Sorting Algorithms Tutorial Notes

Overview

• This tutorial covers different sorting algorithms in data structures and algorithms.
• Focus on understanding inversions, sorting algorithms, and their implementations in C++.

Key Concepts

Inversions

• Definition: An inversion in an array is an ordered pair of indices (i, j) such that:
  • i < j
  • array[i] > array[j]
• Purpose: Measure how unsorted an array is. The more inversions, the more unsorted the array.
• Example: An unsorted list can have multiple inversions; in the example discussed, there are 9 inversions.

Sorting Purpose

• Reduce the number of inversions in an array.
• Ideal sorted array has zero inversions.
• Average number of inversions for n distinct integers can be calculated using a specific formula.

Sorting Algorithms

Quadratic Time Complexity Algorithms (O(n^2))

• Suitable when dealing with small datasets or when only a few inversions exist.

1. Insertion Sort

   • Places an unsorted element into its correct position in each iteration.
   • Efficient for small lists.

2. Selection Sort

   • Selects the smallest element from the unsorted portion and places it at the start.

3. Bubble Sort

   • Compares adjacent elements and swaps them if they are in the wrong order.
   • Requires multiple passes through the list.
     • For example, if 5 > 1, swap them.
   • Continue until the entire list is sorted.

Better Time Complexity Algorithms (O(n log n))

• More efficient for larger datasets.

1. Merge Sort

   • Divide and conquer approach.
   • Divide the array into two halves, sort each half, and merge them back together.
     • Recursively divide until single elements remain, then merge.

2. Heap Sort

   • Utilizes a binary tree structure (Max Heap or Min Heap).
   • Remove the largest element (root) and re-heapify the tree.

3. Quick Sort

   • Selects a pivot element and partitions the array into sub-arrays.
   • Recursively sort the sub-arrays.
   • Often the default sorting algorithm in C++ STL.

Special Case Algorithm

• Bucket Sort
  • Can achieve linear time complexity (O(n)) in specific scenarios (e.g., sorting positive integers with a known maximum value).

Recap

• Understanding inversions helps determine the best sorting algorithm to use.
• Choose sorting algorithms based on the size of the dataset and the number of inversions:
  • For large datasets, use O(n log n) algorithms.
  • For small datasets or few inversions, O(n^2) algorithms are acceptable.

C++ Implementation Examples

1. Bubble Sort Implementation

   • Uses nested loops to compare and swap elements.

2. Merge Sort Implementation

   • Implements a recursive function to merge sorted sub-arrays.

3. Quick Sort Implementation

   • Utilizes a partition function to organize elements around a pivot.

Built-in Functionality in C++ STL

• Use the sort function from the STL for efficiency and simplicity.
• Can sort in ascending or descending order using the std::reverse function or custom lambda functions.

Conclusion

• Understanding different sorting algorithms is essential for optimizing performance in data handling tasks.
• Knowledge of implementation helps in applying the correct algorithm based on specific needs and constraints.