What is the time complexity of quicksort and mergesort algorithms?

The time complexity of quicksort and mergesort algorithms is an important consideration when selecting a sorting algorithm for a particular use case. Both algorithms are commonly used in computer science and offer different trade-offs in terms of performance and memory usage.

Quicksort:

Quicksort is a popular sorting algorithm that uses a divide-and-conquer approach to sorting by partitioning the array into two sub-arrays based on a pivot element. The time complexity of quicksort depends on the choice of pivot element and the distribution of the input data.

In the best case scenario, where the pivot element is chosen to be the median of the input data, the time complexity of quicksort is O(n log n), where n is the size of the input array. This occurs when the pivot element divides the array into two sub-arrays of roughly equal size, allowing the algorithm to quickly sort the sub-arrays.

In the average case scenario, where the input data is uniformly distributed, the time complexity of quicksort is also O(n log n). However, the constant factor is typically smaller than that of other O(n log n) algorithms, such as mergesort, making it faster in practice.

In the worst case scenario, where the pivot element is chosen to be the minimum or maximum value of the input data, or the input data is already sorted, the time complexity of quicksort is O(n^2), where n is the size of the input array. This occurs when the pivot element divides the array into sub-arrays of vastly different sizes, causing the algorithm to repeatedly partition the same sub-array.

Mergesort:

Mergesort is a sorting algorithm that uses a divide-and-conquer approach to sorting by recursively dividing the input array into smaller sub-arrays and merging them back together in sorted order. The time complexity of mergesort is always O(n log n), regardless of the distribution of the input data.

The time complexity of mergesort is determined by the number of comparisons required to merge the sub-arrays. In each merge step, the algorithm compares the smallest elements of each sub-array and adds the smallest element to a new sorted array until all elements have been added. The number of comparisons required to merge two sub-arrays of size n/2 is n, and the number of merge steps required is log n. Therefore, the total number of comparisons required for mergesort is n log n.

In terms of performance, mergesort is generally slower than quicksort for small arrays due to its higher constant factor and memory overhead. However, mergesort is more suitable for larger arrays and is always guaranteed to have a time complexity of O(n log n), making it a more predictable and reliable sorting algorithm than quicksort.

In summary, the time complexity of quicksort and mergesort algorithms depends on the choice of pivot element and the distribution of the input data. Quicksort has a best-case and average-case time complexity of O(n log n) and a worst-case time complexity of O(n^2), while mergesort has a time complexity of O(n log n) regardless of the distribution of the input data. The choice of sorting algorithm depends on the specific use case and the trade-offs between performance and memory usage.