Question

Use mergosort to sort the list \(6,3,1,9\). Count the total number-of comparisons, including

Short answer

The sorted list is [1, 3, 6, 9], with a total of 5 comparisons made.

Step-by-step solution

Understand the Merge Sort Algorithm

Merge sort is a divide and conquer algorithm that splits a list down to very small pieces and then merges them back together in order. The core of the algorithm involves splitting the list continually until each sublist consists of a single element and then merging them back together in sorted order.

Split the List

Start with the list \(6, 3, 1, 9\) and split it into two halves. First, split the list into \([6, 3]\) and \([1, 9]\).

Split the Sublists

Continue splitting each sublist until each contains only one element. Thus, \([6, 3]\) splits into \([6]\) and \([3]\), while \([1, 9]\) splits into \([1]\) and \([9]\).

Merge Sublists and Count Comparisons

Begin merging the sublists and count comparisons:1. Merge \([6]\) and \([3]\): Compare 6 and 3. Choose 3, followed by 6; This counts as 1 comparison.2. Merge \([1]\) and \([9]\): Compare 1 and 9. Choose 1, followed by 9; This counts as 1 comparison.

Merge Sorted Sublists

Merge the two sorted sublists \([3, 6]\) and \([1, 9]\):1. Compare 3 and 1. Choose 1. 2. Compare 3 and 9. Choose 3.3. Compare 6 and 9. Choose 6, followed by 9.This step involves 3 comparisons.

Final Sorted List and Total Comparisons

The final sorted list is \([1, 3, 6, 9]\). By counting the number of comparisons made throughout the merge sort process, we find that we have made a total of 5 comparisons.

Key concepts

Merge Sort: Divide and Conquer

The merge sort algorithm is a classic example of the divide and conquer strategy, which breaks down a problem into smaller, more manageable subproblems. In the context of sorting a list like \([6, 3, 1, 9]\), this strategy involves dividing the list into two halves repeatedly until each sublist consists of a single element. This step ensures that when we start merging, the smallest tasks are already practically sorted.

Think of it like breaking a complicated puzzle into smaller, easy-to-solve sections. Each piece, or in this case, each single-element sublist, is easy to manage. Once all sections are completed or in the algorithm's case, once each sublist contains only one element, you can start the process of piecing them together efficiently.

This method not only simplifies sorting but also enhances efficiency, especially for larger datasets. By leveraging the divide and conquer principle, merge sort ensures that the sorting happens in a more systematic and less chaotic manner.

Understanding Comparison Counting

Comparison counting in the merge sort algorithm is a crucial part of understanding its efficiency. Each time two elements are compared and a decision is made, that counts as one comparison. It's an indicator of the algorithm's performance, showing us how many evaluations it made to sort a list.

In the case of sorting the list \([6, 3, 1, 9]\), counting comparisons helps to measure how the algorithm behaves step-by-step. For each merge operation, comparisons occur:

• Merging \([6]\) and \([3]\) requires 1 comparison.
• Merging \([1]\) and \([9]\) requires 1 comparison.
• Merging \([3, 6]\) with \([1, 9]\) involves 3 comparisons.

When all the merging and sorting finish, the total number of comparisons is summed up, ending up with 5 in this example. This aspect makes merge sort predictable and allows for easy analysis of its efficiency in comparison with other sorting algorithms.

Step-by-Step: Algorithm Execution

Merge sort progresses through a series of well-defined steps to ensure orderly sorting. Understanding these steps aids in visualizing how the divide and conquer strategy is implemented practically:

1. **Initial Split**: Begin by dividing the list \([6, 3, 1, 9]\) into halves: \([6, 3]\) and \([1, 9]\).
2. **Sublist Division**: Repeat the division until sublists have one element each: \([6]\), \([3]\), \([1]\), \([9]\).
3. **First Merge**: Start merging by comparing elements in small sublists: - Compare 6 with 3, choose the smaller (3), then 6, resulting in a partially sorted sublist \([3, 6]\). - Similarly, compare 1 with 9, choose 1 followed by 9, forming \([1, 9]\).
4. **Final Merge**: Take the sorted sublists \([3, 6]\) and \([1, 9]\) and merge them. Comparisons are made between the first elements of each: - Compare 3 and 1: choose 1. - Compare 3 and 9: choose 3. - Compare 6 and 9: choose 6, then remaining 9.

Finally, you have the sorted list \([1, 3, 6, 9]\). By adhering to these steps, merge sort efficiently creates a sorted version of any list, reflecting its robust but straightforward mechanism.