1. Analysis of Bubble Sort and Loop Invariant

2. N-1 Iteration 77 12 35 42 5
101
N - 1 5 42 12 35 77
101 42 5 35 12 77
101 42 35 5 12 77
101 42 35 12 5 77
101

3. Bubble Sort Algorithm Bubble-Sort( A ): for i = 1 … (N – 1):
for k = 1 … (N – 1):
if A[k] > A[k+1]:
Swap( A, k, k+1 )
To do N-1 iterations
To bubble a value
Inner loop
Outer loop
Python Code in

4. Bubble Sort Time Complexity
Best-Case Time Complexity
The scenario under which the algorithm will do the least amount of work (finish the fastest)
Worst-Case Time Complexity
The scenario under which the algorithm will do the largest amount of work (finish the slowest)

5. Bubble Sort Time Complexity
Called Linear Time
O(N2)
Order-of-N-square
Best-Case Time Complexity
Array is already sorted
(N-1) + (N-2) + (N-3) + …. + (1) = (N-1)* N / 2
comparisons
Worst-Case Time Complexity
Need N-1 iterations
Called Quadratic Time
O(N2)
Order-of-N-square

6. Loop Invariant for i = 1 … (N – 1): for k = 1 … (N – 1):
It is a condition or property that is guaranteed to be correct with each iteration in the loop
Usually used to prove the correctness of the algorithm
for i = 1 … (N – 1):
for k = 1 … (N – 1):
if A[k] > A[k+1]:
Swap( A, k, k+1 )
Inner loop
Outer loop
To bubble a value
To do N-1 iterations

7. Loop Invariant for Bubble Sort
for i = 1 … (N – 1):
for k = 1 … (N – 1):
if A[k] > A[k+1]:
Swap( A, k, k+1 )
Inner loop
Outer loop
To bubble a value
To do N-1 iterations
By the end of iteration i  the right-most i items (largest) are sorted and in place

8. N-1 Iterations 77 12 35 42 5
101
N - 1
1st 5 42 12 35 77
101
2nd 42 5 35 12 77
101
3rd 42 35 5 12 77
101
4th 42 35 12 5 77
101
5th

9. Correctness of Bubble Sort (using Loop Invariant)
Bubble sort has N-1 Iterations
Invariant: By the end of iteration i  the right-most i items (largest) are sorted and in place
Then: After the N-1 iterations  The right-most N-1 items are sorted
This implies that all the N items are sorted