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Algorithms Lecture 5: Sorting Algorithms II

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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.

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Algorithms Lecture 5: Sorting Algorithms II

  1. 1. Analysis and Design of Algorithms Sorting Algorithms II
  2. 2. Analysis and Design of Algorithms Counting Sort Radix Sort Merge Sort
  3. 3. Analysis and Design of Algorithms Counting Sort
  4. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 65. Analysis and Design of Algorithms  Array is now sorted 1 2 2 3 3 4 5
  66. 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. 67. Analysis and Design of Algorithms  Python Code
  68. 68. Analysis and Design of Algorithms
  69. 69. Analysis and Design of Algorithms
  70. 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. 71. Analysis and Design of Algorithms Radix Sort
  72. 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. 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. 74. Analysis and Design of Algorithms  Assume the following Array: 170 45 75 90 802 24 2 66
  75. 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. 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. 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. 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. 79. Analysis and Design of Algorithms  Array is now sorted 2 24 45 66 75 90 170 802
  80. 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. 81. Analysis and Design of Algorithms
  82. 82. Analysis and Design of Algorithms  Python Code
  83. 83. Analysis and Design of Algorithms  Python Code
  84. 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. 85. Analysis and Design of Algorithms Merge Sort
  86. 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. 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. 88. Analysis and Design of Algorithms  Assume the following Array: 85 24 63 45 17 31 96 50
  89. 89. Analysis and Design of Algorithms  Divide 85 24 63 45 17 31 96 50 85 24 63 45 17 31 96 50
  90. 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. 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. 92. Analysis and Design of Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
  93. 93. Analysis and Design of Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
  94. 94. Analysis and Design of Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
  95. 95. Analysis and Design of Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
  96. 96. Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
  97. 97. Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
  98. 98. Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
  99. 99. Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
  100. 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. 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. 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. 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. 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. 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. 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. 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. 108. Analysis and Design of Algorithms  Array is now sorted 17 24 31 45 50 63 85 96
  109. 109. Analysis and Design of Algorithms  Example 2
  110. 110. Analysis and Design of Algorithms
  111. 111. Analysis and Design of Algorithms
  112. 112. Analysis and Design of Algorithms
  113. 113. Analysis and Design of Algorithms Time Complexity: O(n * log(n) )
  114. 114. Analysis and Design of Algorithms facebook.com/mloey mohamedloey@gmail.com twitter.com/mloey linkedin.com/in/mloey mloey@fci.bu.edu.eg mloey.github.io
  115. 115. Analysis and Design of Algorithms www.YourCompany.com © 2020 Companyname PowerPoint Business Theme. All Rights Reserved. THANKS FOR YOUR TIME

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