def f(n):
    if n < 100000:
        return 0;
    for(int i = 0; i < n*n; i++){
        return f(n-1)

What is the time complexity?

My answer is $O((n!)^2)$. Here's my thought process:

  1. The for loop will be running $n^2$ the first time.

  2. However, during the first loop (i.e., $i = 0$), it will call $f(n-1)$, hence the next for loop will be $(n-1)^2$.

  3. This will keep going until $n <10000$ (base case). Assuming $n$ is very huge, the number of calls for each function to base case is essentially $n$ times.

  4. Now, considering all the for loops, the total number of calls is essentially $n^2 \cdot (n-1)^2 \cdot (n-2)^2 \cdots 1!$ (there will be a total of $n$ times multiplication, and each multiplication will be $n-1$ of the previous one because of $f(n-1)$ call).

  • 3
    $\begingroup$ There are clearly some braces missing in your code, so nobody knows what that function does - but my best guess is it is O(n). $\endgroup$
    – gnasher729
    Jul 15 '19 at 15:54
  • $\begingroup$ What kind of programming language is this? The first 3 lines look like Python but then line 4 takes a sharp turn towards C (or towards C sharp?—pun intended). $\endgroup$
    – dkaeae
    Jul 15 '19 at 16:06
  • $\begingroup$ @dkaeae It's pseudocode. $\endgroup$ Jul 15 '19 at 18:47
  • 1
    $\begingroup$ Possible duplicate of Is there a system behind the magic of algorithm analysis? $\endgroup$
    – xskxzr
    Jul 16 '19 at 1:33

It runs in time $O(n)$. Remember that a function only returns once. In each call to f, the for loop is immediately terminated at i=0 by the return statement, so the function body is equivalent to

if n < 100000
    return 0;
    return f(n-1);

However, your answer of $O((n!)^2)$ is not wrong: $(n!)^2$ is a huge overestimate of the running time, but big-$O$ essentially means "at most this much."

  • $\begingroup$ ohh yeah I didn't see that the for loop is useless because of the return statement $\endgroup$ Jul 16 '19 at 1:42
  • $\begingroup$ what if i take away the return part? what will be the time complexity $\endgroup$ Jul 16 '19 at 11:30

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