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Time Complexity of Loop with Powers

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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

<script>
 
// JavaScript program for the above approach
function fun(n, k)
{
    for(let i = 1; i <= n; i++)
    {
        int p = Math.pow(i, k);
        for (let j = 1; j <= p; j++)
        {
            // Some O(1) work
        }
    }
}
 
// This code is contributed by Shubham Singh
 
</script>

                    

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) + ?(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)



Last Updated : 30 Oct, 2023
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