There is no single best sorting algorithm for asymptotic runtime complexity. Depending on the size of the array to be sorted and the number of elements to be sorted, various widespread sorting algorithms can be used. Some of the most common algorithms use insertion, selection, bubble, merge, shell and quick sort.

To compare the time complexity of these sorting algorithms, let's look at the runtime complexity as a function of the size of the array to be sorted (n). The time complexity is typically expressed in Big-O notation, which describes the worst-case runtime of a function while ignoring any constant factors.

## Insertion Sort

Insertion sort is a popular algorithm used when the size of the array to be sorted is relatively small. When sorting an array of length n, the insertion sort algorithm requires n2 comparisons, making it a slow algorithm for large datasets. The best case scenario is when the array is already sorted or nearly sorted, in which case the insertion sort algorithm requires only n comparisons.

## Selection Sort

Selection sort is also a popular algorithm used for small data sets. The runtime of the selection sort algorithm is n2 because it requires n2 comparisons. However, the selection sort algorithm does not require additional memory and can run faster in certain cases.

## Bubble Sort

Bubble sort is an algorithm that involves repeatedly swapping adjacent elements. It provides a runtime of n2 for arrays of length n. Bubble sort is ideal for small datasets but should not be used for larger datasets since it is inefficient and time-consuming.

## Merge Sort

Merge sort is a divide-and-conquer algorithm which divides an array in half, recursively sorts each half, and then merges the sorted halves. It is an efficient sorting algorithm and its worst-case runtime is n log(n).

## Shell Sort

Shell sort is an improvement on insertion sort. It is based on the gap sequence and involves exchanging elements that are far enough apart. Its runtime is O(n^2^) in the worst case, but it runs faster than insertion sort.

## Quick Sort

Quick sort is another popular algorithm with a runtime of n log(n). It is a divide-and-conquer algorithm that works by choosing a pivot element from the array and then recursively sorting the sub-arrays on either side of the pivot. It is an efficient algorithm for large datasets.

## Conclusion

In conclusion, there is no single best sorting algorithm for asymptotic runtime complexity. Depending on the size of the array and the number of elements, various widespread sorting algorithms can be used. For small data sets, insertion sort and selection sort are popular algorithms. For large datasets, quick sort and merge sort are more suitable.

## FAQs

### What is Big-O notation?

Big-O notation is a way of describing the worst-case performance of an algorithm as a function of its input size. It is a mathematical notation which describes the upper bound of a function’s complexity by ignoring any constant factors.

### How does insertion sort work?

Insertion sort is a sorting algorithm that involves repeatedly shifting elements to the right until they are in the correct position. It is ideal for small data sets but is not recommended for larger datasets as it has a runtime of n2.

### What is selection sort?

Selection sort is an algorithm for sorting an array that repeatedly finds the minimum element from the unsorted part and puts it at the beginning of the array. Its worst-case runtime is n2 and it does not require additional memory, making it a suitable algorithm for small data sets.

### What is the runtime of merge sort?

Merge sort is an efficient sorting algorithm with a worst-case runtime of n log(n). It is a divide-and-conquer algorithm which recursively sorts each half of an array and then merges the sorted halves.

### What is the difference between insertion sort and bubble sort?

Insertion sort requires n2 comparisons in the worst case. Bubble sort is an algorithm that involves repeatedly swapping adjacent elements and its runtime is also n2. Bubble sort is useful for small datasets but should not be used for larger datasets as it is inefficient and time-consuming.

### What is quick sort?

Quick sort is a divide-and-conquer algorithm which works by choosing a pivot element from the array and then recursively sorting the sub-arrays on either side of the pivot. Its worst-case runtime is n log(n). It is an efficient algorithm for large datasets.