Sorting.ppt read only

Sorting
-ByVarsha kumari
GLA University - CSE 4070

Sorting : Outline
Bubble Sort
Insertion Sort
Selection Sort
Quick Sort
Two-way Merge sort
Heap Sort

Bubble Sort
It sort the array elements by repeatedly moving
the largest element to the highest index position
of the array[for ascending order].
Consecutive adjacent pair of elements in the
array are compared with each other.
Two elements are interchanged if lower index
value is greater than upper index value.
Process is continued till list of unsorted elements
exhausted.
It is called bubble sort because larger elements
‘bubble’ to the end of the list.

Algorithm
Bubble_sort(a[ ], n)
{
Repeat for i = 0 to n-1
Repeat for j = 0 to n-i-2
if(a[j]>a[j+1])
swap a[j] and a[j+1]
end if
end for
end for
}

Bubble Sort : 1st
pass
30 < 52 no swapping
52 >25 swap 52 and 25
52 < 84 no swapping
84 > 65 swap 84 and 65
Max value 84 moved to
last index

2nd
pass
30 > 25 swap 30 and 25
30 < 52 no swapping
52 < 65 no swapping
2nd
Max value 65 moved
to 2nd
last index

Insertion Sort
Algorithm works by moving the current
data element past the already sorted
values and repeatedly interchanging it
with preceding value until it is it’s correct
place.
less efficient when compared with other
algorithms such as quick sort, heap sort
and merge sort.

Algorithm
Insertion_sort(a[ ], n)
{
set temp = a[i]
set j=i-1
Repeat while temp< a[j] and j>=0
set a[j+1]= a[j]
set j=j-1
end while
set a[j+1] = temp
end for
}

Insertion Sort

Selection Sort
It takes n-1 passes to sort the entire
array and works as follows:
◦ Find smallest value in the array and place it in
it’s position.
◦ Repeat this process untill array is sorted.

Algorithm
Selection_sort(a[ ], n)
{
Repeat for j = i+1 to n-1
if(a[i]>a[j])
swap a[i] and a[j]
end if
end for
end for
}

Selection Sort : 1st
pass

2nd
pass

Quick Sort
Quick sort works s follows:
◦ Select a pivot element from the array
elements.
◦ Rearrange the array elements int the array
such that all the elements less than the pivot
element appears before the pivot and all the
elements greater than the pivot element
comes after the pivot.(this process is called
partition)
◦ Recursively sort two sub arrays thus
obtained.

Algorithm
quickSort(array, first, last)
{
pivot = partition(array, first,last)
quickSort(array,first,pivot-1)
quickSort(array,pivot+1,last)
}

Partition algorith
partition (arr[], first, last)
{
set pivot = first, i = first , j=last
Repeat while i < j
Repeat while arr[i] < arr[pivot]
set i= i+1
end while
Repeat while arr[j] > arr[pivot]
set j= j-1
end while
if i<j
swap arr[i] and arr[j]
end if
End while
Swap arr[pivot] and arr[j]
Return pivot }

quickSort(arr,0,5)
Partition Initialization...
first moves to the right until
Array element > pivot element
last moves to the left until
Array element < pivot element
swap arr[first] and arr[last]

quickSort(arr,0,5)
first moves to the right
until
Array element > pivot
element
last moves to the left
until
Array element < pivot
element
swap arr[first] and arr[last]
pivot=6
partition(arr,0,5)

quickSort(arr,0,5)
repeat right/left scan
UNTIL left & right cross
first & last CROSS!!!
Swap pivot and arr[last]
Return new
location of pivot
to quicksort
function
= return 3

Recursive calls to quickSort()

Recursive calls to quickSort()
first & llast CROSS!!!
1 - Swap pivot element and arr[last]
2 - Return new location of pivot element
to quicksort function
Return 0 GLA University - CSE 4070

Quicksort

Mergesort
It focuses on divide, conquer and
combine
Divide means partitioning the array to be
sorted into two equal sub-arrays
Conquer means sorting two sub-array
recursively
Combine means merge two sorted sub-
arrays.

Algorithm
MergeSort(arr[], l, r)
{ if r>l
set mid = (l+r)/2
Call mergeSort(arr, l, mid)
Call mergeSort(arr, mid+1, r)
Call merge(arr, l, m, r)
end if
}

Mergesort

Algorithm
Merge(arr[], l, mid, r)
{ compute n1 and n2
copy n1 elements in left[]
copy n2 elements in right[]
set left[n1]= 9999, rightt[n1]= 9999
set i=0,j=0,k=0
for k = l to r
if left[i]<right[j]
set arr[k]= left[i]
i=i+1
else arr[k]=right[j]
j=j+1
end if
end for
}

Merge
array arr[ ]

Merge

Heap
A heap is a data structure that stores a
collection of keys , and has the following
properties:
◦ Complete Binary tree
◦ Satisfies Heap property which states that
 child key value >= parent key value for min - heap
 child key value <= parent key value for max - heap
Each node of the binary tree corresponds to an
element of the array. The array is completely
filled on all levels except possibly lowest.

Heap
Max-heap :
A[Parent(i)] ≥ A[i]
Min-heap : A[Parent(i)] ≤ A[i]
PARENT (i) where i is index of any key
        return floor(i/2)
LEFT (i)
        return 2i
RIGHT (i)
        return 2i + 1

Insertion in heap
1. Add the new element to the next available position at the
lowest level
2. Restore the max-heap property if violated
 Let the new rise to its appropriate place, if the
parent of the element is smaller than the
element, then interchange the parent and child.
OR
Restore the min-heap property if violated
 Sink down the new element, if the parent of the
element is larger than the element, then
interchange the parent and child.

Insert 17 in max-heap
maintain the heap property

Delete from heap
Delete max
◦ Replace the last element with root value (delete
last element).
◦ Restore the max heap property by sinking down
the new root value.
Delete min
◦ Replace the last element with root value (delete
last element).
◦ Restore the min heap property by sinking down
the new root value.

Delete from heap
Swap 25
and 7
Swap 7 and 17
Swap 7 and 15

Heap Sort
 Heapify method picks the largest child key and compare
it to the parent key. If parent key is larger than heapify
quits, otherwise it swaps the parent key with the
largest child key. So that the parent is now becomes
larger than its children.
The heap algorithm works as:
◦ Build a heap from the given array using heapify
method
◦ Repeatedly delete the root element using max
heap(for descending order)
◦ Repeatedly delete the root element using min
heap(for ascending order)

Algorithm
Heapsort(A,n)
{
   Buildheap(A)
   for i = 0 to n-1
call Insert_heap(A,n,A[i])
End for
   Repeat while n>0
call Delete_heap(A,n,value)
set n = n – 1
end while loop
}
// heapify
// Get max value after
deletion (for
descending order)

Complexities
Sortings best case Average
case
Worst
case
Bubble sort O(n ) O(n2
) O(n2
)
Insertion sort O(n ) O(n2
) O(n2
)
Selection sort O(n2
) O(n2
) O(n2
)
Quick sort O(nlogn) O(nlogn) O(n2
)
Merge sort O(nlogn) O(nlogn) O(nlogn)
Heap sort O(nlogn) O(nlogn) O(nlogn)

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