QuickSort QuickSort is often called Partition Sort. It is a recursive method, in which the unsorted...
77
QuickSort QuickSort is often called Partition Sort. It is a recursive method, in which the unsorted array is first rearranged so that there is some record, somewhere in the middle of the array, whose key is greater than all the keys to its left & less than or equal to all the keys to its right. Once this “middle is found, the same method can be used to sort the section of the array to the left, then sort the
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Transcript of QuickSort QuickSort is often called Partition Sort. It is a recursive method, in which the unsorted...
QuickSort
QuickSort is often called Partition Sort.
It is a recursive method, in which the unsorted array is first rearranged so that there is some record, somewhere in the middle of the array, whose key is greater than all the keys to its left & less than or equal to all the keys to its right.
Once this “middle is found, the same method can be used to sort the section of the array to the left, then sort the section to the right.
QuickSort Algorithm
1. If First < Last then 2. Partition the elements in the array (First, Last) so that the pivot value is in place ( position PivIndex). 3. Apply QuickSort to the first subarray (First, PivIndex -1) 4. Apply QuickSort to the second subarray (PivIndex + 1, Last).
The two stopping Cases are: 1. (First = Last) - only one value in subarray, so sorted. 2. (First > Last) - no values in subarray, so sorted.
How do we Partition?
1. Define the pivot value as the content of Table[First]2. Initialize Up to First and Down to last3. Repeat 4. Increment Up until Up selects the first element greater than the pivot value 5. Decrement Down until it selects the first element less than or equal to the pivot value. 6. If Up < Down exchange their values.
Until Up meets or passes Down.
7. Exchange Table[First] and Table[Down]8. Define PivIndex as Down
QuickSort Example
44 75 23 64 331243 7755
0 1 2 3 4 5 6 7 8
First Last
Has First exceeded Last? No!
Pivot 44
Define the value in position First to be the Pivot.
QuickSort Example
44 75 23 64 331243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Define Up To be First and Down to be last
Up Down
QuickSort Example
44 75 23 64 331243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Up to the first value > Pivot
Up Down
QuickSort Example
44 75 23 64 331243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Down to the first value <= Pivot
Up Down
QuickSort Example
44 33 23 64 751243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
If Up < Down, exchange their values
Up Down
QuickSort Example
44 33 23 64 751243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Up to the first value > Pivot
UpDown
QuickSort Example
44 33 23 64 751243 7755
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Down to the first value <= Pivot
Up Down
QuickSort Example
44 33 23 64 755543 7712
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
If Up < Down, exchange their values.
Up Down
QuickSort Example
44 33 23 64 755543 7712
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Up to the first value > Pivot
Up Down
QuickSort Example
44 33 23 64 755543 7712
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Move Down to the first value <= Pivot
Up Down
Up and Down have passed each other, so exchange the pivot value and the value in Down.
QuickSort Example
12 33 23 64 755543 7744
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
Up Down
Up and Down have passed each other, so exchange the pivot value and the value in Down.
QuickSort Example
12 33 23 64 755543 7744
0 1 2 3 4 5 6 7 8
First Last
Pivot 44
PivIndex
Note that all values below PivIndex are <= Pivot andall values above PivIndex are > Pivot.
QuickSort Example
12 33 23 64 755543 7744
0 1 2 3 4 5 6 7 8
First1 Last2
Pivot 44
PivIndex
This gives us two new subarrays to Partition
Last1First2
QuickSort Procedure Code
void QuickSort(int Table[], int First, int Last){ int PivIndex; if (First < Last) { PivIndex = Partition(Table, First, Last); QuickSort(Table, First, PivIndex - 1); QuickSort(Table, PivIndex + 1, Last); }}
Heap Sort
How can a tree be represented in an array?
0 1 2 3 4 5 6
12
57 19
87 15 44 23
Heap Sort
How can a tree be represented in an array?
12
0 1 2 3 4 5 6
12
57 19
87 15 44 23
Place the root of the tree in element 0 of the array (RootPos = 0).
Heap Sort
How can a tree be represented in an array?
12 57 19
0 1 2 3 4 5 6
12
57 19
87 15 44 23
Place the root of the tree in element 0 of the array (RootPos = 0).Place the root’s left child in element 1 : (RootPos*2 + 1) = 1Place the root’s right child in element 2: (RootPos*2 + 2) = 2
Heap SortHow can a tree be represented in an array?
