Dynamic Programming – Subset Sum Problem

Objective: Given a set of positive integers, and a value sum S, find out if there exist a subset in array whose sum is equal to given sum S.

Example:

int[] A = { 3, 2, 7, 1}, S = 6

Output: True, subset is (3, 2, 1}

We will first discuss the recursive approach and then we will improve it using Dynamic Programming.

Recursive Approach:

For every element in the array has two options, either we will include that element in subset or we don’t include it.

• So if we take example as int[] A = { 3, 2, 7, 1}, S = 6
• If we consider another int array with the same size as A.
• If we include the element in subset we will put 1 in that particular index else put 0.
• So we need to make every possible subsets and check if any of the subset makes the sum as S.
• If we think carefully this problem is quite similar to “ Generate All Strings of n bits“
• See the code for better explanation.

Time Complexity: O(2ⁿ).

Approach: Dynamic Programming (Bottom-Up)

Base Cases:

• If no elements in the set then we can’t make any subset except for 0.
• If sum needed is 0 then by returning the empty subset we can make the subset with sum 0.

Given – Set = arrA[], Size = n, sum = S

• Now for every element in he set we have 2 options, either we include it or exclude it.
• for any i^th element-
• If include it => S = S-arrA[i], n=n-1
• If exclude it => S, n=n-1.

Recursive Equation:
Base Cases:
SubsetSum(arrA, n, S)= false, if sum > 0 and n == 0 SubsetSum(arrA, n, S)= true, if sum == 0 (return empty set)
Rest Cases
SubsetSum(arrA, n, S) = SubsetSum(arrA, n-1, S)|| SubsetSum(arrA, n-1, S-arrA[n-1])

How to track the elements.

• Start from the bottom-right corner and backtrack and check from the True is coming.
• If value in the cell above if false that means current cell has become true after including the current element. So include the current element and check for the sum = sum – current element.

Complete Code:

Output: From DP: true