Question

Use Mergesort (Algorithms 2.2 and 2.4 ) to sort the following list. Show the actions step by step. \\[ 123 \quad 34 \quad 189 \quad 56 \quad 150 \quad 12 \quad 9 \quad 240 \\]

Short answer

The sorted list is \[9, 12, 34, 56, 123, 150, 189, 240\]

Step-by-step solution

Split the list into individual elements

In this step, split the original list: \[123, 34, 189, 56, 150, 12, 9, 240\] into individual elements, this will create 8 sublists: each containing a single element.

Merge and sort

Start merging the sub-lists, ensuring that the result is sorted at each step. First merge the individual elements back into sub-lists of two: \[(123, 34), (189, 56), (150, 12), (9, 240)\] and then sort these as you do: \[(34, 123), (56, 189), (12, 150), (9, 240)\]

Continue to Merge and sort

Now merge the pairs into lists of four, while keeping them sorted: \[(34, 56, 123, 189), (9, 12, 150, 240)\]

Final Merge and sort

Finally, merge the two sorted sub-lists from earlier into the final sorted list: \[9, 12, 34, 56, 123, 150, 189, 240\] This step marks the end of the merge sort process.

Key concepts

Divide and Conquer

The Mergesort algorithm relies heavily on the concept of 'Divide and Conquer'. This approach breaks down a problem into smaller, more manageable sub-problems, solves each sub-problem individually, and finally combines the solutions to form a complete solution to the original problem.
In the context of sorting, 'Divide and Conquer' involves breaking down the list into smaller parts, sorting each of these parts, and then combining them back together. This method is efficient because sorting smaller sublists is generally easier and faster.

• **Divide**: Split the list into two halves until each sublist contains a single element.
• **Conquer**: Sort each of the sublists recursively.
• **Combine**: Merge the sorted sublists to produce sorted lists, ultimately resulting in one completely sorted list.

By continuously dividing the list and using the straightforward process of merging, Mergesort efficiently organizes data.

Sorting Algorithms

Sorting algorithms, such as Mergesort, are designed to arrange elements in a specific order, typically ascending or descending. Understanding sorting is crucial, as we frequently need to organize data for various applications.
Mergesort's main advantage over some other sorting algorithms is its predictable performance. It consistently runs in O(n log n) time complexity, making it efficient even for large datasets. Unlike simpler algorithms like bubble sort or selection sort, Mergesort is particularly good for large lists due to its divide-and-conquer nature.

• **Stability**: Mergesort is a stable sort, meaning it maintains the relative order of records with equal keys.
• **Performance**: It provides consistent performance with large datasets.
• **Recursive**: The algorithm employs recursion, dividing the list repeatedly before sorting and merging.

However, the primary trade-off is memory usage, as it requires additional space for the temporary arrays used in merging.

Algorithm Steps

Following specific steps ensures clarity and correctness when implementing Mergesort. These steps highlight the structured approach and recursive nature of the process. 1. **Splitting the List**: Start by splitting the original list into individual elements. This involves continuously dividing the list until each element is isolated. This builds a foundation for the recursive structure of Mergesort. 2. **Initial Merge and Sort**: Next, begin merging the individual elements back into pairs while sorting them. For example, elements like (123, 34) become (34, 123) post sort. This progression into sorted pairs sets the stage for more extensive merging. 3. **Merging Larger Groups**: As the pairs of sorted sublists are combined into larger lists, the divide and conquer strategy is highlighted. For instance, merging pairs into (34, 56, 123, 189) continues the process, ensuring each merge step maintains order. 4. **Final Merge**: The last step is to merge the two larger sorted sublists into a single, fully sorted list. This final merge consolidates the previously divided sections into a coherent, ordered whole. By following these structured steps, Mergesort efficiently organizes data, proving why it's such a pivotal algorithm in computer science.