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Quick Sort:

Quick Sort is a popular sorting algorithm based on the divide-and-conquer approach. It works by selecting a ‘pivot’ element from the array and partitioning the other elements into two sub-arrays according to whether they are less than or greater than the pivot. The sub-arrays are then recursively sorted.

Algorithm:

  1. Choose a pivot element from the array.
  2. Partition the array into two sub-arrays: elements less than the pivot and elements greater than the pivot.
  3. Recursively apply the same process to the sub-arrays.
  4. Concatenate the sorted sub-arrays to get the final sorted array.

Complexity:

  • Average Case: O(n log n), where n is the number of elements in the array.
  • Worst Case: O(n^2), when the pivot selection leads to unbalanced partitions.
  • Best Case: O(n log n), when the pivot selection leads to balanced partitions.

Two-Way Merge Sort:

Merge Sort is another efficient sorting algorithm based on the divide-and-conquer approach. It divides the array into two halves, recursively sorts the sub-arrays, and then merges the sorted sub-arrays.

Algorithm:

  1. Divide the array into two halves.
  2. Recursively apply Merge Sort to the two halves.
  3. Merge the sorted halves to produce the final sorted array.

Complexity:

  • Average Case: O(n log n), where n is the number of elements in the array.
  • Worst Case: O(n log n), regardless of the input.
  • Best Case: O(n log n), regardless of the input.

Heap Sort:

Heap Sort is a comparison-based sorting algorithm that uses a binary heap data structure. It first builds a heap from the input array, then repeatedly extracts the maximum element from the heap and rebuilds the heap until the array is sorted.

Algorithm:

  1. Build a max heap from the input array.
  2. Repeatedly extract the maximum element from the heap and adjust the heap.
  3. Continue until the heap is empty, each extracted element forms part of the sorted array.

Complexity:

  • Average Case: O(n log n), where n is the number of elements in the array.
  • Worst Case: O(n log n), regardless of the input.
  • Best Case: O(n log n), regardless of the input.

Comparison and Analysis:

  • Quick Sort and Heap Sort are in-place sorting algorithms, while Merge Sort typically requires O(n) additional space for merging.
  • Quick Sort has better average-case performance compared to Merge Sort and Heap Sort.
  • Heap Sort guarantees O(n log n) time complexity in all cases, making it suitable for large datasets.
  • Merge Sort is stable and can be used for sorting linked lists efficiently.
  • Quick Sort is widely used in practice due to its simplicity and average-case efficiency.

Quick Sort, Two-Way Merge Sort, and Heap Sort are all efficient sorting algorithms with different characteristics and performance profiles. The choice of algorithm depends on factors such as the size of the dataset, stability requirements, and memory constraints.