Quick Sort - Lecture Notes

Introduction to Quick Sort

• QuickSort is a sorting algorithm based on the Divide and Conquer principle.
• It picks an element as a pivot and partitions the given array around this pivot.

How QuickSort Algorithm Works

1. Choose a Pivot:
   • Select an element from the array as the pivot.
   • The choice of pivot can vary: first element, last element, random element, or median.
2. Partition the Array:
   • Rearrange the array around the pivot.
   • After partitioning, all elements smaller than the pivot are on its left, and all elements greater are on its right.
   • The pivot is placed in its correct position.
3. Recursively Call:
   • Apply the same process to the two partitioned sub-arrays (left and right of the pivot).
4. Base Case:
   • Recursion stops when there is only one element left in the sub-array.

Choice of Pivot

• First or Last Element as Pivot:
  • Simple but can lead to worst-case scenarios if the array is already sorted.
• Random Element as Pivot:
  • Preferred as it avoids worst-case patterns.
• Median Element as Pivot:
  • Ideal for time complexity but has high constants for median finding.

Partition Algorithm

• Key process in QuickSort is partition(), with O(n) time complexity.
• Naive Partition:
  • Creates a copy of the array, placing smaller elements first, followed by larger ones.
• Lomuto Partition:
  • Simpler, used in most examples due to its simplicity.
• Hoare’s Partition:
  • Faster, traverses from both ends and swaps elements.

Illustration of QuickSort Algorithm

• Recursively call for smaller sub-arrays around the pivot until sorted.
• Once elements are in their correct positions, the array is sorted.

QuickSort Implementations

• Code examples in C++, Java, Python, C#, JavaScript, and PHP provided.
• Key Functions:
  • Partition Function: Chooses the pivot and re-arranges elements.
  • Swap Function: Used to swap elements during partition.
  • QuickSort Function: Recursively sorts the array.

Complexity Analysis

• Best Case: O(n log n)
• Average Case: O(n log n)
• Worst Case: O(n^2) when the pivot is poorly chosen.
• Auxiliary Space: O(n) due to recursive call stack.

Advantages of Quick Sort

• Efficient on large datasets.
• Low overhead, small memory requirements.
• Cache friendly.
• Fastest general-purpose sorting algorithm.

Disadvantages of Quick Sort

• Worst-case time complexity can be O(n^2).
• Not stable (does not preserve the relative order of equal elements).
• Not ideal for small datasets.

Applications of Quick Sort

• Sorting large datasets efficiently.
• Used in partitioning problems.
• Beneficial in randomized algorithms.
• Integral to cryptography for random permutations.
• Parallelizable for multi-core systems.

Additional Resources

• QuickSort Algorithm Details
• Time and Space Complexity Analysis
• Applications of QuickSort

These notes provide a comprehensive overview of QuickSort, its workings, applications, and implementations in various programming languages.