Explain how to solve sum of subset problem.

I. Introduction The sum of subset problem is a classic problem in computer science, specifically in the field of algorithms and data structures. It is a type of combinatorial problem, which deals with the combination of elements in a set in certain ways. The problem is to determine whether there is a subset of the given set with sum equal to a given sum. This problem has a wide range of applications in real life scenarios such as in financial modeling, resource allocation, and scheduling problems.

II. Problem Definition In the sum of subset problem, we are given a set of n numbers and a value sum. The problem is to determine whether there is a subset of the given set that adds up to the given sum. The input to the problem is the set of numbers and the sum, and the output is a boolean value indicating whether such a subset exists or not.

III. Approach to solve the problem The sum of subset problem can be solved using the concept of backtracking. Backtracking is a general algorithm for finding all (or some) solutions to some computational problems, notably constraint satisfaction problems. It incrementally builds candidates for the solutions, and abandons a candidate as soon as it determines that the candidate cannot possibly be extended to a valid solution.

The process of solving the sum of subset problem using backtracking involves the following steps:

1. Start from the first element in the set.
2. Add the current element to the current sum.
3. If the current sum is equal to the given sum, then we have found a subset that adds up to the given sum.
4. If the current sum is greater than the given sum, then we backtrack and remove the current element from the current sum.
5. If the current sum is less than the given sum, then we move on to the next element in the set.
6. Repeat the above steps until we have checked all elements in the set.

IV. Algorithm The algorithm to solve the sum of subset problem using backtracking is as follows:

1. Initialize an array to keep track of the elements in the current subset.
2. Start from the first element in the set.
3. Add the current element to the current subset and update the current sum.
4. If the current sum is equal to the given sum, then print the current subset.
5. If the current sum is less than the given sum, then recursively call the algorithm for the next element in the set.
6. Backtrack and remove the current element from the current subset and update the current sum.
7. Repeat the above steps until all elements in the set have been processed.

V. Program Here is a Python program to implement the above algorithm:

def subset_sum(set, n, subset, sum):
    if sum == 0:
        print(subset)
        return
    if n == 0:
        return
    subset_sum(set, n-1, subset, sum)
    if sum >= set[n-1]:
        subset_sum(set, n-1, subset + [set[n-1]], sum - set[n-1])

set = [3, 1, 5, 9, 12]
sum = 9
subset_sum(set, len(set), [], sum)

VI. Example Let's take an example to demonstrate how the problem can be solved using the above algorithm and program. Let's say we have a set {3, 1, 5, 9, 12} and we want to find a subset that adds up to 9. The program will output the subsets {3, 1, 5} and {9}.

VII. Conclusion The sum of subset problem is a classic problem in computer science that can be solved using the concept of backtracking. The time complexity of the algorithm is O(2^n), as in the worst case, the algorithm needs to explore all subsets of the set. The space complexity is O(n), as in the worst case, the maximum depth of recursion can be n. The advantage of this approach is that it is simple and easy to understand. However, the disadvantage is that it can be quite slow for large inputs.

Diagram: Not necessary for this question.

Summary

The sum of subset problem is a classic problem in computer science that involves determining whether there is a subset of a given set with a sum equal to a given sum. It can be solved using the concept of backtracking, which is a general algorithm for finding all (or some) solutions to computational problems. The time complexity of the algorithm is O(2^n), and the space complexity is O(n).

Analogy

Solving the sum of subset problem is like trying to find a combination of numbers from a given set that adds up to a specific target sum. It's similar to trying to find a combination of ingredients that creates a specific dish.