Question

Modify the merge sort algorithm to remove duplicates in the merging step to obtain an algorithm that removes duplicates from a list. Note that the resulting list does not have the same ordering as the original one. What is the cfficiency of this algorithm?

Short answer

The modified merge sort still operates at O(n log n) complexity while removing duplicates.

Step-by-step solution

Understanding Merge Sort

Merge Sort is a divide-and-conquer algorithm that divides the list into two halves, recursively sorts both halves, and then merges them back together. Normally, it preserves duplicates while merging. In this exercise, we need to modify the merge step to remove duplicates.

Modify the Merge Function

In the merge function, as we compare elements of the two halves, we not only merge them but also check if an element has already been added to the resultant list. We skip adding any repeated elements to ensure that duplicates are removed. This involves checking the last inserted element of the merged list before insertion.

Implement Merge Sort with Modified Merge

Implement the Merge Sort algorithm with any required sorting logic in the core functions. In the recursive calls to sort the halves, use the modified merge function to construct the sorted, duplicate-free list.

Analyze Efficiency

The efficiency of Merge Sort is O(n log n) because it consistently halves the array and then merges the sorted halves. The modified merge that skips duplicates doesn’t add any additional complexity, so the efficiency remains O(n log n).

Key concepts

Merge Sort

Merge Sort is a powerful and commonly used algorithm that is part of the group of sorting algorithms known as "divide-and-conquer" algorithms. It works by breaking down a list into sublists until each sublist consists of a single element. Single-element lists are naturally sorted, so from there, it continuously merges these sublists to produce increasingly larger sorted lists. This merging process continues until the whole list is sorted.

One of the advantages of Merge Sort is its predictable time complexity of \( O(n \log n) \), making it efficient for large datasets.

• First, the list is divided into two roughly equal parts,
• Each part is sorted recursively via Merge Sort,
• Then, the sorted sublists are merged back together to form a single sorted list.

Merge Sort is stable; it maintains the original order of equivalent elements. This property is important in many applications where duplicates are intentionally kept.

Duplicate Removal

In this exercise, we make a key modification to the classic Merge Sort by incorporating duplicate removal. This means changing how elements are merged in the algorithm's merging step. Normally, Merge Sort doesn't remove duplicates.

To enhance it, we need a check during the merging process. When we add an element from the sublists into the merged list, we ensure that it's not already present in the merged list. This is done by comparing the current element with the last inserted element. If they are the same, the new element is not added, thus eliminating duplicates.

• Check each element before adding it to the result list,
• Skip the element if it matches the last added element,
• Continue merging with the next element.

This method efficiently ensures that our final list contains unique, sorted elements.

Sorting Algorithms

Sorting algorithms are procedures that put elements of a list in a certain order. They are critical for optimizing operations with data and for making other algorithms operate effectively. There are numerous sorting algorithms, each with pros and cons, depending on the dataset size, type, and required order.

Some popular sorting algorithms include Quick Sort, Bubble Sort, Insertion Sort, and of course, Merge Sort.

• Quick Sort: Often faster in practice than Merge Sort but has a worse worst-case time complexity \( O(n^2) \) if not implemented carefully.
• Bubble Sort: Extremely simple but inefficient for large lists, having \( O(n^2) \) complexity.
• Insertion Sort: Good for small datasets or largely sorted datasets, with a time complexity of \( O(n^2) \).
• Merge Sort: Consistent \( O(n \log n) \) efficiency and being stable.

Choosing the right sorting algorithm depends on the specific requirements, like stability, performance, and memory usage.

Algorithm Efficiency

Algorithm efficiency considers both time complexity and space complexity. It determines how well an algorithm performs in terms of time and space as the input size grows. Efficiency is crucial because it impacts the real-world usability of algorithms on large datasets.

Merge Sort offers a predictable time complexity of \( O(n \log n) \), which is efficient and suitable for large data. This efficiency is what makes it a preferred choice for many sorting tasks.

• Time Complexity: The amount of time an algorithm takes to complete, based on input size,
• Space Complexity: The amount of memory an algorithm uses during its computation,
• Big O Notation: A mathematical notation that describes the upper limit of the time and space requirements of an algorithm

By not altering the time complexity in the modified Merge Sort for duplicate removal, the algorithm retains its high level of efficiency. This is crucial for performance in practical applications.