MergesortQuickSort.ppt

Analysis of Algorithms
CS 477/677
Sorting – Part B
Instructor: George Bebis
(Chapter 7)

2
Sorting
• Insertion sort
– Design approach:
– Sorts in place:
– Best case:
– Worst case:
• Bubble Sort
– Sorts in place:
– Running time:
Yes
(n)
(n2)
incremental
Yes
(n2)
incremental

3
Sorting
• Selection sort
– Sorts in place:
– Running time:
• Merge Sort
– Sorts in place:
– Running time:
Yes
(n2)
incremental
No
Let’s see!!
divide and conquer

4
Divide-and-Conquer
• Divide the problem into a number of sub-problems
– Similar sub-problems of smaller size
• Conquer the sub-problems
– Solve the sub-problems recursively
– Sub-problem size small enough  solve the problems in
straightforward manner
• Combine the solutions of the sub-problems
– Obtain the solution for the original problem

5
Merge Sort Approach
• To sort an array A[p . . r]:
• Divide
– Divide the n-element sequence to be sorted into two
subsequences of n/2 elements each
• Conquer
– Sort the subsequences recursively using merge sort
– When the size of the sequences is 1 there is nothing
more to do
• Combine
– Merge the two sorted subsequences

6
Merge Sort
Alg.: MERGE-SORT(A, p, r)
if p < r Check for base case
then q ← (p + r)/2 Divide
MERGE-SORT(A, p, q) Conquer
MERGE-SORT(A, q + 1, r) Conquer
MERGE(A, p, q, r) Combine
• Initial call: MERGE-SORT(A, 1, n)
1 2 3 4 5 6 7 8
6
2
3
1
7
4
2
5
p r
q

7
Example – n Power of 2
1 2 3 4 5 6 7 8
q = 4
6
2
3
1
7
4
2
5
1 2 3 4
7
4
2
5
5 6 7 8
6
2
3
1
1 2
2
5
3 4
7
4
5 6
3
1
7 8
6
2
1
5
2
2
3
4
4
7 1
6
3
7
2
8
6
5
Divide

8
Example – n Power of 2
1
5
2
2
3
4
4
7 1
6
3
7
2
8
6
5
1 2 3 4 5 6 7 8
7
6
5
4
3
2
2
1
1 2 3 4
7
5
4
2
5 6 7 8
6
3
2
1
1 2
5
2
3 4
7
4
5 6
3
1
7 8
6
2
Conquer
and
Merge

9
Example – n Not a Power of 2
6
2
5
3
7
4
1
6
2
7
4
1 2 3 4 5 6 7 8 9 10 11
q = 6
4
1
6
2
7
4
1 2 3 4 5 6
6
2
5
3
7
7 8 9 10 11
q = 9
q = 3
2
7
4
1 2 3
4
1
6
4 5 6
5
3
7
7 8 9
6
2
10 11
7
4
1 2
2
3
1
6
4 5
4
6
3
7
7 8
5
9
2
10
6
11
4
1
7
2
6
4
1
5
7
7
3
8
Divide

10
Example – n Not a Power of 2
7
7
6
6
5
4
4
3
2
2
1
1 2 3 4 5 6 7 8 9 10 11
7
6
4
4
2
1
1 2 3 4 5 6
7
6
5
3
2
7 8 9 10 11
7
4
2
1 2 3
6
4
1
4 5 6
7
5
3
7 8 9
6
2
10 11
2
3
4
6
5
9
2
10
6
11
4
1
7
2
6
4
1
5
7
7
3
8
7
4
1 2
6
1
4 5
7
3
7 8
Conquer
and
Merge

11
Merging
• Input: Array A and indices p, q, r such that
p ≤ q < r
– Subarrays A[p . . q] and A[q + 1 . . r] are sorted
• Output: One single sorted subarray A[p . . r]
1 2 3 4 5 6 7 8
6
3
2
1
7
5
4
2
p r
q

12
Merging
• Idea for merging:
– Two piles of sorted cards
• Choose the smaller of the two top cards
• Remove it and place it in the output pile
– Repeat the process until one pile is empty
– Take the remaining input pile and place it face-down
onto the output pile
1 2 3 4 5 6 7 8
6
3
2
1
7
5
4
2
p r
q
A1 A[p, q]
A2 A[q+1, r]
A[p, r]

