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 last 3. 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 First Last Has First exceeded Last? No! Pivot 44 Define the value in position First to be the Pivot.

QuickSort Example FirstLast Pivot 44 Define Up To be First and Down to be last Up Down

QuickSort Example FirstLast Pivot 44 Move Up to the first value > Pivot Up Down

QuickSort Example FirstLast Pivot 44 Move Down to the first value <= Pivot Up Down

QuickSort Example FirstLast Pivot 44 If Up < Down, exchange their values Up Down

QuickSort Example FirstLast Pivot 44 Move Up to the first value > Pivot Up Down

QuickSort Example FirstLast Pivot 44 Move Down to the first value <= Pivot UpDown

QuickSort Example FirstLast Pivot 44 If Up < Down, exchange their values. UpDown

QuickSort Example FirstLast Pivot 44 Move Up to the first value > Pivot UpDown

QuickSort Example FirstLast Pivot 44 Move Down to the first value <= Pivot UpDown Up and Down have passed each other, so exchange the pivot value and the value in Down.

QuickSort Example FirstLast Pivot 44 UpDown Up and Down have passed each other, so exchange the pivot value and the value in Down.

QuickSort Example FirstLast Pivot 44 PivIndex Note that all values below PivIndex are <= Pivot and all values above PivIndex are > Pivot.

QuickSort Example First 1 Last 2 Pivot 44 PivIndex This gives us two new subarrays to Partition Last 1 First 2

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?

Heap Sort How can a tree be represented in an array? 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? 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) = 1 Place the root’s right child in element 2: (RootPos*2 + 2) = 2

Heap Sort How can a tree be represented in an array? 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* = 3 Right Child (15): ParentPos*2 + 2 = 1 *2 + 2 = 4

Heap Sort How can a tree be represented in an array? 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* = 5 Right Child (15): ParentPos*2 + 2 = 2 *2 + 2 =

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 Sort How to build a heap? Add Table[0] to tree Since it has no parent, we have a heap. 4423

Heap Sort How to build a heap? Add Table[1] to tree Since 12 < 57, it is not a heap. Bubble it up as high as it can go. Exchange

Heap Sort How to build a heap? Exchange Since 57 >12 57 is as high as it can go, so we have a heap

Heap Sort How to build a heap? Add Table[2] to tree Since 57 >19 so, we have a heap

Heap Sort How to build a heap? Add Table[3] to tree Since 87 >12 so, not a heap

Heap Sort How to build a heap? Add Table[3] to tree Since 87 >12 so, not a heap. Bubble it up. Exchange. Again 87 > 57, so not a heap. Bubble it up

Heap Sort How to build a heap? Again 87 > 57, so not a heap. Bubble it up. Exchange. We now have a heap again

Heap Sort How to build a heap? Add Table[4] to tree 15 > 57, so a heap

Heap Sort How to build a heap? Add Table[5] to tree 44 > 19, so not a heap

Heap Sort How to build a heap? > 19, so not a heap. Bubble it up. Exchange. 44<87 Again we have a heap

Heap Sort How to build a heap? Add Table[6] to tree 23 <44 so, we have a heap

Heap Sort How to build a heap? The whole table is now a heap!

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 Here is the heap!

Heap Sort Exchange the root with the last value in the tree

Heap Sort Exchange the root with the last value in the tree

Heap Sort Drop this last value from the tree -- it is now in the array in its sorted position!

Heap Sort Drop this last value from the tree -- it is now in the array in its sorted position! The sorted list The tree

Heap Sort Reform the heap The sorted list The tree

Heap Sort Find the largest child of the current “root” and exchange with “root” The sorted list The tree

Heap Sort Now 23 is larger than both of its children so we have a heap again The sorted list The tree

Heap Sort Exchange the root of the heap with the last value in the tree The sorted list The tree

Heap Sort Exchange the root of the heap with the last value in the tree The sorted list The tree

Heap Sort Drop the last element from the tree since the value is now in its sorted position The sorted list The tree

Heap Sort Drop the last element from the tree since the value is now in its sorted position The sorted list The tree

Heap Sort Reform the heap The sorted list The tree

Heap Sort Find the largest child of the current “root”. Exchange the values The sorted list The tree

Heap Sort Since 19 has no children, we now have a heap again The sorted list The tree

Heap Sort Swap the root with the last position in the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree 44

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Reform the heap by exchanging the root with its largest child The sorted list The tree

Heap Sort Reform the heap by exchanging the root with its largest child The sorted list The tree

Heap Sort Exchange the root with the last value in the tree The sorted list The tree

Heap Sort Exchange the root with the last value in the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Reform the heap The sorted list The tree

Heap Sort Exchange the root with the largest child The sorted list The tree

Heap Sort We have a heap The sorted list The tree

Heap Sort Exchange the root with the last position in the tree The sorted list The tree

Heap Sort Exchange the root with the last position in the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Reform the heap The sorted list The tree

Heap Sort Exchange the root with the largest child The sorted list The tree

Heap Sort We have a heap The sorted list The tree

Heap Sort Exchange the root with the last value in the tree The sorted list The tree

Heap Sort Exchange the root with the last value in the tree The sorted list The tree

Heap Sort Drop the last value from the tree The sorted list The tree

Heap Sort Drop the last value from the tree 5787 The sorted list The tree

Heap Sort The tree consists of a single value at this point which is in its proper place within the sorted list The sorted list The tree

Heap Sort The tree consists of a single value at this point which is in its proper place within the sorted list. The array is sorted The sorted list The tree