0

I'm trying to analyze an algorithm that looks like this:

def foo(L):
    for n in L:
        for x in range(n):
            ...

What would be its time complexity? First I thought $O(n)$, where $n$ is the number of elements in $L$, but that seems a little optimistic given that numbers can become very large. Then I thought $O(n^2)$ but that seem quite arbitrary too. $O(nm)$ where $m$ is the maximum value in $L$ perhaps?

Raphael
  • 72,336
  • 29
  • 179
  • 389
Stand with Gaza
  • 1,234
  • 1
  • 9
  • 15
  • Time complexity on magnitude of input? Can you define it more? – Mr. Sigma. Dec 02 '18 at 23:52
  • Yes, $O(nm)$, assuming the time complexity of the operations represented by "..." on the last line of the code is $O(1)$ – John L. Dec 03 '18 at 04:45
  • Depends on $L$ and what's in the loop, obviously! (And what range means.) – Raphael Dec 03 '18 at 16:56
  • I think our reference question is enough to answer your question, or at least to guide you towards asking a more informed question (right now, you're attempting to pattern-match, not analyzing). (The second answer is probably more helpful here.) You can also check out questions tagged [tag:runtime-analysis+loops]. – Raphael Dec 03 '18 at 16:57
  • Relevant for identifying the worst case: is L a list or set? – Raphael Dec 03 '18 at 17:01

1 Answers1

1

Your result of seems correct, assuming none of the omitted operations modify the values of the variables n, L or x.

For a fixed value $k$ of n, the inner loop (for x in range(n):) loops over the set (or list; it is irrelevant for the analysis) $\{ 0, \dots, k - 1\}$, which contains $k$ elements. The outer loop (for n in L:) simply loops over each element of $L$ for a total of $|L|$ iterations, $|L|$ being the length of $L$. Thus, assuming the omitted operations are executed in $O(1)$ time, we obtain a time complexity of $O(m|L|)$ (as you propounded), where $m$ is the maximum value in $L$.

Incidentally, if you know the average value for the elements of $L$ you could also use the above reasoning to derive the average time complexity as well. (Just plug in the average value for $k$.)

As pointed out by @Raphael in the comments, of course, there is plenty of room for improvements if more assumptions are made about the contents of $L$.

dkaeae
  • 4,997
  • 1
  • 15
  • 31