algorithm - Why is bubble sort O(n^2)?

Question

for (int front = 1; front < intArray.length; front++)
{
    for (int i = 0; i  < intArray.length - front; i++)
    {
        if (intArray[i] > intArray[i + 1])
        {
            int temp = intArray[i];
            intArray[i] = intArray[i + 1];
            intArray[i + 1] = temp;
        }
    }
}

The inner loop is iterating: n + (n-1) + (n-2) + (n-3) + ... + 1 times.

The outer loop is iterating: n times.

So you get n * (the sum of the numbers 1 to n)

Isn't that n * ( n*(n+1)/2 ) = n * ( (n^2) + n/2 )

Which would be (n^3) + (n^2)/2 = O(n^3) ?

I am positive I am doing this wrong. Why isn't O(n^3)?

Answer 1

You are correct that the outer loop iterates n times and the inner loop iterates n times as well, but you are double-counting the work. If you count up the total work done by summing the work done across each iteration of the top-level loop you get that the first iteration does n work, the second n - 1, the third n - 2, etc., since the ith iteration of the top-level loop has the inner loop doing n - i work.

Alternatively, you could count up the work done by multiplying the amount of work done by the inner loop times the total number of times that loop runs. The inner loop does O(n) work on each iteration, and the outer loop runs for O(n) iterations, so the total work is O(n²).

You're making an error by trying to combine these two strategies. It's true that the outer loop does n work the first time, then n - 1, then n - 2, etc. However, you don't multiply this work by n to to get the total. That would count each iteration n times. Instead, you can just sum them together.

Hope this helps!