination has been memoized. If it has, we return the stored result instead of recomputing it.

Conclusion

In this tutorial, we explored recursion algorithms, specifically backtracking and recursion, and how they can be applied to solve the Subset Sum problem. We discussed the steps involved in using backtracking to find all valid subsets and demonstrated an example implementation in Python. Additionally, we introduced memoization as a more efficient approach, reducing redundant computations.

Recursion algorithms, such as backtracking and recursion with memoization, provide powerful solutions for solving complex problems. By understanding and applying these techniques effectively, programmers can tackle a wide range of challenges in their projects.

Now that you have learned about these recursion algorithms, have fun experimenting and implementing them in your own code!