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===Stable sorting=== | ===Stable 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 $u<v$. | + | 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. |
{{Template:FBD}} | {{Template:FBD}} | ||
[[Category:Notes & Ebooks for GATE Preparation]] | [[Category:Notes & Ebooks for GATE Preparation]] | ||
[[Category: Algorithms & Data Structures]] | [[Category: Algorithms & Data Structures]] | ||
| 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 (when array is already sorted) | $\theta(n^{2})$ |
| Insertion | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n)$ (when array is sorted) | 0 (when array is already sorted) | $\theta(n^{2})$ |
| Quick | $\theta(n^{2})$ (when array is sorted, and pivot taken from one end) | $\theta(n\lg n)$ | $\theta(n\lg n)$ | 0 (when array is already sorted) | $\theta(n^{2})$ |
| 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 except, merge sort are in-place sorting. In merge sort, we use an auxiliary array for merging two sub-arrays
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 (when array is already sorted) | $\theta(n^{2})$ |
| Insertion | $\theta(n^{2})$ | $\theta(n^{2})$ | $\theta(n)$ (when array is sorted) | 0 (when array is already sorted) | $\theta(n^{2})$ |
| Quick | $\theta(n^{2})$ (when array is sorted, and pivot taken from one end) | $\theta(n\lg n)$ | $\theta(n\lg n)$ | 0 (when array is already sorted) | $\theta(n^{2})$ |
| 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 except, merge sort are in-place sorting. In merge sort, we use an auxiliary array for merging two sub-arrays
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.
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