Data Structure-Merge Sort

1. Mergesort
Mergesort (divide-and-conquer)
 Divide array into two halves.
A L G O R I T H M S
divideA L G O R I T H M S

2. Mergesort
Mergesort (divide-and-conquer)
 Divide array into two halves.
 Recursively sort each half.
sort
A L G O R I T H M S
divideA L G O R I T H M S
A G L O R H I M S T

3. Mergesort
Mergesort (divide-and-conquer)
 Divide array into two halves.
 Recursively sort each half.
 Merge two halves to make sorted whole.
merge
sort
A L G O R I T H M S
divideA L G O R I T H M S
A G L O R H I M S T
A G H I L M O R S T

4. auxiliary array
smallest smallest
A G L O R H I M S T
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
A

5. auxiliary array
smallest smallest
A G L O R H I M S T
A
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
G

6. auxiliary array
smallest smallest
A G L O R H I M S T
A G
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
H

7. auxiliary array
smallest smallest
A G L O R H I M S T
A G H
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
I

8. auxiliary array
smallest smallest
A G L O R H I M S T
A G H I
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
L

9. auxiliary array
smallest smallest
A G L O R H I M S T
A G H I L
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
M

10. auxiliary array
smallest smallest
A G L O R H I M S T
A G H I L M
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
O

11. auxiliary array
smallest smallest
A G L O R H I M S T
A G H I L M O
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
R

12. auxiliary array
first half
exhausted smallest
A G L O R H I M S T
A G H I L M O R
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
S

13. auxiliary array
first half
exhausted smallest
A G L O R H I M S T
A G H I L M O R S
Merging
Merge.
 Keep track of smallest element in each sorted half.
 Insert smallest of two elements into auxiliary array.
 Repeat until done.
T

14. Do by yourself
 10,3,54,23,25,5,75,1
 10,5,7,6,1,4,8,3,2,9

15. What is radix?
The radix or base is the number of unique digits,
including zero, that a positional numeral system
uses to represent numbers.
recall:
What is the radix of decimal number system?
What is the radix of binary number system?

16. RADIX SORT

17. RadixSort
•Radix = “The base of a number system”
•Idea: Bucket Sort on each digit, bottom up.
17

18. What is a Radix?

19. What is a Radix?
The radix or base is the number of unique
digits, including zero, that a positional
numeral
system uses to represent numbers.
Examples:
1 BINARY 2
2 DECIMAL 10
3 OCTAL 8
4 HEXADECIMAL 16

20. RADIX SORT
• non - comparative sorting algorithm (the
keys are not compared as such)
• sorts the keys by soring individual digits of
the keys which share the same significant
position and value

21. EXAMPLE
INPUT Pass 1 Pass 2 Pass 3 Pass 4
3746 5743 5743 2653 1999
5743 2653 3746 4657 2653
4657 3746 2653 5743 3746
2653 4657 4657 3746 4657
1999 1999 1999 1999 5743

22. RADIX SORT - PSEUDO CODE
void radixsort()
{
 max = A[0], exp = 1;
 for (i = 0; i < n; i++) { if (A[i] > max) max = A[i]; }

 while (max / exp > 0)
 {
 for (i=0; i<=9; i++) bucket [ i ] =0;
 for (i = 0; i < n; i++) bucket [ A[i] / exp % 10 ]++;
 for (i = 1; i <=9; i++) bucket [ i ] += bucket [i - 1];
 for (i = n-1; i >= 0; i--) B [--bucket[A[i] / exp % 10]] =
A[i];
 for (i = 0; i < n; i++) A [ i ] = B [ i ];

27. Radix Sort
A[0], A[1], …, A[N-1] are strings
Very common in practice
Each string is:
cd-1cd-2…c1c0,
where c0, c1, …, cd-1 ∈{0, 1, …, M-1}
M = 128
Other example: decimal numbers
27

28. The Magic of RadixSort
 Input list:
126, 328, 636, 341, 416, 131, 328
 BucketSort on lower digit:
341, 131, 126, 636, 416, 328, 328
 BucketSort result on next-higher digit:
416, 126, 328, 328, 131, 636, 341
 BucketSort that result on highest digit:
126, 131, 328, 328, 341, 416, 636
28

29. Radix Sort 29
35 53 55 33 52 32 25
Q[0] Q[1] Q[2] Q[3] Q[4] Q[5] Q[6] Q[7] Q[8] Q[9]
53 33 35 55 25
52 32 53 33 35 55 25
52 32
Q[0] Q[1] Q[2] Q[3] Q[4] Q[5] Q[6] Q[7] Q[8] Q[9]
32 33 3525 52 53 55
25 32 33 35 52 53 55
A=
A=
A=

30. Running time of Radixsort
 N items, d digit keys of max value M
 How many passes?
 How much work per pass?
 Total time?
30

31. Running time of Radixsort
 N items, d digit keys of max value M
 How many passes? d
 How much work per pass? N + M
just in case M>N, need to account for time to empty
out buckets between passes
 Total time? O( d(N+M) )
31

32. Your Turn
1) 45834,62837,56323,92634,78332,55111
2) 478, 537, 9, 721, 3, 38, 123, 67
3) 126, 328, 636, 341, 416, 131, 328

33. How many squares can you create in this figure by connecting any 4
dots (the corners of a square must lie upon a grid dot?
TRIANGLES:
How many triangles are located in the image below?

34. There are 11 squares total; 5 small, 4 medium, and 2 large.
27 triangles. There are 16 one-cell triangles, 7 four-cell triangles, 3 nine-cell
triangles, and 1 sixteen-cell triangle.

35. GUIDED READING

37. ASSESSMENT
1._____ sort is also known as bin sort.
a. merge sort
b. radix sort
c. bubble sort
d. selection sort

38. CONTD..
2.Digital sort is an example of _________ sort.
a. merge sort
b. bubble sort
c. selection sort
d. radix sort

39. CONTD..
3.Least significant digit in radix sort processes the
digit from _______.
a. right to left
b. left to right
c. top to bottom
d. bottom to up