Big O: How to determine runtime for a for loop incrementation based on outer for loop?

Question

I have the following algorithm and the runtime complexity is O(N^2) but I want to have a deeper understanding of it rather than just memorizing common runtimes.

What wo

Answer 1

What would be the right approach to break it down and analyze it

Take pencil and paper and put down some loops unwraped:

     i        inner loops per i
-------------------------------
     1               length - 1  
     2               length - 2
    ..                       ..  
     k               length - k 
    ..                       ..
length - 1                    1
length                        0

Now, in order to obtain the total time required, let's sum up the inner loops:

 (length - 1) + (length - 2) + ... + (length - k) ... + 1 + 0

It's an arithmetic progression, and its sum is

 ((length - 1) + 0) / 2 * length == length**2 / 2 - length / 2 = O(length**2)