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[\<- 06/01](06-01.md)
---
## Heap Sort
- An improved version of Selection sort
- How it works?
- Build a max-heap (O(nlog(n)))
- Select the largest element based on the heap and swap it with the last element in the unsorted list, and reheap (O(nlog(n)))
- Partitioned array: left side is an unsorted heap, right side is sorted
- total O(nlog(n)) (much faster than O(n^2))
- theoretically the fastest algorithm
- Example: 32, 78, 45, 8, 56, 23
|Heap |to be inserted|to be inserted|to be inserted|to be inserted|to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|32 |78 |45 |8 |56 |23 |
- Insert 78
|Heap |Heap |to be inserted|to be inserted|to be inserted|to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|32 |78 |45 |8 |56 |23 |
- To maintain the heap, swap 32 and 78
|Heap |Heap |to be inserted|to be inserted|to be inserted|to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |32 |45 |8 |56 |23 |
- Insert 45
|Heap |Heap |Heap |to be inserted|to be inserted|to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |32 |45 |8 |56 |23 |
- Insert 8
|Heap |Heap |Heap |Heap |to be inserted|to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |32 |45 |8 |56 |23 |
- Insert 56
|Heap |Heap |Heap |Heap |Heap |to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |32 |45 |8 |56 |23 |
- Swap 32 and 56
|Heap |Heap |Heap |Heap |Heap |to be inserted|
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |56 |45 |8 |32 |23 |
- Insert 23
|Heap |Heap |Heap |Heap |Heap |Heap |
|--------------|--------------|--------------|--------------|--------------|--------------|
|78 |56 |45 |8 |32 |23 |
- Now the heap has been created, but we still need to sort it
- Swap 23 and 56
|Heap |Heap |Heap |Heap |Heap |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|23 |56 |45 |8 |32 |78 |
- Reheap down to get
|Heap |Heap |Heap |Heap |Heap |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|56 |32 |45 |8 |23 |78 |
- Swap 56 and 23
|Heap |Heap |Heap |Heap |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|23 |32 |45 |8 |56 |78 |
- Reheap Down
|Heap |Heap |Heap |Heap |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|45 |32 |23 |8 |56 |78 |
- Swap 45 and 8
|Heap |Heap |Heap |Sorted |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|8 |32 |23 |45 |56 |78 |
- Reheap Down
|Heap |Heap |Heap |Sorted |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|32 |8 |23 |45 |56 |78 |
- Swap 32 and 23
|Heap |Heap |Sorted |Sorted |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|23 |8 |32 |45 |56 |78 |
- Reheap Down (same)
- Swap 23 and 8
|Heap |Sorted |Sorted |Sorted |Sorted |Sorted |
|--------------|--------------|--------------|--------------|--------------|--------------|
|8 |23 |32 |45 |56 |78 |
- Done!
- Is it stable?
- No (too much swapping to easily predict where things will be)
- This is the only downside of Heap Sort
- 78, 56, 32, **32**, 8, 23, 45
- Big O
- Worst case: O(nlog(n))
- Best case: O(nlog(n))
- Average case: O(nlog(n))
- Space overhead? O(1)
## Shell Sort
- An improved version of the insertion sort in which **diminishing partitions** are used to sort the data
- How does it work?
- A list of N elements is divided into **K segments**, where K is known as the increment
- For each segment, do insertion sort
- Reduce the value of K and repeat the process, until K=1
### Demonstration
- We do insertion sort on each segment
### Example
- 77, 62, 14, 9, 30, 21, 80, 25, 70, 55
- We set k= 5, 3, 1 progressively
|77|62|14|9 |30|21|80|25|70|55|
|--|--|--|--|--|--|--|--|--|--|
- Initial Sub-Arrays
- 77, 21
- 62, 80
- 14, 25
- 9, 70
- 30, 55
- After sorting the segments, we get
- 21, 77
- 62, 80
- 14, 25
- 9, 70
- 30, 55
|21|62|14|9 |30|77|80|25|70|55|
|--|--|--|--|--|--|--|--|--|--|
- Now, k=3, the new sub-arrays are
- 21, 9, 80, 55
- 62, 30, 25
- 14, 77, 70
- After sorting the segments, we get
- 9, 21, 55, 80
- 25, 30, 62
- 14, 70, 77
|9 |25|14|21|30|70|55|62|77|80|
|--|--|--|--|--|--|--|--|--|--|
- Now k=1, so the new sub-arrays will be length 1, and we will effectively just be doing an insertion sort
- This will actually be pretty fast, since by this point the array is nearly sorted
### Code
```
void shellSort(int a[], int n){
int i, j, temp, k;
foreach value of k do{
for(i=k; i<n; i+=k){
temp = a[i];
for(j = i-k; j>=0 && temp<a[j]; j-=k){
a[j+k] = a[j];
}
a[j+k] = temp;
}
}
}
```
- Notice that the code inside the for loop is just insertion sort
- How to choose K?
- Prime numbers
- A power of 2 minus 1 (i.e. k = 2^i - 1)
- Fibonacci series
- Is it stable?
- No
- 5 4 **4** 2 3 6
- Big O
- Worst case - depends, O(n^(3/2)) if right k sequence chosen
- Average case - depends, O(n^(4/3)) (as above)
- Best case - O(nlog(n))
- Space Overhead O(1)
### Additional Notes
- Best-case Big-O analysis:
- comparisons =
- n, for 1 sort with elements 1-apart (last step)
- \+ 3 * n/3, for 3 sorts with elements 3-apart (next-to-last step)
- \+ 7 * n/7, for 7 sorts with elements 7-apart
- \+ 15 * n/15, for 15 sorts with elements 15-apart
- Each term is n. The question is how many terms are there? The number of terms is the value k such that (2^i - 1)\<n; So i\<log(n+1), meaning that the sorting time in the best case is less than n * log(n+1) = O(n\*log(n))
### Summary
- The method starts by sorting pairs of elements **far apart from each other**, then **progressively reducing the gap** between elements to be compared. Starting with far apart elements can move some out-of-place elements into position faster than a simple nearest neighbor exchange
## Tree Sort
- Two steps:
- Insert each value from the array into a binary search tree
- Traverse the binary search tree in order and transfer the data back into the array
- Worst case: O(n^2) if BST, O(nlog(n)) if AVL
- Average case: O(nlog(n))
- Best case: O(nlog(n))
- Space overhead? O(n) (high!)
- You need to build a binary tree for this sort
- It is stable!
---
[06/06 ->](06-05.md)
|
