Algorithms | Analysis of Algorithms (Recurrences) | Question 3
What is the worst case time complexity of following implementation of subset sum problem.
C
// Returns true if there is a subset of set[] with sum 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)
Answer: (D)
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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Last Updated :
28 Jun, 2021
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