how to calculate Bubble sort Time Complexity

Question

I was trying to understand the Data Structure and different algorithm, then i got confused to measure the Bubble sort time complexity.

for (c = 0; c < ( n         


        

Answer 1

Best case: This time complexity can occur if the array is already sorted. That means no swapping occurs and only 1 iteration of n elements will be there.

So time complexity is O(n).

Worst case: This time complexity can occur if the array is already sorted but is descending order.

In 1st iteration, number of comparison = n-1
In 2nd iteration, number of comparison = n-2
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In (n-2)th iteration, number of comparison = 2
In (n-1)th iteration, number of comparison = 1

for n elements total number of iteration= n-1
Total number of comparison S = (n-1)+ (n-2) +........ +    2    +    1
We can write this also          S =      1 +      2 + ........+(n-2) + (n-1)
................................................................................................................................                                           2S =       n +     n + ......... +     n +       n .... [Adding both line]
                                          2S = n(n-1) ..... [as total no of iteration = n-1]
                                           S = n(n-1)/2
In polynomial function, highest order of n is considered as time complexity.
So, Time Complexity is O(n^2)