Derive a while loop runs in $\Theta( \sqrt{n} )$

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I know for a fact that algorithm A runs in $\Theta(\sqrt{n})$, but how does one derive that fact?

Algorithm A

i = 0
s = 0
while s <= n:
    s += i
    i += 1

Here is what I am thinking. We know that A is upper-bounded by O(n), as we increment $s$ by more than 1 in each iteration. We also know that it must be lower-bounded by $\log n$, as we increment $s$ with by something less than $2^i$ in each iteration. (Correct me if I am wrong, these are just my thoughts ...).

Now, what else can we say about A? How do we derive that its time complexity is $\Theta(\sqrt{n})$?

Each time we increment i, we add s to i. This thus means that after k steps, s has grown to:

s = /   i

This sequence is known. After the k-th step, there are Tk numbers, with Tk a triangular number [wiki]. A shorter formula for Tk is Tk=i×(i+1)/2. Tk thus scales quadratic with k.

The process will thus stop when Tk is higher than n. We can thus determine that: Tk > n, and thus k×(k+1)/2 > Tk and thus k2/2 + k/2 - n > 0. This is a quadratic equation with as discriminant d = 1/4 + 2×n, and thus as (positive) solution k0=-1/2 + √d. It thus scales with √2n.

Derive a while loop runs in $\Theta( \sqrt{n} )$, Each time we increment i , we add s to i . This thus means that after k steps, s has grown to: k --- \ s = / i --- i=0. This sequence is known. After the  Photo by apoorv mittal on Unsplash. Big O Notation allows us to measure the time and space complexity of our code. Think of the example of a for loop. You can run it over an array of 5 items and it will run pretty quickly, but if you ran it over an array of 10,000 items then the execution time will be much slower.

To help you in your reasoning you could do experimental tests to count the number of iterations. For example:

for n in range(100000000, 1000000000, 100000000) :   
    i = 0
    s = 0
    while s <= n:
        s += i
        i += 1

and get the following results:

    n       | iterations|   sqrt(n)
100000000   |   14143   |   10000.00
200000000   |   20001   |   14142.14
300000000   |   24496   |   17320.51
400000000   |   28285   |   20000.00
500000000   |   31624   |   22360.68
600000000   |   34642   |   24494.90
700000000   |   37418   |   26457.51
800000000   |   40001   |   28284.27
900000000   |   42427   |   30000.00


As recommended by @WillNess You can learn more about it here

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As you can see from the code s = 1 + 2 + 3 + .... + i (1) and s <= n (2). The first equation can also be written as s = i * (i + 1) / 2 which means that in i is approximately sqrt(s * 2) and sqrt(n * 2), and as we see from the code the while loop runs i time, each does a O(1) calculation. Therefore the overall complexity is O(sqrt(n))

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