My teacher claimed that this code block is run in O(n) time… why is that?

Question

My teacher claimed that this code block will take O(n) time to execute. I'm trying to figure out why. I realize that the two for loops bundled together would be an arithmetic series ..... my logical approach was if K = 3, then the inner loop would run three times, then two times, and then once. If K = 2, then the inner loop would run twice, once, and then stop .

In mathematical terms, it would be N, N-1, N-2 for k = 3

I was later able to use the arithmetic series formula and got N*(N + (N-(N-1))/2..

I don't know how to approach the while loop.

All I can surmise is that when N =4 , the loop runs twice and until N = 9, the loop runs thrice... How would I set this up mathematically?

The end result is N*(N+(N-(N-1)))/2 + O(while loop) to get O(N)?

Any advice would be greatly appreciated.

void doit(int n) {
     int k = 0; int m = n; int s = 0;
     while (m <= n) {
         k = k + 1;
         m = k * k;
     }

     for (int i = 0; i < k; i++) {
         for (int j = i; j < k; j++) {
             s = s + m;
             m = m - 1;
         }
     }
 }

Answer 1

The loops itself are O(k^2) , but the thing is what comes before it. The while loop finds the smallest square number m = k^2 that is larger than n , so basically because m relates to n the end result becomes practically O(n) .

Answer 2

Well as you can see the while loop and the nested for loop are separate of each other. Therefore we'll calculate the time complexity for each one separately.

While Loop

Inside the while loop we have instructions that are independent of n . The problem here is how we calculate the number of times the while loop runs. If we 'cheat' a bit and look at it in terms of k instead of n we notice that at once we exit the loop, k holds the number of times the loop has ran for. It's just like having a counter really. At that point k^2 is roughly equal to n therefore the complexity is O(sqrt(n))

For Loops

Here's where the real magic happens as this part will turn out to be O(n) overcoming the O(sqrt(n)) complexity of the while loop.

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The time taken by each for loop can be seen as a sum of a constant number of instructions. Therefore we'd have a nested sum. I'll try to describe it for you as StackOverflow won't let me post images. You basically have two sums, one starting with i=0 and going up to (k-1) and another starting from j=1 and going up to (k-1) and all you're adding is a constant number c of commands.

You can easily calculate this sum and it turns out that the complexity is O(k^2) but again we have to think in terms of n and just like before it becomes O(n) .