Sorting

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G BHARATHKUMAR, AssistantProfessor,CSEDepartment
MODULE-4
Objectives of Sorting:
The objective of the sorting algorithm is to rearrange the records so that their keys are
ordered according to some well-defined ordering rule.
Problem: Given an array of n real number A[1.. n].
Objective: Sort the elements of A in ascending order of their values.
Internal Sort
If the file to be sorted will fit into memory or equivalently if it will fit into an array, then the
sorting method is called internal. In this method, any record can be accessed easily.
External Sort
 Sorting files from tape or disk.
 In this method, an external sort algorithm must access records sequentially, or at least
in the block.
Memory Requirement
1. Sort in place and use no extra memory except perhaps for a small stack or table.
2. Algorithm that use a linked-list representation and so use N extra words of memory
for list pointers.
3. Algorithms that need enough extra memory space to hold another copy of the array to
be sorted.
Stability
A sorting algorithm is called stable if it is preserves the relative order of equal keys in the
file. Most of the simple algorithm are stable, but most of the well-known sophisticated
algorithms are not.
There are two classes of sorting algorithms namely, O(n2)-algorithms and O(n log n)-
algorithms. O(n2)-class includes bubble sort, insertion sort, selection sort and shell sort. O(n
log n)-class includes heap sort, merge sort and quick sort.
O(n2) Sorting Algorithms

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O(n log n) Sorting Algorithms
SORTING PROPERTIES
Sorting Properties
Property Description
Adaptive A sort is adaptive if it runs faster on a partially sorted array.
Stable A sort is stable if it preserves the relative order of equal keys in the
database.
In Situ An in situ (“in place”) sort moves the items within the array itself and, thus,
requires only a small O(1) amount of extra storage.
Online An online sort can process its data piece-by-piece in serial fashion without
having the entire array available from the beginning of the algorithm.

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Properties Of Sorting Algorithms
Adaptive Stable In Situ Online
Linear Insertion Yes Yes Yes Yes
Mergesort No Yes No Yes
Quicksort No† No Yes No
†Quicksort actually runs more slowly on a partially sorted array.
Runtime Properties Of Sorting Algorithms
Linear Insertion o Average case Ο(n2)
o Worst-case Ο(n2)
o Runs in O(n) time on a sorted array
Mergesort o Average case Ο(n lg n)
o Worst-case Ο(n lg n)
o Runtime is not affected by the array contents, only the array size
Quicksort o Average case Ο(n lg n)
o Worst-case Ο(n2) on a sorted array
o Median-of-three partitioning guarantees Ο(n lg n) runtime
Choosing a Sorting Algorithm
To choose a sorting algorithm for a particular problem, consider the running time, space
complexity, and the expected format of the input list.
Algorithm Best-case Worst-case Average-case Space Complexity Stable?
Merge Sort O(n log
n)O(nlogn)
O(n log
n)O(nlogn)
O(n log
n)O(nlogn)
O(n)O(n) Yes
Insertion
Sort
O(n)O(n) O(n^2)O(n2) O(n^2)O(n2) O(1)O(1) Yes

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Bubble Sort O(n)O(n) O(n^2)O(n2) O(n^2)O(n2) O(1)O(1) Yes
Quicksort O(n log
n)O(nlogn)
O(n^2)O(n2) O(n log
n)O(nlogn)
log
nlogn best, nn avg
Usuall
y not*
Heapsort O(n log
n)O(nlogn)
O(n log
n)O(nlogn)
O(n log
n)O(nlogn)
O(1)O(1) No
Counting
Sort
O(k+n)O(k+n) O(k+n)O(k+n) O(k+n)O(k+n) O(k+n)O(k+n) Yes
*Most quicksort implementations are not stable, though stable implementations do exist.
When choosing a sorting algorithm to use, weigh these factors. For example, quicksort is a
very fast algorithm but can be pretty tricky to implement; bubble sort is a slow algorithm but
is very easy to implement. To sort small sets of data, bubble sort may be a better option since
it can be implemented quickly, but for larger datasets, the speedup from quicksort might be
worth the trouble implementing the algorithm.
1. Bubble sort
It is also called as exchange sort. In this sort the comparison of adjacent elements is done whenever
the 1st
element is greater than the 2nd
element. Then the swapping will be done. For any kind of sorting
if we have n number elements there will be n-1 iterations. The efficiency of bubble sort is O(n).
Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent
elements if they are in wrong order.
Example:
First Pass:
( 5 1 4 2 8 ) –> ( 1 5 4 2 8 ), Here, algorithm compares the first two elements, and swaps since 5 >
1.
( 1 5 4 2 8 ) –> ( 1 4 5 2 8 ), Swap since 5 > 4
( 1 4 5 2 8 ) –> ( 1 4 2 5 8 ), Swap since 5 > 2
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 ), Now, since these elements are already in order (8 > 5), algorithm does
not swap them.
Second Pass:
( 1 4 2 5 8 ) –> ( 1 4 2 5 8 )
( 1 4 2 5 8 ) –> ( 1 2 4 5 8 ), Swap since 4 > 2

