When can an algorithm have square root(n) time complexity?

Question

Can someone give me example of an algorithm that has square root(n) time complexity. What does square root time complexity even mean?

Answer 1

There are many cases. These are the few problems which can be solved in root(n) complexity [better may be possible also].

• Find if a number is prime or not.
• Grover's Algorithm: allows search (in quantum context) on unsorted input in time proportional to the square root of the size of the input.link
• Factorization of the number.

There are many problems that you will face which will demand use of sqrt(n) complexity algorithm.

As an answer to second part:

sqrt(n) complexity means if the input size to your algorithm is n then there approximately sqrt(n) basic operations ( like **comparison** in case of sorting). Then we can say that the algorithm has sqrt(n) time complexity.

Let's analyze the 3rd problem and it will be clear.

let's n= positive integer. Now there exists 2 positive integer x and y such that
     x*y=n;
     Now we know that whatever be the value of x and y one of them will be less than sqrt(n). As if both are greater than sqrt(n) 
  x>sqrt(n) y>sqrt(n) then x*y>sqrt(n)*sqrt(n) => n>n--->contradiction.

So if we check 2 to sqrt(n) then we will have all the factors considered ( 1 and n are trivial factors).

Code snippet:

   int n;
   cin>>n;
   print 1,n;
   for(int i=2;i<=sqrt(n);i++) // or for(int i=2;i*i<=n;i++)
     if((n%i)==0)
       cout<

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Note: You might think that not considering the duplicate we can also achieve the above behaviour by looping from 1 to n. Yes that's possible but who wants to run a program which can run in O(sqrt(n)) in O(n).. We always look for the best one.

Go through the book of Cormen Introduction to Algorithms.

I will also request you to read following stackoverflow question and answers they will clear all the doubts for sure :)

1. Are there any O(1/n) algorithms?
2. Plain english explanation Big-O
3. Which one is better?
4. How do you calculte big-O complexity?