Analysis of Bubble Sort and Loop Invariant

1. Analysis of Bubble Sort and Loop Invariant

2. 1 2 3 4 5 6 77 5 101 42 35 12 1 2 3 4 5 6 5 77 101 35 12 42 1 2 3 4 5 6 42 77 101 12 35 5 N - 1 1 2 3 4 5 6 42 77 101 12 5 35 1 2 3 4 5 6 42 77 101 5 12 35 N-1 Iteration

3. To do N-1 iterations procedure Bubblesort(A isoftype in/out Arr_Type) to_do, index isoftype Num to_do <- N – 1 loop exitif(to_do = 0) index <- 1 loop exitif(index > to_do) if(A[index] > A[index + 1]) then Swap(A[index], A[index + 1]) endif index <- index + 1 endloop to_do <- to_do - 1 endloop endprocedure // Bubblesort To bubble a value Outer loop Inner loop

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

6. Loop Invariant • 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

7. Loop Invariant for Bubble Sort • By the end of iteration i the right-most i items (largest) are sorted and in place

8. 1 2 3 4 5 6 77 5 101 42 35 12 1 2 3 4 5 6 5 77 101 35 12 42 1 2 3 4 5 6 42 77 101 12 35 5 N - 1 1 2 3 4 5 6 42 77 101 12 5 35 1 2 3 4 5 6 42 77 101 5 12 35 N-1 Iterations 1st 2nd 3rd 4th 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

10. Correctness of Bubble Sort (using Loop Invariant) • What if we stop early (after iteration K) • Remember the Boolean “did_swap” flag • From the invariant  the right-most K items are sorted • A[N-(K-1)] < A[N-2] < A[N-1] < A[N] • Since we stopped early, then no swaps are done, Then • A[1] < A[2] < ….. < A[N-K] • By merging these two, then the entire array is sorted