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Algorithms Lecture 5: Sorting Algorithms II

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.

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Analysis and Design of Algorithms Sorting Algorithms II
Analysis and Design of Algorithms Counting Sort Radix Sort Merge Sort
Analysis and Design of Algorithms Counting Sort
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.
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
Analysis and Design of Algorithms  Example 1 Assume the following Array in range of 0 to 5: 1 4 3 2 3 5 2
Analysis and Design of Algorithms  Create a count array to store the count of each unique object: 1 4 3 2 3 5 2
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Analysis and Design of Algorithms  Array is now sorted 1 2 2 3 3 4 5
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
Analysis and Design of Algorithms  Python Code
Analysis and Design of Algorithms
Analysis and Design of Algorithms
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
Analysis and Design of Algorithms Radix Sort
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.
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
Analysis and Design of Algorithms  Assume the following Array: 170 45 75 90 802 24 2 66
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
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
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
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
Analysis and Design of Algorithms  Array is now sorted 2 24 45 66 75 90 170 802
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
Analysis and Design of Algorithms
Analysis and Design of Algorithms  Python Code
Analysis and Design of Algorithms  Python Code
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.
Analysis and Design of Algorithms Merge Sort
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.
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
Analysis and Design of Algorithms  Assume the following Array: 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Divide 85 24 63 45 17 31 96 50 85 24 63 45 17 31 96 50
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
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
Analysis and Design of Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
Analysis and Design of Algorithms  Sort & Merge 24 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
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
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
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
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
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
Analysis and Design of Algorithms  Array is now sorted 17 24 31 45 50 63 85 96
Analysis and Design of Algorithms  Example 2
Analysis and Design of Algorithms
Analysis and Design of Algorithms
Analysis and Design of Algorithms
Analysis and Design of Algorithms Time Complexity: O(n * log(n) )
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
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In this document
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Overview of sorting algorithms: Counting Sort, Radix Sort, and Merge Sort.

Counting Sort is based on element keys in a specific range. It counts unique object occurrences and positions.

Steps to implement Counting Sort: Create count array, modify counts, create output array.

Demonstration using an array to create a count array, showcasing its values incrementally.

In the Counting Sort process, modifying the count array to reflect cumulative counts for positioning.

Final stages of Counting Sort, where elements are output based on updated counts.

Final output of sorted array, verifying Counting Sort, includes demonstration with example.

Time complexities: Counting Sort O(n+k), Radix Sort O(n+k/d), and Merge Sort O(n log n).

Introduction to Radix Sort, a digit-based sorting using Counting Sort as a subroutine.

Radix Sort algorithm: Sorts from least significant to most significant digit with examples.

Introduction to Merge Sort, a Divide and Conquer sorting algorithm.

Steps of Merge Sort: Divide and merge in sorted order until complete.

Examples and step-by-step process of Merge Sort, culminating in the sorted array output.

Time complexities: Counting Sort O(n+k), Radix Sort O(n+k/d), and Merge Sort O(n log n).

Algorithms Lecture 5: Sorting Algorithms II

  • 1.
    Analysis and Designof Algorithms Sorting Algorithms II
  • 2.
    Analysis and Designof Algorithms Counting Sort Radix Sort Merge Sort
  • 3.
    Analysis and Designof Algorithms Counting Sort
  • 4.
    Analysis and Designof 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 Designof 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 Designof Algorithms  Example 1 Assume the following Array in range of 0 to 5: 1 4 3 2 3 5 2
  • 7.
    Analysis and Designof Algorithms  Create a count array to store the count of each unique object: 1 4 3 2 3 5 2
  • 8.
    Analysis and Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof 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 Designof Algorithms  Array is now sorted 1 2 2 3 3 4 5
  • 66.
    Analysis and Designof 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 Designof Algorithms  Python Code
  • 68.
    Analysis and Designof Algorithms
  • 69.
    Analysis and Designof Algorithms
  • 70.
    Analysis and Designof 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 Designof Algorithms Radix Sort
  • 72.
    Analysis and Designof 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 Designof 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 Designof Algorithms  Assume the following Array: 170 45 75 90 802 24 2 66
  • 75.
    Analysis and Designof 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 Designof 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 Designof 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 Designof 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 Designof Algorithms  Array is now sorted 2 24 45 66 75 90 170 802
  • 80.
    Analysis and Designof 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 Designof Algorithms
  • 82.
    Analysis and Designof Algorithms  Python Code
  • 83.
    Analysis and Designof Algorithms  Python Code
  • 84.
    Analysis and Designof 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 Designof Algorithms Merge Sort
  • 86.
    Analysis and Designof 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 Designof 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 Designof Algorithms  Assume the following Array: 85 24 63 45 17 31 96 50
  • 89.
    Analysis and Designof Algorithms  Divide 85 24 63 45 17 31 96 50 85 24 63 45 17 31 96 50
  • 90.
    Analysis and Designof 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 Designof 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 Designof Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
  • 93.
    Analysis and Designof Algorithms  Sort & Merge 85 24 63 45 17 31 96 50
  • 94.
    Analysis and Designof Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
  • 95.
    Analysis and Designof Algorithms  Sort & Merge 24 85 85 24 63 45 17 31 96 50
  • 96.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
  • 97.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 85 24 63 45 17 31 96 50
  • 98.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
  • 99.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 17 31 85 24 63 45 17 31 96 50
  • 100.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
  • 101.
    Analysis and Designof Algorithms  Sort & Merge 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
  • 102.
    Analysis and Designof Algorithms  Sort & Merge 24 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
  • 103.
    Analysis and Designof Algorithms  Sort & Merge 24 45 24 85 45 63 17 31 50 96 85 24 63 45 17 31 96 50
  • 104.
    Analysis and Designof 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 Designof 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 Designof 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 Designof 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 Designof Algorithms  Array is now sorted 17 24 31 45 50 63 85 96
  • 109.
    Analysis and Designof Algorithms  Example 2
  • 110.
    Analysis and Designof Algorithms
  • 111.
    Analysis and Designof Algorithms
  • 112.
    Analysis and Designof Algorithms
  • 113.
    Analysis and Designof Algorithms Time Complexity: O(n * log(n) )
  • 114.
    Analysis and Designof Algorithms facebook.com/mloey mohamedloey@gmail.com twitter.com/mloey linkedin.com/in/mloey mloey@fci.bu.edu.eg mloey.github.io
  • 115.
    Analysis and Designof Algorithms www.YourCompany.com © 2020 Companyname PowerPoint Business Theme. All Rights Reserved. THANKS FOR YOUR TIME