Bubble Sort

The worst-case time complexity of bubble sort is O(n^2), where n is the number of elements in the array being sorted. This means that as the size of the array being sorted increases, the time taken by the algorithm to sort it grows exponentially.

Bubble sort involves iterating through the array multiple times, comparing adjacent elements and swapping them if they are in the wrong order. In the worst-case scenario, where the array is in reverse order, the algorithm would need to iterate through the array n times, with each iteration involving n-1 comparisons and swaps. This gives a total of n(n-1) operations, which simplifies to O(n^2) time complexity.

It is worth noting that bubble sort is not an efficient algorithm for large datasets and should only be used for small datasets or for educational purposes. There are more efficient sorting algorithms like merge sort and quicksort that have a better time complexity of O(n log n) in the worst case.

Selection Sort

The worst-case time complexity of selection sort is also O(n^2), where n is the number of elements in the array being sorted. Like bubble sort, selection sort involves iterating through the array multiple times, but instead of swapping adjacent elements, it selects the minimum value and places it at the beginning of the array, then repeats the process for the remaining unsorted elements.

In the worst-case scenario, selection sort would need to make n-1 comparisons and swaps in the first iteration, followed by n-2 comparisons and swaps in the second iteration, and so on, until the last iteration with only one comparison and swap. This gives a total of n(n-1)/2 operations, which simplifies to O(n^2) time complexity.

Like bubble sort, selection sort is not very efficient for large datasets and should only be used for small datasets or for educational purposes. There are more efficient sorting algorithms like merge sort and quicksort that have a better time complexity of O(n log n) in the worst case.

Insertion Sort

The worst-case time complexity of insertion sort is O(n^2), where n is the number of elements in the array being sorted. Insertion sort works by iterating through the array and inserting each element in its proper place within the sorted part of the array.

In the worst-case scenario, where the array is in reverse order, each element would need to be compared to every element in the sorted part of the array before it can be inserted in the correct position. This would result in n-1 comparisons and swaps in the first iteration, followed by n-2 comparisons and swaps in the second iteration, and so on, until the last iteration with only one comparison and swap. This gives a total of n(n-1)/2 operations, which simplifies to O(n^2) time complexity.

However, in the best-case scenario, where the array is already sorted, the time complexity of insertion sort is O(n), because each element only needs to be compared to the previous element in the sorted part of the array before it can be placed in its proper position.

In practice, insertion sort is often faster than other O(n^2) sorting algorithms like bubble sort and selection sort, especially for small arrays or partially sorted arrays. Additionally, because of its simple implementation and low overhead, insertion sort can be useful in certain specialized cases, such as when sorting small arrays in memory-constrained environments or in some embedded systems.

Quick Sort

The worst-case time complexity of quick sort is O(n^2), but its average time complexity is O(n log n), where n is the number of elements in the array being sorted. Quick sort is a divide-and-conquer algorithm that works by selecting a "pivot" element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The pivot is then placed in its correct position in the final sorted array, and the process is repeated recursively for the two sub-arrays.

In the worst-case scenario, where the selected pivot is the largest or smallest element in the array, and the array is already sorted or nearly sorted, quick sort may partition the array in a way that one of the sub-arrays has n-1 elements, while the other sub-array is empty. This would result in a worst-case time complexity of O(n^2).

However, in most cases, the choice of the pivot is randomized or based on a median-of-three algorithm to avoid the worst-case scenario. In the average case, quick sort has a time complexity of O(n log n), which is faster than other O(n^2) sorting algorithms like bubble sort, selection sort, and insertion sort.

Quick sort is a popular and efficient algorithm for sorting large datasets and is widely used in practice. However, it may not be suitable for datasets with a large number of duplicate values, as it may result in skewed partitions and sub-optimal performance. In such cases, other algorithms like merge sort or heap sort may be more appropriate.

Heap Sort

The worst-case and average time complexity of heap sort is O(n log n), where n is the number of elements in the array being sorted. Heap sort is a comparison-based sorting algorithm that works by first building a binary heap from the input array, and then repeatedly extracting the maximum element from the heap and placing it at the end of the sorted portion of the array.

Building the heap takes O(n) time complexity, while each extraction takes O(log n) time complexity. Since the maximum element is extracted n times, the total time complexity of heap sort is O(n log n).

Heap sort has the advantage of being an in-place sorting algorithm, meaning that it does not require additional memory beyond the input array. It is also relatively easy to implement and has a predictable performance, making it a popular choice for sorting large datasets in practice. However, it may not be as efficient as other O(n log n) sorting algorithms like merge sort or quick sort, especially for partially sorted datasets or datasets with a high degree of duplication.