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I'm debating with a friend whether a particular function I wrote is $O(N^3)$ or $O(N \times M \times X)$

I believe it is the latter since all 3 variables differ in size. $N = 100, M = 50, X = 10000$

for i in range(len(N)):
  for j in range(len(M)):
    for p in range(len(X)):
      if statement:
        count += 1
        list.append(count)

The outer loop executes $N$ times, the inner loop executes $M$ times, and the most inner loop executes $X$. Hence giving $N \times M \times X$. His theory is that because $X$ is so much greater than the other two variables it makes it $O(N^3)$

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  • $\begingroup$ What is the variable $n$? How much time does the append function take? Are $N$, $M$, and $X$ constants or variables? $\endgroup$ – ryan Jun 6 '19 at 17:14
  • $\begingroup$ @ryan Assume append is $O(1)$ and $N, M$ and $X$ are all variables. $\endgroup$ – bogdboa Jun 6 '19 at 17:17
  • $\begingroup$ @ryan Sorry that was a typo, $n$ should actually be $N$ $\endgroup$ – bogdboa Jun 6 '19 at 17:19
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Obviously you can't say it's $O(N^3)$ because X might grow a lot faster than N. But you can't even say it's O (N x M x X), because you don't know how often the "list.append(count)" is executed and what the time complexity of that operation is.

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As you have mentioned that $ N, M, X $ are variables, the time complexity would be $ O(N*M*X) $.
A possible counter argument for the complexity not equal to $ O(N^3) $ is, what if the rate of increase of $ X $ is greater than rate of increase of $ N $ ?

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  • $\begingroup$ Would it still be $O(N \times M \times X)$ if they were constants? $\endgroup$ – bogdboa Jun 6 '19 at 17:41
  • $\begingroup$ If they were constants, you can just say that, the program takes constant time. If only N is a variable, you can say that program takes $ M * X * O(N) $ time, which is equivalent to $ O(N) $ $\endgroup$ – SiluPanda Jun 6 '19 at 17:43

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