# Algorithms | Analysis of Algorithms (Recurrences) | Question 3

• Difficulty Level : Hard
• Last Updated : 28 Jun, 2021

What is the worst case time complexity of following implementation of subset sum problem.

 `// Returns true if there is a subset of set[] with sun equal to given sum``bool` `isSubsetSum(``int` `set[], ``int` `n, ``int` `sum)``{``   ``// Base Cases``   ``if` `(sum == 0)``     ``return` `true``;``   ``if` `(n == 0 && sum != 0)``     ``return` `false``;``  ` `   ``// If last element is greater than sum, then ignore it``   ``if` `(set[n-1] > sum)``     ``return` `isSubsetSum(set, n-1, sum);``  ` `   ``/* else, check if sum can be obtained by any of the following``      ``(a) including the last element``      ``(b) excluding the last element   */``   ``return` `isSubsetSum(set, n-1, sum) || ``          ``isSubsetSum(set, n-1, sum-set[n-1]);``}`

(A) O(n * 2^n)
(B) O(n^2)
(C) O(n^2 * 2^n)
(D) O(2^n)

Explanation: Following is the recurrence for given implementation of subset sum problem

T(n) = 2T(n-1) + C1
T(0) = C1

Where C1 and C2 are some machine specific constants.

The solution of recurrence is O(2^n)

We can see it with the help of recurrence tree method

```           C1
/       \
T(n-1)     T(n-1)

C1
/       \
C1           C1
/     \        /    \
T(n-2)  T(n-2)   T(n-2)  T(n-2)

C1
/       \
C1           C1
/     \        /    \
C1     C1      C1     C1
/   \   /  \    /  \   /  \

If we sum the above tree level by level, we get the following series
T(n) = C1 + 2C1 + 4C1 + 8C1 + ...
The above series is Geometrical progression and there will be n terms in it.
So T(n) = O(2^n)
```
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