Time Complexity of Loop with Powers
What is the time complexity of below function?
void fun(int n, int k) { for (int i=1; i<=n; i++) { int p = pow(i, k); for (int j=1; j<=p; j++) { // Some O(1) work } } } |
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Time complexity of above function can be written as 1k + 2k + 3k + … n1k.
Let us try few examples:
k=1
Sum = 1 + 2 + 3 ... n
= n(n+1)/2
= n2 + n/2
k=2
Sum = 12 + 22 + 32 + ... n12.
= n(n+1)(2n+1)/6
= n3/3 + n2/2 + n/6
k=3
Sum = 13 + 23 + 33 + ... n13.
= n2(n+1)2/4
= n4/4 + n3/2 + n2/4
In general, asymptotic value can be written as (nk+1)/(k+1) + Θ(nk)
Note that, in asymptotic notations like Θ we can always ignore lower order terms. So the time complexity is Θ(nk+1 / (k+1))
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