13
Example: MERGE(A, 9, 12, 16)
p r
q

14
Example: MERGE(A, 9, 12, 16)

18
Merge - Pseudocode
Alg.: MERGE(A, p, q, r)
1. Compute n1 and n2
2. Copy the first n1 elements into
L[1 . . n1 + 1] and the next n2 elements into R[1 . . n2 + 1]
3. L[n1 + 1] ← ; R[n2 + 1] ← 
4. i ← 1; j ← 1
5. for k ← p to r
6. do if L[ i ] ≤ R[ j ]
7. then A[k] ← L[ i ]
8. i ←i + 1
9. else A[k] ← R[ j ]
10. j ← j + 1
p q
7
5
4
2
6
3
2
1
r
q + 1
L
R


1 2 3 4 5 6 7 8
6
3
2
1
7
5
4
2
p r
q
n1 n2

19
Running Time of Merge
(assume last for loop)
• Initialization (copying into temporary arrays):
– (n1 + n2) = (n)
• Adding the elements to the final array:
- n iterations, each taking constant time  (n)
• Total time for Merge:
– (n)

20
Analyzing Divide-and Conquer Algorithms
• The recurrence is based on the three steps of
the paradigm:
– T(n) – running time on a problem of size n
– Divide the problem into a subproblems, each of size
n/b: takes D(n)
– Conquer (solve) the subproblems aT(n/b)
– Combine the solutions C(n)
(1) if n ≤ c
T(n) = aT(n/b) + D(n) + C(n) otherwise

21
MERGE-SORT Running Time
• Divide:
– compute q as the average of p and r: D(n) = (1)
• Conquer:
– recursively solve 2 subproblems, each of size n/2
 2T (n/2)
• Combine:
– MERGE on an n-element subarray takes (n) time
 C(n) = (n)
(1) if n =1
T(n) = 2T(n/2) + (n) if n > 1

22
Solve the Recurrence
T(n) = c if n = 1
2T(n/2) + cn if n > 1
Use Master’s Theorem:
Compare n with f(n) = cn
Case 2: T(n) = Θ(nlgn)

23
Merge Sort - Discussion
• Running time insensitive of the input
• Advantages:
– Guaranteed to run in (nlgn)
• Disadvantage
– Requires extra space N

24
Sorting Challenge 1
Problem: Sort a file of huge records with tiny
keys
Example application: Reorganize your MP-3 files
Which method to use?
A. merge sort, guaranteed to run in time NlgN
B. selection sort
C. bubble sort
D. a custom algorithm for huge records/tiny keys
E. insertion sort

25
Sorting Files with Huge Records and
Small Keys
• Insertion sort or bubble sort?
– NO, too many exchanges
• Selection sort?
– YES, it takes linear time for exchanges
• Merge sort or custom method?
– Probably not: selection sort simpler, does less swaps

26
Sorting Challenge 2
Problem: Sort a huge randomly-ordered file of
small records
Application: Process transaction record for a
phone company
Which sorting method to use?
A. Bubble sort
B. Selection sort
C. Mergesort guaranteed to run in time NlgN
D. Insertion sort

27
Sorting Huge, Randomly - Ordered Files
• Selection sort?
– NO, always takes quadratic time
• Bubble sort?
– NO, quadratic time for randomly-ordered keys
• Insertion sort?
– NO, quadratic time for randomly-ordered keys
• Mergesort?
– YES, it is designed for this problem

28
Sorting Challenge 3
Problem: sort a file that is already almost in
order
Applications:
– Re-sort a huge database after a few changes
– Doublecheck that someone else sorted a file
Which sorting method to use?
A. Mergesort, guaranteed to run in time NlgN
B. Selection sort
C. Bubble sort
D. A custom algorithm for almost in-order files
E. Insertion sort

29
Sorting Files That are Almost in Order
• Selection sort?
– NO, always takes quadratic time
• Bubble sort?
– NO, bad for some definitions of “almost in order”
– Ex: B C D E F G H I J K L M N O P Q R S T U V W X Y Z A
• Insertion sort?
– YES, takes linear time for most definitions of “almost
in order”
• Mergesort or custom method?
– Probably not: insertion sort simpler and faster

