Problem 16 - Discrete Mathematics with Applications

Question

Using the merge sort algorithm, arrange each list into ascending order. $$9,11,6,2,12,3,8,5,31,13$$

Step-by-Step Solution

Answer

The final sorted list in ascending order using the merge sort algorithm is: \[2, 3, 5, 6, 8, 9, 11, 12, 13, 31\]

Step 1: Divide the list into two halves

To begin with the merge sort algorithm, divide the original list into two equal halves: Left half: \[9, 11, 6, 2, 12\] Right half: \[3, 8, 5, 31, 13\]

Step 2: Continue dividing the lists until there is only one element left

Keep dividing the lists into halves until the lists have only one element left: Left half: - Sublist 1: \(9, 11\) - Sublist 2: \(6, 2\) - Sublist 3: \(12\) Right half: - Sublist 1: \(3, 8\) - Sublist 2: \(5, 31\) - Sublist 3: \(13\)

Step 3: Merge the sublists by sorting

Start from the smallest sublists and merge them while sorting the elements. Continue the merging process until all the sublists have been combined into a single sorted list: First, merge sublists of size two elements: - Left half: \(2, 6, 9, 11, 12\) - Right half: \(3, 5, 8, 13, 31\) Finally, merge the two sorted halves together: - Sorted list: \(2, 3, 5, 6, 8, 9, 11, 12, 13, 31\)

Step 4: The sorted list

The final sorted list in ascending order, using the merge sort algorithm, is: \[2, 3, 5, 6, 8, 9, 11, 12, 13, 31\]

Key Concepts

Recursive Division

In merge sort, the process begins by splitting the original list into smaller parts. This is known as recursive division. The idea is to break down the problem into more manageable chunks. Imagine you have a large box of items, and you start by dividing this box into two smaller boxes repeatedly. That's what recursive division does!
We keep splitting the list until each segment contains just one element. At this point, the list segments are as simple as they can be. Here, we had our original list:

• Left half: \(9, 11, 6, 2, 12\)
• Right half: \(3, 8, 5, 31, 13\)

By continuously dividing these, we break them down even further:

• Left half breaks into smaller sublists: \(9, 11\), \(6, 2\), \(12\)
• Right half breaks into smaller sublists: \(3, 8\), \(5, 31\), \(13\)

Each step makes it easier to handle the individual elements, setting the stage for sorting next.

Sorting Sublists

Once we have our single element sublists, we need to sort them. This is the second core principle of the merge sort algorithm. We take these tiny unsorted segments and start organizing them. Sorting sublists is like arranging pieces of a puzzle before putting them all together. Since each sublist has only one or two elements, sorting them is straightforward and helps us build up to a fully sorted list.
We need to compare the elements of these sublists and sort them:

• Sublist \(9, 11\) becomes \(9, 11\) (already sorted)
• Sublist \(6, 2\) becomes \(2, 6\)
• Sublist \(3, 8\) becomes \(3, 8\) (already sorted)
• Sublist \(5, 31\) remains \(5, 31\)

After sorting, we're ready for the next step, which involves merging these sorted sublists.

Combining Sorted Lists

After sorting the sublists, the next crucial step in the merge sort algorithm is to combine them together. This process of recombination is where the power of the merge sort shines. When combining sorted lists, we look at the first element of each sublist and compare them. The smaller of the two gets placed into the new sorted list first.
This is an efficient and orderly way to put the elements together:

• Start by merging the smallest sorted pairs, gradually bringing all elements back together.
• For example, combine \(2\) and \(6\) from the left and \(3\) and \(8\) from the right.
• This eventually leads to the fully sorted list \(2, 3, 5, 6, 8, 9, 11, 12, 13, 31\).

With each merge, the sublists integrate into larger, ordered lists, making it simple to finalize a fully sorted sequence without confusion.

Divide and Conquer

The merge sort algorithm is a classic example of the divide and conquer approach. This technique involves breaking down a problem into smaller subproblems, solving each one, and combining them to form a complete solution.
Divide and conquer works in three steps:

• Divide the complex problem into simpler parts (e.g., split the list into sublists).
• Conquer by solving each small problem efficiently (e.g., sort the sublists individually).
• Combine these solutions to solve the initial problem comprehensively (e.g., merge the sublists into one sorted list).

This strategy is highly efficient for sorting large data sets, as it allows handling lots of information logically and systematically. Therefore, the merge sort algorithm effectively uses the divide and conquer method, providing an efficient structure for sorting tasks.