下面 3 个嵌套循环的时间复杂度是多少？

Question

Here's the code:

for (int i = 0; i < n; i++) {
    for (int j = 0; j < n * n; j++) {
        for (int k = 0; k < j; k++) {
            sum++;
        }
    }
}

I need to evaluate the Time complexity in Big-O notation of the nested loops above.

Is it just O(n) * O(n) * O(n) + O(1) to make O(n^3) ? Or is there more to it?

Answer 1

The most inner loop is executed in quadratic time (not constant), hence it should be O(n) * O(n^2) * O(n^2) = O(n^5) .

Here are all the costs:

• Most outer loop - O(n)

• The second loop - O(n^2) for each element for the outer loop

• The most inner loop - O(n^2) for each element for the second loop

Answer 2

for (int i = 0; i < n; i++) -> runs n times.
 for (int j = 0; j < n * n; j++) -> runs n² times.
  for (int k = 0; k < j; k++) -> runs n² times (k == j == n²)

n * n² * n² = n^5.

sum+++ is an operation of constant runtime (1) and can therefore be ignored.

Answer 3

The second loop isn't O(n) , it's O(n^2) by itself.