Question

Trace a walkthrough of merge sort with these lists: \(\begin{array}{llllllllllll}\text { a. } & 5 & 11 & 7 & 3 & 5 & 4 & 7 & 11 & 4 & 9 & \\ \text { b. } & 9 & 0 & 11 & 10 & 5 & 8 & -7 & 6 & 8 & 7 & 5\end{array}\)

Short answer

List a: [3, 4, 4, 5, 5, 7, 7, 9, 11]. List b: [-7, 0, 5, 5, 6, 7, 8, 8, 9, 10, 11].

Step-by-step solution

Understand the Problem

Merge sort is a divide-and-conquer algorithm that breaks the list into individual elements and then merges them back in sorted order. Our task is to apply merge sort to two separate lists.

Break Down the List a

Start with the list a: [5, 11, 7, 3, 5, 4, 7, 11, 4, 9]. 1. Divide the list into two halves: [5, 11, 7, 3, 5] and [4, 7, 11, 4, 9]. 2. Repeat the division until each sublist has one element.

Merge Sublist a

1. Merge pairs: [5] and [11] merge to [5, 11]; [7] stays [7]. 2. Merge sorted pairs: [3] and [5] merge to [3, 5]. 3. Continue merging sorted sublists until fully sorted: [3, 5, 5, 7, 11].

Break Down the List b

Now process list b: [9, 0, 11, 10, 5, 8, -7, 6, 8, 7, 5]. 1. Divide the list into two halves: [9, 0, 11, 10, 5] and [8, -7, 6, 8, 7, 5]. 2. Keep dividing until you get sublists with one element each.

Merge Sublist b

1. Merge pairs in sorted order: [0, 9, 11] from [0], [9] and [11]. 2. Continue merging until all are sorted: [-7, 5, 5, 6, 7, 8, 8, 9, 10, 11].

Final Sorted Lists

Both lists are now sorted. List a: [3, 4, 4, 5, 5, 7, 7, 9, 11]. List b: [-7, 0, 5, 5, 6, 7, 8, 8, 9, 10, 11].

Key concepts

Divide and Conquer

The essence of the divide and conquer approach is to simplify a complex problem by breaking it into smaller, more manageable parts. In the case of merge sort, this strategy involves dividing an unsorted list into individual elements. Each element is inherently sorted because it stands alone. Then, we conquer by merging these elements back into a single, sorted list.

• Initially, the list is repeatedly divided until each sub-list has only one element.
• This breaking down process reduces the problem to its simplest form.
• Finally, each small part is combined in a systematic order, producing the complete sorted list.

This strategy not only makes solving the problem easier but ensures each portion of the data is handled efficiently.

Algorithm Analysis

To fully understand how an algorithm like merge sort performs, we need to conduct an analysis of its efficiency in terms of time and space.

• Time Complexity: Merge sort has a time complexity of O(n log n), where _n_ is the number of elements in the list. This is because the list is split log n times, and each level of splits involves _n_ operations to merge.
• Space Complexity: It requires additional space to hold the temporary lists as the merge process proceeds, making its space complexity O(n).

This analysis helps us compare merge sort with other algorithms like bubble sort or quicksort and determine its suitability for different tasks based on how much time and memory it consumes.

Sorting Algorithms

Sorting algorithms are methods used to reorder elements within a list. Merge sort is just one type among several others:

• Bubble Sort: This algorithm swaps adjacent elements if they are in the wrong order, iterating over the list as many times as necessary.
• Insertion Sort: It builds a final sorted array one item at a time, assuming the first element is already sorted.
• Quick Sort: Another divide and conquer algorithm, which selects a 'pivot' element and partitions the list into elements less than and greater than the pivot.

Each sorting algorithm has its own characteristics, time complexities, and use-cases, with merge sort being particularly advantageous for larger lists due to its stable and efficient O(n log n) performance.

Problem-Solving Steps

Solving a problem like sorting a list using merge sort involves several clear steps. Let's break it down:

• Understand the Problem: Know that the task is to reorder the elements in an ascending or descending sequence.
• Devise a Plan: Follow the divide and conquer approach, planning to break down, conquer by merging, and combine efficiently.
• Execute the Plan: Implement the merge sort algorithm step-by-step, ensuring each element is part of the correct subdivision and correctly merged.
• Review the Result: Lastly, verify that the list is sorted and satisfies the initial problem requirements.

These steps guide you through applying merge sort systematically, enhancing both understanding and execution efficiency.