关于您计算它的方式，冒泡排序算法的时间复杂度如何导致 O(n^2)？

Question

I understand that why bubble sort is O(n^2).

However in many explanations I see something like this:

(n-1) + (n-2) + (n-3) + ..... + 3 + 2 + 1
Sum = n(n-1)/2

How do you calcuate Sum from this part:

(n-1) + (n-2) + (n-3) + ..... + 3 + 2 + 1

Can anyone help?

Answer 1

Here's the trick:

If n is even:
  n + (n-1) + (n-2) + … + 3 + 2 + 1
= [n + 1] + [(n-1) + 2] + [(n-2) + 3] + … + [(n - (n/2 - 1)) + n/2]
= (n + 1) + (n + 1) + (n + 1) + … + (n + 1)
= n(n+1)/2

If n is odd:
  n + (n-1) + (n-2) + … + 3 + 2 + 1
= [n + 1] + [(n-1) + 2] + [(n-2) + 3] + … + [(n - (n-1)/2 + 1) + (n-1)/2] + (n-1)/2 + 1
= (n+1) + (n+1) + (n+1) + … + (n+1) + (n-1)/2 + 1
= (n+1)(n-1)/2 + (n-1)/2 + 1
= (n^2 - 1 + n - 1 + 2)/2
= (n^2 + n)/2
= n(n+1)/2

For your case, since you're counting up to n-1 rather than n, replace n with (n-1) in this formula, and simplify:

   x(x+1)/2, x = (n-1)
=> (n-1)((n-1)+1)/2 
 = (n-1)(n)/2 
 = n(n-1)/2

Answer 2

The simplest "proof" to understand without deriving the equation is to imagine the complexity as area:

so if we have sequence:

n+(n-1)+(n-2)...

we can create a shape from it... let consider n=5 :

n     5     *****
n-1   4     ****
n-2   3     ***
n-3   2     **
n-4   1     *

Now when you look at the starts they form a right angle triangle with 2 equal length sides... that is half of nxn square so the area is:

area = ~ n.n / 2 = (n^2)/2

In complexities the constants are meaningless so the complexity would be:

O(n^2)