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Reading wikipedia I found that they give this example

Consider the problem of testing whether a number n is prime, by naively checking whether no number in $\{2,3,\dotsc ,\sqrt {n}\}$ divides $n$ evenly. This approach can take up to $\sqrt {n}-1$ divisions, which is sub-linear in the value of $n$ but exponential in the length of $n$ (which is about $\log(n)$. For example, a number n slightly less than 10,000,000,000 would require up to approximately 100,000 divisions, even though the length of n is only 11 digits.

I dont understand. First of all why dont they teach this way in regular computer science courses? Most of our time orders classification are regularly done with respect to N and not its digit length.

Secondly what I don't understand is, if we were to evaluate each problem with respect to its digits then almost every polynomial problem that I know from my CS course wouldn't be polynomial. Even the simplest of algorithms like counting from 1 to N. Suppose I give you a number you need to count from 1 to n. Usually in my CS course such a problem runtime would be N (therefore polynomial). Just running a loop from 1 to n. But according to wikipedia. This shouldn't be polynomial but rather exponential because the number of digits is log n but number of runs would be n. So fe: for input of 3 digits I have 8 runs. For input of 6 digits I have 64 runs.. this growth is certainly exponential for just counting from 1 to n. How does that make sense? And why this is not the way we (at least I) study computer science?

Inuyasha Yagami
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bilanush
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  • What do you mean by: "Suppose I give you a number you need to count from 1 to n." Is the algorithm suppose to print all the numbers from $1$ to $n$? – Inuyasha Yagami Aug 28 '21 at 09:52
  • Doesn't matter. Let's say print it – bilanush Aug 28 '21 at 10:40
  • Then, it is not polynomial time since input size is $\log n$. However, suppose you want to sort some set of arbitrary $n$ integers, then the input size is $\Omega(n)$. In computer science, we do study the running time complexity of an algorithm based on its input size. I do not understand what is the problem. – Inuyasha Yagami Aug 28 '21 at 12:47
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    If you have studied the DP algorithm for the knapsack problem. Then, it is not polynomial time but pseudo polynomial time. It is just that you are observing this fact for the first time. And, that is perfectly fine. – Inuyasha Yagami Aug 28 '21 at 12:51
  • You might find reading this helpful. – Inuyasha Yagami Aug 28 '21 at 12:55
  • I would say that it $n$ is often based on convention. We could have stated that $n$ is the size of the number we are factoring, instead of stating that $n$ is the size in bits. Then the way that the running times are described would change, but it would not fundamentally alter the speed of the algorithms. It would just be different numbers being used for $n$. – Matt Groff Aug 29 '21 at 03:39

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