| Algorithm | Worst Case | Average Case | Best Case | Min. no. of swaps | Max. no. of swaps |
|---|---|---|---|---|---|
| Bubble | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ |
| Selection | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | 0 | $\theta(n^{2})$ |
| Insertion | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n)$ | 0 | $\theta(n^{2})$ |
| Quick | $\theta(n^{2})$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | 0 | $\theta(n^{2})$ |
| Merge | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | Is not in-place sorting | Is not in-place sorting |
| Heap | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $O(n\lg n)$ | $\theta(n\lg n)$ |
When sorting happens within the same array, its called in-place sorting. All the above sorting algorithms except merge sort are in-place sorting. In merge sort, we use an auxiliary array for merging two sub-arrays and hence its not in-place sorting.
This matters only for arrays with same element repeated. If $A[x] = A[y]$, in the original array such that $x<y$, then in the sorted array, $A[x]$ must be placed at position $u$ and $A[y]$ must be placed at position $v$, such that $u<v$. i.e.; the relative positions of same value elements must be same in the sorted array as in the input array, for the sorting algorithm to be stable. In most cases stability depends on the way in which algorithm is implemented. Still, its not possible to implement heap sort as a stable sort.
| Algorithm | Worst Case | Average Case | Best Case | Min. no. of swaps | Max. no. of swaps |
|---|---|---|---|---|---|
| Bubble | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ |
| Selection | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n^{2})$ | 0 | $\theta(n^{2})$ |
| Insertion | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n)$ | 0 | $\theta(n^{2})$ |
| Quick | $\theta(n^{2})$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | 0 | $\theta(n^{2})$ |
| Merge | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | Is not in-place sorting | Is not in-place sorting |
| Heap | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $\theta(n\lg n)$ | $O(n\lg n)$ | $\theta(n\lg n)$ |
When sorting happens within the same array, its called in-place sorting. All the above sorting algorithms except merge sort are in-place sorting. In merge sort, we use an auxiliary array for merging two sub-arrays and hence its not in-place sorting.
This matters only for arrays with same element repeated. If $A[x] = A[y]$, in the original array such that $x<y$, then in the sorted array, $A[x]$ must be placed at position $u$ and $A[y]$ must be placed at position $v$, such that $u<v$. i.e.; the relative positions of same value elements must be same in the sorted array as in the input array, for the sorting algorithm to be stable. In most cases stability depends on the way in which algorithm is implemented. Still, its not possible to implement heap sort as a stable sort.
GATECSE