1. Lecture 4 Sort(2) Merge sort Quick sort
ACKNOWLEDGEMENTS: Some contents in this lecture source from COS226 of Princeton University by Kevin Wayne and Bob Sedgewick

2. Roadmap
Merge sort
Quick sort

3. Merge sort
Transylvanian-Saxon folk dance

4. Merge sort
Divide array into two halves.
Recursively sort each half.
Merge two halves. 2 13 6 24 43 1 51 9 10 2 13 6 24 1 43 9 10 2 6 13 24 9 10 51 1 9 10 43 51 1 2 6 9 10 13 24 43 51

5. Merging
2, 13, 43, 45, 89
6, 24, 51, 90, 93
2, 6, 13, 24, 43, 45, 51, 89, 90, 93
Assume we need to merge two sorted arrays, with M, N elements respectively. Then
How many comparisons in best case?
How many comparisons in worst case?
A. M+N B. max{M,N} C. min{M,N} D. M+N-1

6. MERGE(A,p,q,r)
..., 2, 13, 43, 45, 89, 6, 24, 51, 90, 93, ... A p q r

7. Complexity D (N) = 2 D (N/2) + N Compute by picture. D (N) D (N / 2)
D (N / 2k)
2 (N/2)
2k (N/2k)
N/2 (2)
...
lg N
N lg N
= N
4 (N/4)

8. Complexity D (N) = 2 D (N/2) + N Compute by expansion.
D (N) / N = 2 D (N/2) / N + 1
= D (N/2) / (N/2) + 1
= D (N/4) / (N/4)
= D (N/8) / (N/8)
. . .
= D (N/N) / (N/N)
= lg N

9. Complexity Good algorithms are better than supercomputers.
Laptop executes 108 compares/second.
Supercomputer executes 1012 compares/second.
insertion sort (N2)
mergesort (N log N)
computer
thousand
million
billion
home
instant
2.8 hours
317 years
1 second
18 min
super
1 week
Good algorithms are better than supercomputers.

10. Discussion
Mergesort is
stable?
in place?
Recursions cost spaces.

11. Bottom-up merge sort 1 2 6 9 10 13 24 43 51 1 2 6 9 13 24 43 51 10 2 6 13 24 1 9 43 51 10 2 13 6 24 1 43 9 51 10 2 13 6 24 43 1 51 9 10

12. Bottom-up merge sort

13. Quiz
Give the array that results immediately after the 7th call to merge() by using:
a) top-down mergesort
b) bottom-up merge sort
M B X V Z Y H U N S I K
A. B M V X Y Z H N U S I K
B. B M V X Y Z H N U S K I
C. B M V X Y Z H U N S I K
D. B M V X Y Z H N U S K I

14. Quiz
Mergesort, BottomupMergesort, which one is more efficient? which one is easier for understanding and debugging? which one you prefer?
A. Mergesort B. BottomupMergesort

15. O(nlogn) sorting algorithms
n=2k
Best case
Worst case
Bottom-up-merge
Merge
Space Θ(n)
Quicksort: worst O(n2), average O(nlogn), space O(logn)
Heapsort: worst O(nlogn), space O(1)

16. Sorting

17. Where are we?
Merge sort
Quick sort

18. Quick sort
Quicksort honored as one of top 10 algorithms of 20th century in science and engineering.
Mergesort.
Java sort for objects.
Perl, C++ stable sort, Python stable sort, Firefox JavaScript, ...
Quicksort.
Java sort for primitive types.
C qsort, Unix, Visual C++, Python, Matlab, Chrome JavaScript, ...

19. Quick sort
Sir Charles Antony Richard Hoare Turing Award

20. Quick sort Shuffle the array. Partition so that, for some j
entry a[j] is in place
no larger entry to the left of j
no smaller entry to the right of j
Sort each piece recursively.

21. Partition#1

22. Partition#1 10 13 6 24 43 1 51 9 i j

23. Partition #2 - by Sedgewick

24. Complexity
Best case: array always split into two equal-sized subarrays.
similar to mergesort, O(NlogN)
Worst case: array always split into a 0-sized and an N-1-sized subarrays
similar to selection sort, O(N2)
Average case:

25. Complexity Good algorithms are better than supercomputers.
Laptop executes 108 compares/second.
Supercomputer executes 1012 compares/second.
insertion sort (N2)
mergesort (N log N)
quicksort (N log N)
computer
thousand
million
billion
home
instant
2.8 hours
317 years
1 second
18 min
0.6 sec
12 min
super
1 week
Good algorithms are better than supercomputers.
Great algorithms are better than good algorithms.

26. Duplicate keys Dijkstra’s 3-way partition B A A B A B B B C C C
A A A A A A A A A A A
Dijkstra’s 3-way partition

27. Quiz
Give the array that results after applying quicksort partitioning to the following array by using:
a) partition #1
b) partition #2
c) Dijkstra’s 3-way partition
H J B R H R H B C V S
A. C B H H B H R J R V S
B. H C B B H H R R J V S
C. C B B H H H R R V S J
D. none of above

28. Sort summary inplace? stable? worst average best remarks selection x
N exchanges
insertion
N 2 / 4 N
use for small N or partially ordered
shell ?
tight code, subquadratic
quick
2 N ln N
N lg N
N log N probabilistic guarantee fastest in practice
3-way quick
improves quicksort in presence of duplicate keys
merge
N log N guarantee, stable

29. Exercise Download hw4.pdf from our course homepage Due on Oct. 11
Watch the week#4 video of Algorithm(Part I) Coursera.
Optional: Programming Assignment 4
4/14/2017
Xiaojuan Cai