Question

In the text, we say that after the merge sort splits the array into two subarrays, it then sorts these two subarrays and merges them. Why might someone be puzzled by our statement that "it then sorts these two subarrays"?

Short answer

The 'sorts' step is implicit in the recursion; merge sort inherently sorts subarrays before merging.

Step-by-step solution

Understanding Merge Sort

Merge sort is a divide-and-conquer algorithm to sort an array. It works by splitting the array into two halves, recursively sorting each half, and then merging the sorted halves back together. The recursion ensures that each subarray is sorted.

Confusion in Sorting Subarrays

The confusion may arise from the use of the term "sorts", implying a separate action is needed to sort the subarrays. Instead, the recursive nature of merge sort inherently sorts the subarrays as part of its process.

Recursive Sorting Mechanism

Merge sort automatically sorts the subarrays through its recursive calls. Each split results in smaller subarrays that are eventually sorted when the recursion reaches the base case of a single element or an empty subarray.

Merging Sorted Subarrays

Once the subarrays are recursively sorted, the merging process comes into play. During merging, two sorted subarrays are combined into a single sorted array by comparing elements and arranging them in order.

Key concepts

Divide-and-Conquer Strategy

The merge sort algorithm is a perfect example of a divide-and-conquer strategy, which is a powerful technique used in algorithms to solve complex problems. This strategy works by:

• Dividing the problem into smaller subproblems.
• Conquering the subproblems by solving them independently.
• Combining the solutions of the subproblems to form the solution to the original problem.

In the case of merge sort, the array is divided into two halves, forming the subproblems. These smaller pieces are easier to manage and solve. By tackling each piece separately, merge sort makes the entire sorting process more manageable and efficient.
The beauty of this approach is its ability to break down a large task into manageable chunks, leading to efficient problem solving. This is particularly beneficial for sorting large datasets where tackle directly can be overwhelming.

Recursive Sorting

Recursive sorting is integral to the merge sort algorithm and is what allows it to sort the array without needing additional sorting actions. In merge sort, once the array is split into two subarrays, each subarray undergoes further splits until the base case is reached.
The base case in recursion refers to the simplest instance of a problem, generally an array with one or zero elements, which is inherently sorted. Thus, when merge sort reaches these base cases, it automatically sorts them.
As the recursive calls return, they naturally construct sorted subarrays from these base cases back up to the larger partial arrays. Every recursive call essentially sorts its current subarray as part of its natural process, which is why sorting might be said to "just happen" as a result of recursion.

Merging Subarrays

The final step in the merge sort algorithm is merging subarrays, which is crucial for forming the fully sorted array. Once recursion has sorted the smallest subarrays, the merge process takes two sorted subarrays and combines them into a single sorted array.
The merge operation involves comparing the smallest elements of both subarrays:

• If the smallest element of the first subarray is less than that of the second, it is placed in the combined array.
• This process repeats, extracting elements from both subarrays and maintaining order, until all input subarray elements have been merged into the final sorted form.

This systematic merging ensures that all elements are in sorted order when finally combined. It showcases the efficiency of divide-and-conquer by leveraging the sorted nature of smaller divisions, which simplifies the merging and makes merge sort a highly effective sorting algorithm.