Time Complexity of Loop with Powers

• Difficulty Level : Medium
• Last Updated : 30 Dec, 2021

What is the time complexity of the below function?

C++

 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        }    }} // This code is contributed by Shubham Singh

C

 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        }    }}

Java

 static void fun(int n, int k){    for (int i = 1; i <= n; i++)    {        int p = Math.pow(i, k);        for (int j = 1; j <= p; j++)        {            // Some O(1) work        }    }} // This code is contributed by umadevi9616

Python3

 def fun(n, k):         for i in range(1, n + 1):        p = pow(i, k)        for j in range(1, p + 1):            # Some O(1) work          # This code is contributed by Shubham Singh

C#

 static void fun(int n, int k){    for (int i = 1; i <= n; i++)    {        int p = Math.Pow(i, k);        for (int j = 1; j <= p; j++)        {            // Some O(1) work        }    }} // This code is contributed by umadevi9616

Javascript



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
= n2/2 + 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)
If  n>=k then the time complexity will be considered in O((nk+1)/(k+1)) and if n<k, then the time complexity will be considered as  in the O(nk)