30
Quicksort
• Sort an array A[p…r]
• Divide
– Partition the array A into 2 subarrays A[p..q] and A[q+1..r], such
that each element of A[p..q] is smaller than or equal to each
element in A[q+1..r]
– Need to find index q to partition the array
≤
A[p…q] A[q+1…r]

31
Quicksort
• Conquer
– Recursively sort A[p..q] and A[q+1..r] using Quicksort
• Combine
– Trivial: the arrays are sorted in place
– No additional work is required to combine them
– The entire array is now sorted
A[p…q] A[q+1…r]
≤

32
QUICKSORT
Alg.: QUICKSORT(A, p, r)
if p < r
then q  PARTITION(A, p, r)
QUICKSORT (A, p, q)
QUICKSORT (A, q+1, r)
Recurrence:
Initially: p=1, r=n
PARTITION())
T(n) = T(q) + T(n – q) + f(n)

33
Partitioning the Array
• Choosing PARTITION()
– There are different ways to do this
– Each has its own advantages/disadvantages
• Hoare partition (see prob. 7-1, page 159)
– Select a pivot element x around which to partition
– Grows two regions
A[p…i]  x
x  A[j…r]
A[p…i]  x x  A[j…r]
i j

34
Example
7
3
1
4
6
2
3
5
i j
7
5
1
4
6
2
3
3
i j
7
5
1
4
6
2
3
3
i j
7
5
6
4
1
2
3
3
i j
7
3
1
4
6
2
3
5
i j
A[p…r]
7
5
6
4
1
2
3
3
i
j
A[p…q] A[q+1…r]
pivot x=5

36
Partitioning the Array
Alg. PARTITION (A, p, r)
1. x  A[p]
2. i  p – 1
3. j  r + 1
4. while TRUE
5. do repeat j  j – 1
6. until A[j] ≤ x
7. do repeat i  i + 1
8. until A[i] ≥ x
9. if i < j
10. then exchange A[i]  A[j]
11. else return j
Running time: (n)
n = r – p + 1
7
3
1
4
6
2
3
5
i j
A:
ar
ap
i
j=q
A:
A[p…q] A[q+1…r]
≤
p r
Each element is
visited once!

37
Recurrence
Alg.: QUICKSORT(A, p, r)
if p < r
then q  PARTITION(A, p, r)
QUICKSORT (A, p, q)
QUICKSORT (A, q+1, r)
Recurrence:
Initially: p=1, r=n
T(n) = T(q) + T(n – q) + n

38
Worst Case Partitioning
• Worst-case partitioning
– One region has one element and the other has n – 1 elements
– Maximally unbalanced
• Recurrence: q=1
T(n) = T(1) + T(n – 1) + n,
T(1) = (1)
T(n) = T(n – 1) + n
=
2 2
1
1 ( ) ( ) ( )
n
k
n k n n n

 
       
 
 

n
n - 1
n - 2
n - 3
2
1
1
1
1
1
1
n
n
n
n - 1
n - 2
3
2
(n2)
When does the worst case happen?

39
Best Case Partitioning
• Best-case partitioning
– Partitioning produces two regions of size n/2
• Recurrence: q=n/2
T(n) = 2T(n/2) + (n)
T(n) = (nlgn) (Master theorem)

40
Case Between Worst and Best
• 9-to-1 proportional split
Q(n) = Q(9n/10) + Q(n/10) + n

41
How does partition affect performance?

42
How does partition affect performance?

43
Performance of Quicksort
• Average case
– All permutations of the input numbers are equally likely
– On a random input array, we will have a mix of well balanced
and unbalanced splits
– Good and bad splits are randomly distributed across throughout
the tree
Alternate of a good
and a bad split
Nearly well
balanced split
n
n - 1
1
(n – 1)/2
(n – 1)/2
n
(n – 1)/2
(n – 1)/2 + 1
• Running time of Quicksort when levels alternate
between good and bad splits is O(nlgn)
combined partitioning cost:
2n-1 = (n)
partitioning cost:
n = (n)

MergesortQuickSort.ppt

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