while循环内排序的时间复杂度

Question

I am trying to understand the time complexity of an algorithm with below pseudo code:

let nums has ArrayList of numbers

sort(nums)
while(nums.size() > 1) {
   // remove two elements
   nums.remove(0);
   nums.remove(0);

   nums.add(some_number);
   sort(nums);
}

sort(nums) is (N)Log(N) . nums.remove(0) is O(N) nums.add() is O(1)

Now what is the time complexity for this algorithm.

Answer 1

The final complexity is O(n² log n) since you do n times the operation O(n log n) .

Keep in mind that estimating complexity ( O(...) ) is not the same as establishing the total number of operations (usually the time function T(...) is given by total operations), they are two different concepts. A great introduction could be Analysis of Algorithms

Thus, the O(...) notation is a upper bound but T(...) is the real steps.

You can try to calculate exactly the T function, or you can go down the upper bound by improving O , but they will always be different functions, as O applies to the worst case of all possible entries.

In your code, we unknown T for the sort function, only their upper bound wich is O(n log n) , then:

T(n) ≤ O(n log n) + T(3) + O((n - 1) log (n - 1)) + T(3) + O((n - 2) log (n - 2) + ...
T(n) ≤ O(n log n) + n T(3) + n O(n log n)
                               ^^^^^^^^^

Here, we cannot define exactly the T for sorting operations on n-1 , n-2 , ... that is the reason for establishing as a higher level O(n log n) for all of them. Then:

T(n) ≤ O(n log n) + n T(3) + n O(n log n)
T(n) ≤ O(n log n) + O(n) + O(n² log n)
T(n) ≤ O(n² log n)

In the second expression, we have a fixed number of upper bounds , in which case the upper bound will be the highest of the upper bounds.

(remove, remove and add is T(3) , goto and comparisons are ignored)