We will discuss the following: Sorting Algorithms, Counting Sort, Radix Sort, Merge Sort.Algorithms, Time Complexity & Space Complexity, Algorithm vs Pseudocode, Some Algorithm Types, Programming Languages, Python, Anaconda.

1. Analysis and Design of Algorithms
Sorting Algorithms II

2. Analysis and Design of Algorithms
Counting Sort
Radix Sort
Merge Sort

3. Analysis and Design of Algorithms
Counting Sort

4. Analysis and Design of Algorithms
Counting sort is a sorting technique based on keys
between a specific range. It works by counting the
number of objects having distinct key values (kind of
hashing). Then doing some arithmetic to calculate
the position of each object in the output sequence.

5. Analysis and Design of Algorithms
Algorithm:
 Step1: Create a count array to store the count of each unique
object
 Step2 : Modify count array by adding the previous number.
 Step3 : Create output array by decrease count array

6. Analysis and Design of Algorithms
 Example 1 Assume the following Array in range of 0 to 5:
1 4 3 2 3 5 2

7. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2

8. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 0 0 0 0

9. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 0 0 0 0

10. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 0 0 0 0

11. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 0 0

12. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 0 0

13. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 0 0

14. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 0 0

15. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 1 0

16. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 1 0

17. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 0 1 0

18. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 1 1 0

19. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 1 1 0

20. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 0 1 1 0

21. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 1 1 0

22. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 1 1 0

23. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 0

24. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 0

25. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 0

26. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 1

27. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 1

28. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 1 2 1 1

29. Analysis and Design of Algorithms
 Create a count array to store the count of each unique object:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 2 2 1 1

30. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 2 2 1 1

31. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 2 2 1 1

32. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 2 1 1

33. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 1 1

34. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 1

35. Analysis and Design of Algorithms
 Modify count array by adding the previous number :
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7

36. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7

37. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7
1 2 3 4 5 6 7

38. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7
1 2 3 4 5 6 7

39. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7
1 2 3 4 5 6 7

40. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 1 3 5 6 7
1 2 3 4 5 6 7
1

41. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 6 7
1 2 3 4 5 6 7
1

42. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 6 7
1 2 3 4 5 6 7
1

43. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 5 7
1 2 3 4 5 6 7
1 4

44. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 5 7
1 2 3 4 5 6 7
1 4

45. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 5 7
1 2 3 4 5 6 7
1 4

46. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 5 5 7
1 2 3 4 5 6 7
1 3 4

47. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 4 5 7
1 2 3 4 5 6 7
1 3 4

48. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 4 5 7
1 2 3 4 5 6 7
1 3 4

49. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 4 5 7
1 2 3 4 5 6 7
1 3 4

50. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 3 4 5 7
1 2 3 4 5 6 7
1 2 3 4

51. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 4 5 7
1 2 3 4 5 6 7
1 2 3 4

52. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 4 5 7
1 2 3 4 5 6 7
1 2 3 4

53. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 4 5 7
1 2 3 4 5 6 7
1 2 3 4

54. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 4 5 7
1 2 3 4 5 6 7
1 2 3 3 4

55. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 7
1 2 3 4 5 6 7
1 2 3 3 4

56. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 7
1 2 3 4 5 6 7
1 2 3 3 4

57. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 7
1 2 3 4 5 6 7
1 2 3 3 4

58. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 7
1 2 3 4 5 6 7
1 2 3 3 4 5

59. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 6
1 2 3 4 5 6 7
1 2 3 3 4 5

60. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 6
1 2 3 4 5 6 7
1 2 3 3 4 5

61. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 6
1 2 3 4 5 6 7
1 2 3 3 4 5

62. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 2 3 5 6
1 2 3 4 5 6 7
1 2 2 3 3 4 5

63. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 1 3 5 6
1 2 3 4 5 6 7
1 2 2 3 3 4 5

64. Analysis and Design of Algorithms
 Output each object from the input sequence followed by
decreasing its count by 1:
1 4 3 2 3 5 2
0 1 2 3 4 5
0 0 1 3 5 6
1 2 3 4 5 6 7
1 2 2 3 3 4 5

65. Analysis and Design of Algorithms
 Array is now sorted
1 2 2 3 3 4 5

66. Analysis and Design of Algorithms
2 4 1 1 3
 Example 2:
0 1 2 3 4
0 2 1 1 1
0 2 3 4 5
2
Count
Range=[0-4]
Add
1 2 3 4 5
Output
0 1 2 3 4
0 2 2 4 5Reduce
0 1 2 3 4
2 4
1 2 3 4 5
Output
0 2 2 4 4Reduce
0 1 2 3 4
1 2 4
1 2 3 4 5
Output
0 1 2 4 4Reduce
0 1 2 3 4
1 1 2 4
1 2 3 4 5
Output
0 0 2 4 4Reduce
0 1 2 3 4
1 1 2 3 4
1 2 3 4 5
Output
1 1 2 3 4
1 2 3 4 5
Sorted

67. Analysis and Design of Algorithms
 Python
Code

68. Analysis and Design of Algorithms

69. Analysis and Design of Algorithms

70. Analysis and Design of Algorithms
 Time Complexity: O(n+k) where n is the number of elements in input
array, and k is the range of input.
 Example of worst case
 Range between 1 to 10K
10 5 10k 5k 200

71. Analysis and Design of Algorithms
Radix Sort

72. Analysis and Design of Algorithms
 Radix sort is an algorithm that sorts numbers by processing digits
of each number either starting from the least significant digit (LSD)
or starting from the most significant digit (MSD).
 The idea of Radix Sort is to do digit by digit sort starting from
least significant digit to most significant digit. Radix sort uses
counting sort as a subroutine to sort.

