Question

Show the steps in merging $A$ and $B$ into $C$ where $$A=8,12,19,34B=3,5,15,21$$

Short answer

The merged and sorted list is $[3,5,8,12,15,19,21,34]$.

Step-by-step solution

Understand the task

We need to merge two lists, $A=[8,12,19,34]$ and $B=[3,5,15,21]$, into a single sorted list $C$.

Start with the first elements

Look at the first elements of both lists. Compare 8 (from $A$) and 3 (from $B$). The smaller element is 3, so we'll add 3 to list $C$. Now, $C=[3]$.

Move to the next elements

Now compare the first element from $A$ (8) and the next element in $B$, which is 5. The smaller element is 5, so we add 5 to $C$. Now, $C=[3,5]$.

Continue sorting

Compare 8 (from $A$) and 15 (from $B$). The smaller element is 8, so add 8 to $C$. Now, $C=[3,5,8]$.

Move forward in list B

Compare 12 (from $A$) and 15 (from $B$). The smaller element is 12, so add 12 to $C$. Now, $C=[3,5,8,12]$.

Add next element from A

Compare the next element 19 (from $A$) and 15 (from $B$). 15 is smaller, so add 15 to $C$. Now, $C=[3,5,8,12,15]$.

Insert element from A

Compare 19 (from $A$) to 21 (from $B$). 19 is smaller, so add 19 to $C$. Now, $C=[3,5,8,12,15,19]$.

Add next from B

Now compare 34 (from $A$) with 21 (from $B$). 21 is smaller, so add 21 to $C$. Now, $C=[3,5,8,12,15,19,21]$.

Append remaining element from A

There is remaining element 34 in $A$ with no further elements in $B$ to compare. Add 34 to $C$. Now final $C=[3,5,8,12,15,19,21,34]$.

Key concepts

Sorting Algorithms

Sorting algorithms are methods used to reorganize a set of elements in a specific order, typically in ascending or descending order.
One popular sorting algorithm is Merge Sort, which is based on the divide-and-conquer principle.
This means the algorithm divides arrays or lists into two halves, sorts each half, and then merges them back together.
The Merge Sort process is highly efficient for large data sets, as it keeps dividing the data to smaller chunks that are easier to sort.
During the merge step, it processes two sorted halves and combines them into a single sorted list.
Key features of Merge Sort:

• Stable sort, meaning elements with the same values appear in the same order as they were in the input.
• Efficient for large datasets due to its time complexity.
• Works well with linked lists and other data structures connected by pointers.

These characteristics make Merge Sort an appealing choice for students learning about sorting algorithms and their implementations.

Data Structures

Data structures are foundational elements used for storing and organizing data.
In computer science, selecting the appropriate data structure is crucial for efficient algorithm implementation.
Merging two sorted lists, such as exercise lists A and B, into list C is an example that highlights the integral role of data structures. In this exercise:

• Lists are used to hold sequences of numbers that are sorted incrementally during the merge process.
• The merging considers each list sequentially, maintaining their sorted order in the final list C.

Understanding lists and how they represent data efficiently is important not just for this exercise but for others involving similar operations.
The proper manipulation of these structures leads to effective problem-solving by implementing algorithms like Merge Sort.

Algorithm Analysis

When analyzing algorithms like Merge Sort, we look at its performance, particularly its time and space complexity.
Time complexity measures the speed of the algorithm, commonly represented as a function of input size.Merge Sort operates in $O(n\log n)$ time complexity.
This is faster than other simpler sorting algorithms such as bubble sort or insertion sort, which typically have $O(n^2)$ time complexity.
Despite this, Merge Sort's space complexity is $O(n)$ due to its requirement to store additional data during the merging step.
This means that while it is efficient in terms of time, it consumes extra space for utilizing temporary storage.Understanding these concepts helps make informed decisions about when to use Merge Sort based on data size and available system resources.
It balances the need for speed and memory, making it a strong candidate for handling large data sets efficiently.