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( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
Now, the array is already sorted, but our algorithm does not know if it is completed. The algorithm
needs one whole pass without any swap to know it is sorted.
Third Pass:
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
( 1 2 4 5 8 ) –> ( 1 2 4 5 8 )
// C program for implementation of Bubble sort
#include<stdio.h>
#include<conio.h>
int main()
{
int n,i,j,a[100],t;
printf("Enter number of elements: "); -
scanf("%d",&n);
printf("Enter the elements into an array: ");
for(i=0;i<n;i++)
scanf("%d",&a[i]);
for(i=0;i<n;i++)
{
for(j=0;j<n-i-1;j++)
{
if(a[j]>a[j+1])
{
t=a[j];
a[j]=a[j+1];
a[j+1]=t;
}
}
}
printf("After performed the bubble sort the sorted array is : ");
for(i=0;i<n;i++)
printf("%dt",a[i]);
}
Worst and Average Case Time Complexity: O(n*n). Worst case occurs when array is reverse
sorted.
Best Case Time Complexity: O(n). Best case occurs when array is already sorted.
Auxiliary Space: O(1)
Boundary Cases: Bubble sort takes minimum time (Order of n) when elements are already sorted.
Sorting In Place: Yes

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Stable: Yes
Due to its simplicity, bubble sort is often used to introduce the concept of a sorting algorithm.
In computer graphics it is popular for its capability to detect a very small error (like swap of just
two elements) in almost-sorted arrays and fix it with just linear complexity (2n). For example, it is
used in a polygon filling algorithm, where bounding lines are sorted by their x coordinate at a
specific scan line (a line parallel to x axis) and with incrementing y their order changes (two
elements are swapped) only at intersections of two lines
Recurrence form of Bubble-Sort
T(n)=T(n-1) + n
2. Selection Sort:
 Selection sort is a simple sorting algorithm. This sorting algorithm is an in-place
comparison-based algorithm in which the list is divided into two parts, the sorted part
at the left end and the unsorted part at the right end. Initially, the sorted part is empty
and the unsorted part is the entire list.
 The smallest element is selected from the unsorted array and swapped with the
leftmost element, and that element becomes a part of the sorted array. This process
continues moving unsorted array boundary by one element to the right.
 This algorithm is not suitable for large data sets as its average and worst case
complexities are of Ο(n2), where n is the number of items.
Algorithm for Selection Sort:
Step 1 − Set min to the first location
Step 2 − Search the minimum element in the array
Step 3 – swap the first location with the minimum value in the array
Step 4 – assign the second element as min.
Step 5 − Repeat the process until we get a sorted array.
Source Code:
#include<stdio.h>
#include<conio.h>
int main()
{
int n,i,a[100],j,min,t;
printf("enter the number of elements: ");
scanf("%d",&n);
printf("enter the elements into an array: ");

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for(i=0;i<n;i++)
scanf("%d",&a[i]);
for(i=0;i<n-1;i++)
{
min=i;
for(j=i+1;j<n;j++)
{
if(a[min]>a[j])
{
min=j;
}
}
t=a[i];
a[i]=a[min];
a[min]=t;
}
printf("after performed the selection sort,the sort list aren");
for(i=0;i<n;i++)
printf("%dn",a[i]);
return 0;
}

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How Selection Sort Works?
Consider the following depicted array as an example.
For the first position in the sorted list, the whole list is scanned sequentially. The first
position where 14 is stored presently, we search the whole list and find that 10 is the lowest
value.
So we replace 14 with 10. After one iteration 10, which happens to be the minimum value in
the list, appears in the first position of the sorted list.
For the second position, where 33 is residing, we start scanning the rest of the list in a linear
manner.
We find that 14 is the second lowest value in the list and it should appear at the second
place. We swap these values.
After two iterations, two least values are positioned at the beginning in a sorted manner.
The same process is applied to the rest of the items in the array.Following is a pictorial
depiction of the entire sorting process

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recurrence relation of selection sort
T(n)=T(n-1) + n-1
3. Insertion sort
Insertion sort is a simple sorting algorithm that works similar to the way you sort
playing cards in your hands. The array is virtually split into a sorted and an unsorted part.
Values from the unsorted part are picked and placed at the correct position in the sorted
part.
Example:
How Insertion Sort Works?
We take an unsorted array for our example.
Insertion sort compares the first two elements.
It finds that both 14 and 33 are already in ascending order. For now, 14 is in sorted sub-list.