12 57 19 87 15
0 1 2 3 4 5 6
12
57 19
87 15 44 23
Place the root of the tree in element 0 of the array (RootPos = 0).Place the root’s left child in element 1 : (RootPos*2 + 1)Place the root’s right child in element 2: (RootPos*2 + 2)
The Children of 57 are placed in:Left Child (87): ParentPos*2 +1 = 1* 2 + 1 = 3Right Child (15): ParentPos*2 + 2 = 1 *2 + 2 = 4
Heap SortHow can a tree be represented in an array?
12 57 19 87 15
0 1 2 3 4 5 6
12
57 19
87 15 44 23
Place the root of the tree in element 0 of the array (RootPos = 0).Place the root’s left child in element 1 : (RootPos*2 + 1)Place the root’s right child in element 2: (RootPos*2 + 2)
The Children of 19 are placed in:Left Child (44): ParentPos*2 +1 = 2* 2 + 1 = 5Right Child (15): ParentPos*2 + 2 = 2 *2 + 2 = 6
44 23
Heap Sort
To perform the heap sort we must:
1. Create a heap (a tree with all nodes greater than their children)
2. Remove the root element from the heap one at a time, recomposing the heap.
Building the Heap
1. For each value in the array(0, n)
2. Place the value into the “tree” 3. Bubble the value as high as it can go (push the largest values to highest position)
Heap SortHow to build a heap?
12 57 19 87 15
0 1 2 3 4 5 6
12
Add Table[0] to treeSince it has no parent, we have a heap.
44 23
Heap SortHow to build a heap?
12 57 19 87 15
0 1 2 3 4 5 6
12
Add Table[1] to treeSince 12 < 57, it is not a heap.Bubble it up as high as it can go. Exchange
44 23
57
Heap SortHow to build a heap?
57 12 19 87 15
0 1 2 3 4 5 6
57
ExchangeSince 57 >12 57 is as high as it can go, so we have a heap.
44 23
12
Heap SortHow to build a heap?
57 12 19 87 15
0 1 2 3 4 5 6
57
Add Table[2] to treeSince 57 >19 so, we have a heap.
44 23
12 19
Heap SortHow to build a heap?
57 12 19 87 15
0 1 2 3 4 5 6
57
Add Table[3] to treeSince 87 >12 so, not a heap.
44 23
12 19
87
Heap SortHow to build a heap?
57 87 19 12 15
0 1 2 3 4 5 6
57
Add Table[3] to treeSince 87 >12 so, not a heap. Bubble it up. Exchange.Again 87 > 57, so not a heap. Bubble it up
44 23
87 19
12
Heap SortHow to build a heap?
87 57 19 12 15
0 1 2 3 4 5 6
87
Again 87 > 57, so not a heap. Bubble it up.Exchange.We now have a heap again.
44 23
57 19
12
Heap SortHow to build a heap?
87 57 19 12 15
0 1 2 3 4 5 6
87
Add Table[4] to tree15 > 57, so a heap.
44 23
57 19
12 15
Heap SortHow to build a heap?
87 57 19 12 15
0 1 2 3 4 5 6
87
Add Table[5] to tree44 > 19, so not a heap.
44 23
57 19
12 15 44
Heap SortHow to build a heap?
87 57 44 12 15
0 1 2 3 4 5 6
87
44 > 19, so not a heap. Bubble it up. Exchange.44<87Again we have a heap.
19 23
57 44
12 15 19
Heap SortHow to build a heap?
87 57 44 12 15
0 1 2 3 4 5 6
87
Add Table[6] to tree23 <44 so, we have a heap.
19 23
57 44
12 15 19 23
Heap SortHow to build a heap?
87 57 44 12 15
0 1 2 3 4 5 6
87
The whole table is now a heap!
23
57 44
12 15 19 23
19
Heap Sort Algorithm
1. Repeat n -1 times 2. Exchange the root value with the last value in the tree 3. “Drop” the last value from the tree 4. Reform the heap 5. Start at the root node of the current tree 6. If the root is larger than its children, stop- you have a heap. 7. If not, exchange the root with the largest child. 8. Consider this child to be the current root and repeat step 4
Heap Sort
87 57 44 12 15
0 1 2 3 4 5 6
87
Here is the heap!
19 23
57 44
12 15 19 23
Heap Sort
87 57 44 12 15
0 1 2 3 4 5 6
87
Exchange the root with the last value in the tree
19 23
57 44
12 15 19 23
Heap Sort
23 57 44 12 15
0 1 2 3 4 5 6
23
Exchange the root with the last value in the tree
19 87
57 44
12 15 19 87
Heap Sort
23 57 44 12 15
0 1 2 3 4 5 6
23
Drop this last value from the tree -- it is now in the array in its sorted position!
