Algorithms Lecture 5: Sorting Algorithms II
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The document discusses sorting algorithms, focusing primarily on counting sort, which is a non-comparison based sorting technique that counts the occurrences of each unique key within a defined range. It outlines the steps for implementing counting sort, including creating a count array, modifying it, and producing the output array based on this count. An illustrative example is provided to demonstrate the counting sort process and its implementation details.
![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](https://image.slidesharecdn.com/algorithms-lecture5-171025144935/85/Algorithms-Lecture-5-Sorting-Algorithms-II-66-320.jpg)
Algorithms Lecture 5: Sorting Algorithms II
- 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 facebook.com/mloey [email protected] twitter.com/mloey linkedin.com/in/mloey [email protected] mloey.github.io
- 115. Analysis and Design of Algorithms www.YourCompany.com © 2020 Companyname PowerPoint Business Theme. All Rights Reserved. THANKS FOR YOUR TIME