Merge Sort Variant

The Merge Sort algorithm can be applied to a variety of problems on LeetCode. Below are a few examples:

1. Sorting Lists or Arrays: Problems that involve sorting a list or array can be solved using Merge Sort. For example:

   • Sort List
   • Merge Sorted Array

2. Finding the Inversion Count: The concept of Merge Sort can be used to find the number of inversion pairs in an array. An inversion count indicates how far the array is from being sorted. For instance:

   • Reverse Pairs
   • Count of Smaller Numbers After Self

3. Merge k Sorted Lists: This problem can be efficiently solved by modifying the standard merge sort algorithm to merge k lists instead of 2.

   • Merge k Sorted Lists

4. External Sorting: While not a common category in LeetCode, external sorting refers to sorting data that is too large to fit into a computer’s main memory at one time. It uses Merge Sort-like algorithms.

Multiple solutions could exist for these problems, and the choice of algorithm often depends on the specific constraints and requirements of the problem.

In some problems, we do not need the full merge sort algorithm. Instead, we might use a variation that solves the specific problem more efficiently. However, you are correct in that a simplified example of merge sort doesn’t necessarily represent a variant.

A common variant of merge sort is the inversion count or count of reverse pairs, which is a modification of the merge sort algorithm that, besides sorting the input array, also counts the number of inversions in the array. This variant is used in problems like LeetCode 493. Reverse Pairs.

Here is a simplified Python code for the inversion count variant:

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def merge_and_count(nums, start, mid, end, temp):
    i, j, count = start, mid, 0
    while i < mid and j < end:
        if nums[i] <= 2 * nums[j]:
            i += 1
        else:
            j += 1
            count += mid - i
    return count

def count_inversions(nums, start, end, temp):
    if start >= end - 1:
        return 0

    mid = (start + end) // 2
    left = count_inversions(nums, start, mid, temp)
    right = count_inversions(nums, mid, end, temp)
    merged = merge_and_count(nums, start, mid, end, temp)
    return left + right + merged

def reverse_pairs(nums):
    return count_inversions(nums, 0, len(nums), [0]*len(nums))

The function merge_and_count performs both the merge operation and counts the inversions. The function count_inversions recursively applies itself to left and right halves of the input, and the reverse_pairs function provides the entry point. This variant leverages the divide-and-conquer nature of merge sort but modifies the merge step to count inversions.

Claude Explanation

Merge sort is often modified and adapted for different use cases. Some variants include:

• Top-down vs bottom-up: Top-down is recursive, bottom-up iterative using subarrays.

• In-place merge: Saves memory by merging in-place instead of extra space.

• Three-way merge: Splits array into three parts for each merge.

• Hybrid merge: Uses insertion sort on small chunks instead of recursion.

Java top-down merge sort:

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void mergeSort(int[] arr, int left, int right) {
  if (left >= right) return;

  int mid = left + (right - left) / 2;
  mergeSort(arr, left, mid);
  mergeSort(arr, mid+1, right);

  merge(arr, left, mid, right); 
}

void merge(int[] arr, int left, int mid, int right) {
  // Standard merging logic
}

C++ bottom-up merge sort:

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void mergeSort(vector<int>& arr) {
  int n = arr.size();

  // Merge subarrays of size 1 to n/2
  for (int size = 1; size < n; size *= 2) {
    for (int left = 0; left < n; left += 2*size) {
      int mid = left + size - 1;
      int right = min(left + 2*size - 1, n-1);
      merge(arr, left, mid, right);
    }
  }
} 

Python in-place merge sort:

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def merge_sort(arr):
  # Recursive base case

  mid = (left+right)//2
  merge_sort(arr, left, mid) # Sort left half
  merge_sort(arr, mid+1, right) # Sort right half

  merge_in_place(arr, left, mid, right) # In-place merge

def merge_in_place(arr, left, mid, right):
  # Merge without extra space

Merge sort is very adaptable and allows optimizations for different situations.