19 87
57 44
12 15 19 87
Heap Sort
23 57 44 12 15
0 1 2 3 4 5 6
23
Drop this last value from the tree -- it is now in the array in its sorted position!
19 87
57 44
12 15 19
The sorted listThe tree
Heap Sort
23 57 44 12 15
0 1 2 3 4 5 6
23
Reform the heap
19 87
57 44
12 15 19
The sorted listThe tree
Heap Sort
23 57 44 12 15
0 1 2 3 4 5 6
23
Find the largest child of the current “root”and exchange with “root”
19 87
57 44
12 15 19
The sorted listThe tree
Heap Sort
57 23 44 12 15
0 1 2 3 4 5 6
57
Now 23 is larger than both of its children so we have a heap again.
19 87
23 44
12 15 19
The sorted listThe tree
Heap Sort
57 23 44 12 15
0 1 2 3 4 5 6
57
Exchange the root of the heap with the last value in the tree.
19 87
23 44
12 15 19
The sorted listThe tree
Heap Sort
19 23 44 12 15
0 1 2 3 4 5 6
19
Exchange the root of the heap with the last value in the tree.
57 87
23 44
12 15 57
The sorted listThe tree
Heap Sort
19 23 44 12 15
0 1 2 3 4 5 6
19
Drop the last element from the tree since the value is nowin its sorted position.
57 87
23 44
12 15 57
The sorted listThe tree
Heap Sort
19 23 44 12 15
0 1 2 3 4 5 6
19
Drop the last element from the tree since the value is nowin its sorted position.
57 87
23 44
12 15
The sorted listThe tree
Heap Sort
19 23 44 12 15
0 1 2 3 4 5 6
19
Reform the heap
57 87
23 44
12 15
The sorted listThe tree
Heap Sort
19 23 44 12 15
0 1 2 3 4 5 6
19
Find the largest child of the current “root”.Exchange the values.
57 87
23 44
12 15
The sorted listThe tree
Heap Sort
44 23 19 12 15
0 1 2 3 4 5 6
44
Since 19 has no children, we now have a heap again.
57 87
23 19
12 15
The sorted listThe tree
Heap Sort
44 23 19 12 15
0 1 2 3 4 5 6
44
Swap the root with the last position in the tree.
57 87
23 19
12 15
The sorted listThe tree
Heap Sort
15 23 19 12 44
0 1 2 3 4 5 6
15
Drop the last value from the tree.
57 87
23 19
12
The sorted listThe tree
44
Heap Sort
15 23 19 12 44
0 1 2 3 4 5 6
15
Drop the last value from the tree.
57 87
23 19
12
The sorted listThe tree
Heap Sort
15 23 19 12 44
0 1 2 3 4 5 6
15
Reform the heap by exchanging the root with its largest child
57 87
23 19
12
The sorted listThe tree
Heap Sort
23 15 19 12 44
0 1 2 3 4 5 6
23
Reform the heap by exchanging the root with its largest child
57 87
15 19
12
The sorted listThe tree
Heap Sort
23 15 19 12 44
0 1 2 3 4 5 6
23
Exchange the root with the last value in the tree
57 87
15 19
12
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Exchange the root with the last value in the tree
57 87
15 19
23
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
15 19
23
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
15 19
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Reform the heap
57 87
15 19
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Exchange the root with the largest child
57 87
15 19
The sorted listThe tree
Heap Sort
19 15 12 23 44
0 1 2 3 4 5 6
19
We have a heap
57 87
15 12
The sorted listThe tree
Heap Sort
19 15 12 23 44
0 1 2 3 4 5 6
19
Exchange the root with the last position in the tree
57 87
15 12
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Exchange the root with the last position in the tree
57 87
15 19The sorted list
The tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
15 19The sorted list
The tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
15
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Reform the heap
57 87
15
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Exchange the root with the largest child
57 87
15
The sorted listThe tree
Heap Sort
15 12 19 23 44
0 1 2 3 4 5 6
15
We have a heap
57 87
12
The sorted listThe tree
Heap Sort
15 12 19 23 44
0 1 2 3 4 5 6
15
Exchange the root with the last value in the tree
57 87
12
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Exchange the root with the last value in the tree
57 87
15
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
15
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
Drop the last value from the tree
57 87
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
12
The tree consists of a single value at this point which is in its proper place within the sorted list.
57 87
The sorted listThe tree
Heap Sort
12 15 19 23 44
0 1 2 3 4 5 6
The tree consists of a single value at this point which is in its proper place within the sorted list.The array is sorted.
57 87
The sorted listThe tree
