1. Quadratic Sorts & Breaking the O(n2) Barrier

2. Sort Terminology
Stable : maintains original ordering for "equal" items
Given A B E B C
B will always be before B

3. Sort Terminology
Adaptive : Performs better when data is already sorted

4. Bubble Sort Size - 1 times:
Walk through array, swap neighbors if out of order

5. Analysis
K = 1 : i repeats size - 1 times K = 2 : i repeats size - 2 times K = 3 : i repeats size - 3 times … K = size - 3 : i = repeats 3 times K = size - 2 : i = repeats 2 times K = size - 1 : i = repeats 1 times
O(1)

6. ( 𝑛−1 2 𝑔𝑟𝑜𝑢𝑝𝑠)(𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛𝑠)= 𝑛−1 2 𝑛 = 𝑛 2 −𝑛 2 = 𝑛 2
Analysis
size - 1 size - 2 size - 3 …
( 𝑛−1 2 𝑔𝑟𝑜𝑢𝑝𝑠)(𝑛 𝑐𝑜𝑚𝑝𝑎𝑟𝑖𝑠𝑜𝑛𝑠)= 𝑛−1 2 𝑛 = 𝑛 2 −𝑛 2 = 𝑛 2

7. Analysis
O(n2) work (n levels of n comparisons & swaps)
Stable

8. Analysis Can make adaptive with done flag
Best case now O(n) (1 pass through n things)
Worst still O(n2)

9. Selection Sort Size - 1 times:
Find largest remaining unsorted and swap to end

10. Selection Sort OR Size - 1 times:
Find smallest remaining unsorted and swap to start

11. Analysis
O(n) swaps
O(n2) comparisons (n passes each that does up to n comparisons)
Overall O(n2)

12. Analysis Not adaptive Not stable
Swap can move items out of original order:

13. Insertion Sort Size - 1 times:
Take first unsorted item and swap it to the left until sorted

14. Analysis
O(n2) swaps & comparisons
Overall O(n2)

15. Analysis Adaptive (If doing n passes of O(1) work) Stable
No big leaps
Good choice for small problems / mostly sorted data

16. Breaking N2
Bubble, Insertion, Selction all N2
Why?

17. Inversions Inversions = out of order elements For:
inversions are: (1,2), (1,3), (1,5), (3,5), (4,5)
Item 1 2 3 4 5
Value 29 10 14 37 13

18. Inversions Worst case inversions:
Every element inverted vs every following element
Inversions
Count
(1,2), (1,3), (1,4), …, (1,n),
n-1
(2,3), (2,4), …, (2,n),
n-2
(3,4), …, (3,n),
n-3 ⋮
(n-1,n) 1

19. Inversions Worst case inversions:
Every element inverted vs every following element
Inversions
Count
(1,2), (1,3), (1,4), …, (1,n)
n-1
(2,3), (2,4), …, (2,n)
n-2
(3,4), …, (3,n)
n-3 ⋮
(n-2, n-1), (n-2, n) 2
(n-1,n) 1
(N-1) / 2 groups of N
= (N2 - N) / 2

20. Limits Array can have n(n-1)/2  n2 inversions
Swap of neighbors can fix only one inversion
Any sort algorithm that swaps adjacent pairs is O(n2)

21. Shell Sort
Shell sort
Insertion sort where items jump multiple spots

22. Shell Sort Multiple passes, descending jump size on each pass
Decrease by at least constant divisor
Big O depends on sequence

23. Shell Sort BigO jumpSize = size / 3 repeat until jumpSize length = 0
For each of jumpSize starting locations
Do insertion sort starting there, steping by jump size

24. Shell Sort BigO Exact big O depends on sequence
Shown sequence O( 𝑛 )
Best known case : O(n*log(n)2)
Lowest possible big O unproven