1. Bubble Sort

2. Bubble Sort: Algorithm
Assume that we use an array of integers A with N elements.
for (p = 0; p < N; p++) // pass p
for (e = 1; e < N - p; e++) // sort sub-array A[0...(N-p)]
if (A[e-1] > A[e]) // sort in non-decreasing order
Swap(A[e-1], A[e]);

3. Bubble-Sort on a Sequence
Original data sequence:
=================
==End of Pass 1====
| 9
| 9
==End of Pass 2====
| 7 9
==End of Pass 3====
2 3 5 | 6 7 9
==End of Pass 4====

4. Analysis: Best Case and Worst Case
If the given list is ordered in reverse order, we do:
N(N-1) comparisons (if statement)
N(N-1) swaps (function Swap())
If the given list is ordered in “random” order, we do:
N(N-1) comparisons
less than N(N-1) swaps
If the given list is already sorted, we do:
no swaps
In all 3 cases, the running time is O(N2) due to N(N-1) comparisons.

5. Optimizing the Bubble Sort
The previous algorithm can be optimized, if the program stops as soon as it is detected that no swap has been made during a pass. In this case, a Boolean variable swapped can be used, that is set to FALSE at the beginning of every pass. swapped will then be set to TRUE whenever a swap is made. At the end of a pass, if swapped still keeps the value FALSE, the program stops.

6. Optimizing the Bubble Sort
swapped = TRUE;
for (p = 0; p < N && swapped; p++) // pass p {
swapped = FALSE;
for (e = 1; e < N - p; e++) // sort sub-array A[0...(N-p)]
if (A[e-1] > A[e])
swap(A[e-1], A[e]); }
If the given list is already sorted, only (N-1) comparisons need to be done before the program stops. So the best-case running time of the optimized Bubble Sort is W(N).