Question

Write a function def mergesorted \((a, b)\) that merges two sorted lists, producing a new sorted list. Keep an index into each list, indicating how much of it has been processed already. Each time, append the smallest unprocessed element from either list, then advance the index. For example, if a is 14916 and \(b\) is \(\begin{array}{lllll}4 & 7 & 9 & 9 & 11\end{array}\) then mergesorted returns a new list containing the values \(\begin{array}{lllllllll}1 & 4 & 4 & 7 & 9 & 9 & 9 & 11 & 16\end{array}\)

Short answer

The function merges two sorted lists by comparing elements step by step and appending the smaller element to the merged list.

Step-by-step solution

Initialize Variables

We start by creating an empty list `merged_list` which will hold the merged result. We also initialize two indices, `index_a` and `index_b`, both set to 0, which will keep track of our position in lists `a` and `b` respectively.

Compare and Merge Elements

We enter a loop that runs as long as there are unprocessed elements in both lists. During each iteration, we compare the current elements of list `a` and list `b` using their respective indices. We append the smaller element to `merged_list` and increment the index of the list from which the element was taken.

Append Remaining Elements from List `a`

After the loop, if there are any remaining elements in list `a`, they are already larger than any elements in list `b`. We append these remaining elements to `merged_list` one by one using the index `index_a`.

Append Remaining Elements from List `b`

Similarly, if there are any remaining elements in list `b` after exiting the loop, we append these to `merged_list` using the index `index_b`.

Return the Merged List

Finally, once all elements from both lists have been processed and appended to `merged_list`, we return `merged_list` as the final sorted list.

Key concepts

List Indexing

List indexing is a fundamental concept in programming that allows you to access elements within a list using their position or 'index'. In Python, indices start at 0. This means the first element is at index 0, the second element is at index 1, and so on.
This concept is crucial for managing the elements as you process them, much like in the merge sort algorithm. By using indices, you can keep track of which elements in your list have already been processed and which are yet to be processed.
For example, if you have two lists 'a' and 'b', you will typically have two separate indices: `index_a` for list 'a' and `index_b` for list 'b'. These indexes help you navigate through each list without losing track of your position, which is essential when merging sorted lists.
Utilizing indices efficiently can make your sorting algorithms more effective and ensure that you don't miss any elements during the merge process.

Step-by-Step Solution

Following a step-by-step approach is a powerful method to methodically solve complex problems, like merging two sorted lists with the merge sort algorithm. It breaks down the process into manageable tasks that you can execute in sequence.
Here’s how you can apply a step-by-step solution to merge sorted lists:

• Initialize Variables: Start with an empty list `merged_list` to store the sorted results, and set both `index_a` and `index_b` to 0 for list 'a' and 'b'. This helps in tracking progress through the lists.
• Compare and Merge: Enter a loop to compare elements from lists 'a' and 'b'. Append the smaller element to `merged_list`, and move the respective index forward. This continues until one of the lists runs out of elements.
• Finalize Merging Remaining Elements: If elements are left in either list 'a' or 'b', append them directly to `merged_list`. Since the lists started sorted, these remaining elements are already in order.
• Return the Result: Once all elements are merged, the resultant sorted list is returned. This final step verifies that all elements have been processed.

By breaking the task into these steps, the problem becomes more approachable and less prone to errors, maintaining clarity throughout the process.

Sorted Lists

A sorted list is a collection of elements arranged in ascending (or descending) order. This can involve numbers, strings, or other data types, all organized according to their value or sort order.
Sorted lists are beneficial in algorithms like merge sort, because they simplify the process of combining different datasets. When two lists are pre-sorted, it becomes easier to merge them into one sorted list.
In the merge sort algorithm, taking advantage of already sorted lists means that you only need to compare the current smallest elements of each list at their indices. This process is far more efficient than sorting a mixed dataset from scratch.
Sorted lists, when used well, reduce the time complexity of sorting operations, meaning you can achieve a sorted result faster and with less computational effort.

Algorithm Implementation

Implementing an algorithm means translating a problem-solving method into a code that a computer can execute. For the merge sort algorithm, this involves strategic planning and coding techniques.
The implementation of a merge sort algorithm is straightforward once the logic is established. You start by initializing variables and indices for the lists. The core function iterates over the lists, continuously merging elements by comparing their current indices.

• Code Structuring: Organize the code clearly, ensure each part is responsible for a distinct task, like initialization, comparison, and finalization.
• Error Handling: Anticipate potential errors such as running out of elements in one list, and ensure the code handles these gracefully.
• Efficiency Considerations: Optimize the loop and conditions to reduce unnecessary computations, like limiting the comparisons and appends to what's absolutely necessary.
• Testing and Debugging: Test with different sorted lists to ensure robustness and accuracy of your implementation.

The goal of your implementation should be a clean, efficient, and correct program that accurately merges any two sorted lists.