Can we improve the performance of a merge sort algorithm?

The merge sort, in essence, is a divide-and-conquer algorithm. It breaks down a problem into multiple sub-problems, solves each individually, and finally combines these sub-problems solutions to form the final solution.

The key operation in merge sort is the merging step. This step is where the algorithm compares and combines the individual elements of the sublists. It takes advantage of the fact that each of the sublists is already sorted. The merging process continues until there is only one sorted list remaining.

This methodological partitioning allows for efficient sorting, even in large datasets.

The algorithm can be divided into two main steps:

• Divide: The divide step computes the midpoint of the indices, taking constant time regardless of the subarray size.
  
• Conquer (Merge): The conquer step recursively sorts two subarrays of approximately n/2 elements each. The time this step takes will be considered when we account for the subproblems. The combined step merges a total of n elements, taking Θ(n) time. This step is where the 'merge' in merge sort comes from.

Overall time complexity of Merge sort is O(nLogn).

Optimizing Merge Sort:

The Fusion of Merge Sort and Insertion Sort for Optimal Performance.

• We can cut the running time of merge sort substantially with some carefully considered modifications to the implementation.
• Use insertion sort for small subarrays. We can improve most recursive algorithms by handling small cases differently. Switching to insertion sort for small subarrays will improve the running time of a typical merge sort.
• This is because insertion sort, while worse in the long run, performs faster on small input sizes. Changing the base case of your merge sort so that if the array to sort is below a certain size threshold (say, 50–100), you switch to insertion sort, which can markedly improve the performance of the algorithm.

Why insertion sort for smaller dataset better than merge sort?

Insertion sort is often considered more efficient than merge sort for smaller datasets due to its lower overhead and simpler implementation. Time complexity of insertion sort is O(n^2) but for small datasets, the quadratic time complexity doesn't impose a significant overhead thus, O(n^2)~ O(n), and the simplicity of the algorithm makes it efficient. While merge sort involves more complex operations and has higher space requirements.

Implementation of merge sort with insertion sort for better complexity and optimal performance.

Sorted Array:
[1, 5, 6, 9, 10, 12]

Understanding and leveraging the merge sort algorithm can significantly enhance your code’s performance, particularly when working with large datasets.