73. Analysis and Design of Algorithms
Algorithm:
 Step1: Take the least significant digit of each element
 Step2 : Sort the list of elements based on that digit
 Step3 : Repeat the sort with each more significant digit

74. Analysis and Design of Algorithms
 Assume the following Array:
170 45 75 90 802 24 2 66

75. Analysis and Design of Algorithms
 The Sorted list will appear after three steps
170 45 75 90 802 24 2 66
170 90 802 2 24 45 75 66
802 2 24 45 66 170 75 90
2 24 45 66 75 90 170 802

76. Analysis and Design of Algorithms
 Step1: Sorting by least significant digit (1s place)
170 45 75 90 802 24 2 66
170 90 802 2 24 45 75 66

77. Analysis and Design of Algorithms
 Step2: Sorting by next digit (10s place)
170 90 802 2 24 45 75 66
802 2 24 45 66 170 75 90

78. Analysis and Design of Algorithms
 Step3: Sorting by most significant digit (100s place)
802 2 24 45 66 170 75 90
2 24 45 66 75 90 170 802

79. Analysis and Design of Algorithms
 Array is now sorted
2 24 45 66 75 90 170 802

80. Analysis and Design of Algorithms
 Example 2
1 2 3
5 8 3
1 5 4
5 6 7
6 8 9
6 2 5
4 5 6
1 2 3
5 8 3
1 5 4
6 2 5
4 5 6
5 6 7
6 8 9
1 2 3
6 2 5
1 5 4
4 5 6
5 6 7
5 8 3
6 8 9
1 2 3
1 5 4
4 5 6
5 6 7
5 8 3
6 2 5
6 8 9
1s 10s 100s

81. Analysis and Design of Algorithms

82. Analysis and Design of Algorithms
 Python Code

83. Analysis and Design of Algorithms
 Python Code

84. Analysis and Design of Algorithms
Time Complexity: O(n+k/d) where n is the number
of elements in input array, k is the range of input,
and d is number of digits.

85. Analysis and Design of Algorithms
Merge Sort

86. Analysis and Design of Algorithms
 Merge Sort is a Divide and Conquer algorithm. It
divides input array in two halves, calls itself for the
two halves and then merges the two sorted halves.

87. Analysis and Design of Algorithms
Algorithm:
 Step1: Divide the list recursively into two halves until it can
no more be divided
 Step2 : Merge (Conquer) the smaller lists into new list in
sorted order

88. Analysis and Design of Algorithms
 Assume the following Array:
85 24 63 45 17 31 96 50

89. Analysis and Design of Algorithms
 Divide
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50

90. Analysis and Design of Algorithms
 Divide
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50

91. Analysis and Design of Algorithms
 Divide
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50
85 24 63 45 17 31 96 50

92. Analysis and Design of Algorithms
 Sort & Merge
85 24 63 45 17 31 96 50

93. Analysis and Design of Algorithms
 Sort & Merge
85 24 63 45 17 31 96 50

94. Analysis and Design of Algorithms
 Sort & Merge
24 85
85 24 63 45 17 31 96 50

95. Analysis and Design of Algorithms
 Sort & Merge
24 85
85 24 63 45 17 31 96 50

96. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63
85 24 63 45 17 31 96 50

97. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63
85 24 63 45 17 31 96 50

98. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63 17 31
85 24 63 45 17 31 96 50

99. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63 17 31
85 24 63 45 17 31 96 50

100. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

101. Analysis and Design of Algorithms
 Sort & Merge
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

102. Analysis and Design of Algorithms
 Sort & Merge
24
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

103. Analysis and Design of Algorithms
 Sort & Merge
24 45
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

104. Analysis and Design of Algorithms
 Sort & Merge
24 45 63
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

105. Analysis and Design of Algorithms
 Sort & Merge
24 45 63 85
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

106. Analysis and Design of Algorithms
 Sort & Merge
24 45 63 85 17 31 50 96
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

107. Analysis and Design of Algorithms
 Sort & Merge
17 24 31 45 50 63 85 96
24 45 63 85 17 31 50 96
24 85 45 63 17 31 50 96
85 24 63 45 17 31 96 50

108. Analysis and Design of Algorithms
 Array is now sorted
17 24 31 45 50 63 85 96

109. Analysis and Design of Algorithms
 Example 2

110. Analysis and Design of Algorithms

111. Analysis and Design of Algorithms

112. Analysis and Design of Algorithms

113. Analysis and Design of Algorithms
Time Complexity: O(n * log(n) )

114. Analysis and Design of Algorithms
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115. Analysis and Design of Algorithms
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