算法复杂度分析混乱

Question

static int sumAfterPos(int[] A) {
  int N = A.length;
  for (int i = 0; i < N; i += 1) {
    if (A[i] > 0) {
      int S;
      S = 0;
      for (int j = i + 1; j < N; j += 1) {
        S += A[j];
      }
      return S;
    }
  }
  return 0;
}

I am having trouble figuring out whether this code runs in O(n^2) or in O(n). I am not sure whether the return S will have a big impact on the runtime.

Answer 1

It is O(N) in time.

For example, A[K] > 0 , then you have already have K steps. Then you run another NK steps and return. So totally you have O(N) .

Let us say, all A[i] < 0 , this will make the inner loop away. So it is O(N) in this case.

Now let us say, A[0] > 0 , this will make the out loop only repeat once and the inner loop will run from 1 to N - 1, so totally you have 1 + (N-1 - 1 + 1) = N.

Now let us say, A[1] > 0 , this will make the out loop only repeat twice and the inner loop will run from 2 to N - 1, so totally you have 2 + (N-1 - 2 + 1) = N.

...

Now let us say, A[k] > 0 , this will make the out loop only repeat k + 1 times and the inner loop will run from k + 1 to N - 1, so totally you have k + 1 + (N-1 - k -1 + 1) = N.

Now let us say, A[N-1] > 0 , this will make the out loop only repeat N and the inner loop will never run, so totally you have N times.

Answer 2

This looks to be O(n) , or rather, exactly N iterations will occur. If the statement (A[i] > 0) is true, then a value would be returned after the inner for loop completes its iterations. If there are N elements in the array A , and the outer for loop is at iteration i , and the conditional is met, then the inner for loop will iterate at most Ni times and then return. If the conditional is never met, then the outer for loop will iterate N times. Thus, exactly N iterations between the outer- and inner- for loops will execute.

Return statements are generally never taken into account in determining the runtime complexity of an algorithm (unless the return statement for some reason is non-trivially complex (eg recursion)).

EDIT: For another perspective, note that as the index i increases linearly, the starting point for the index j increases linearly at the same rate, and thus the possible remaining work is decreasing linearly at the same rate as well. Since the the function is guaranteed to return once the inner loop is reached, the total number of iterations between the two for loops is i+j=N by the end of execution.