Explain multiway merge sort with an example.

Multiway Merge Sort

Multiway merge sort is a generalization of the classic two-way merge sort algorithm. It divides the input sequence into multiple sub-sequences of equal size, sorts them concurrently, and then merges them together to obtain the final sorted sequence. The number of sub-sequences is referred to as the way count.

Algorithm

1. Divide: Split the input sequence into multiple sub-sequences, each containing a roughly equal number of elements. The way count determines the number of sub-sequences. These sub-sequences are sorted independently in parallel.

2. Conquer: Each sub-sequence is sorted using a suitable sorting algorithm, such as multi-threaded merge sort or heap sort. This step ensures that each sub-sequence is individually sorted.

3. Merge: The sorted sub-sequences are merged together to obtain the final sorted sequence. This merging process is performed by repeatedly taking the smallest element from each sorted sub-sequence and appending it to the final sorted sequence until all sub-sequences are empty.

Example

Let's consider an example with a way count of 4 and an input sequence of 12 numbers:

Input: [5, 3, 8, 2, 1, 4, 7, 6, 9, 11, 10, 12]

1. Divide: Divide the input sequence into four sub-sequences:

Sub-sequence 1: [5, 3, 8]
Sub-sequence 2: [2, 1, 4]
Sub-sequence 3: [7, 6, 9]
Sub-sequence 4: [11, 10, 12]

1. Conquer: Sort each sub-sequence independently using merge sort or another sorting algorithm.

Sorted Sub-sequence 1: [3, 5, 8]
Sorted Sub-sequence 2: [1, 2, 4]
Sorted Sub-sequence 3: [6, 7, 9]
Sorted Sub-sequence 4: [10, 11, 12]

1. Merge: Merge the sorted sub-sequences to obtain the final sorted sequence.

Merge [3, 5, 8], [1, 2, 4], [6, 7, 9], [10, 11, 12]:

• Start with four pointers, one pointing to the first element of each sorted sub-sequence.
• Select the smallest element among the four, which is 1 from sub-sequence 2. Append it to the final sorted sequence.
• Move the pointer of the sub-sequence from which the element was selected (sub-sequence 2) to the next element.
• Repeat steps 3 and 4 until all elements are merged.

The final sorted sequence:

[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Advantages and Disadvantages

Advantages of Multiway Merge Sort:

• Scalability: Multiway merge sort can be efficiently parallelized by sorting multiple sub-sequences concurrently, making it suitable for large datasets on multi-core or distributed systems.
• External Memory: It can be adapted to handle external memory scenarios, where data is stored on disk and retrieved in blocks. This allows for sorting massive datasets that cannot fit entirely in main memory.

Disadvantages of Multiway Merge Sort:

• Overhead: Dividing the input sequence into sub-sequences and merging them back together incurs some overhead, which can be significant for small datasets.
• Way Count Selection: Choosing the appropriate way count is crucial for performance. A large way count can lead to excessive overhead, while a small way count may limit the potential for parallelism.

Applications

Multiway merge sort is particularly useful in scenarios where:

• A large dataset needs to be sorted efficiently.
• The dataset can be partitioned into multiple parts that can be sorted independently.
• Multiple processors or cores are available for parallel processing.

Examples of applications include:

• Sorting large files or databases.
• Data warehousing and data analytics.
• Scientific computing.
• Financial analysis.

Overall, multiway merge sort is a powerful sorting algorithm that combines the benefits of merge sort with the ability to exploit multiple processing units or cores for faster sorting.