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Insertion sort moves ahead and compares 33 with 27.
And finds that 33 is not in the correct position.
It swaps 33 with 27. It also checks with all the elements of sorted sub-list. Here we see that
the sorted sub-list has only one element 14, and 27 is greater than 14. Hence, the sorted sub-
list remains sorted after swapping.
By now we have 14 and 27 in the sorted sub-list. Next, it compares 33 with 10.
These values are not in a sorted order.
So we swap them.
However, swapping makes 27 and 10 unsorted.
Hence, we swap them too.
Again we find 14 and 10 in an unsorted order.

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We swap them again. By the end of third iteration, we have a sorted sub-list of 4 items.
This process goes on until all the unsorted values are covered in a sorted sub-list. Now we
shall see some programming aspects of insertion sort.
Example 2:
Example 3:
12, 11, 13, 5, 6
Let us loop for i = 1 (second element of the array) to 4 (last element of the array)
i = 1. Since 11 is smaller than 12, move 12 and insert 11 before 12
11, 12, 13, 5, 6
i = 2. 13 will remain at its position as all elements in A[0..I-1] are smaller than 13
11, 12, 13, 5, 6
i = 3. 5 will move to the beginning and all other elements from 11 to 13 will move one
position ahead of their current position.
5, 11, 12, 13, 6
i = 4. 6 will move to position after 5, and elements from 11 to 13 will move one position
ahead of their current position.
5, 6, 11, 12, 13
Algorithm:
Step 1 − If it is the first element, it is already sorted. return 1;
Step 2 − Pick next element

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Step 3 − Compare with all elements in the sorted sub-list
Step 4 − Shift all the elements in the sorted sub-list that is greater than the value to be sorted
Step 5 − Insert the value
Step 6 − Repeat until list is sorted
#include<stdio.h>
#include<conio.h>
int main()
{
int n,i,j,arr[100],t,key;
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&arr[i]);
for (i = 1; i < n; i++) {
key = arr[i];
j = i - 1;
/* Move elements of arr[0..i-1], that are greater than key, to one position ahead
of their current position */
while (j >= 0 && arr[j] > key) {
arr[j + 1] = arr[j];
j = j - 1;
}
arr[j + 1] = key;
}
printf("After performed the insertion sort the sorted array is : ");
for(i=0;i<n;i++)
printf("%dt",arr[i]);
}
4. Shell Sort
Shell sort is a highly efficient sorting algorithm and is based on insertion sort algorithm.
This algorithm avoids large shifts as in case of insertion sort, if the smaller value is to the
far right and has to be moved to the far left.
This algorithm uses insertion sort on a widely spread elements, first to sort them and then
sorts the less widely spaced elements.

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ShellSort is mainly a variation of Insertion Sort. In insertion sort, we move elements
only one position ahead. When an element has to be moved far ahead, many movements are
involved. The idea of shellSort is to allow exchange of far items.
Insertion sort does not perform well when the close elements are far apart. Shell sort
helps in reducing the distance between the close elements. Thus, there will be less number of
swappings to be performed.
How Shell Sort Works?
1. Suppose, we need to sort the following array.
Initial array
2. We are using the shell's original sequence (N/2, N/4, ...1) as intervals in
our algorithm.
In the first loop, if the array size is N = 8 then, the elements lying at the
interval of N/2 = 4 are compared and swapped if they are not in order.
a. The 0th element is compared with the 4th element.
b. If the 0th element is greater than the 4th one then, the 4th element is first
stored in temp variable and the 0th element (ie. greater element) is stored in
the 4th position and the element stored in temp is stored in the 0th position.
Rearrange the elements at n/2 interval

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This process goes on for all the remaining elements.
Rearrange all the elements at n/2 interval
3. In the second loop, an interval of N/4 = 8/4 = 2 is taken and again the
elements lying at these intervals are sorted.
Rearrange
the elements at n/4 interval
You might get confused at this point.
All the
elements in the array lying at the current interval are compared.
The elements at 4th and 2nd position are compared. The elements
at 2nd and 0th position are also compared. All the elements in the array
lying at the current interval are compared.
4. The same process goes on for remaining elements.
Rearrange all the elements at n/4 interval

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5. Finally, when the interval is N/8 = 8/8 =1 then the array elements lying at the
interval of 1 are sorted. The array is now completely sorted.
Rearrange the elements at n/8 interval
Shell Sort Algorithm
for (interval = n / 2; interval > 0; interval /= 2) {
for (i = interval; i < n; i += 1) {
int temp = a[i];
int j;
for (j = i; j >= interval && a[j - interval] > temp; j -= interval) {
a[j] = a[j - interval];
}

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a[j] = temp;
} }
Source Code:
#include<stdio.h>
#include<conio.h>
int main()
{ int n,i,j,a[100],t, interval;
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&a[i]);
for (interval = n / 2; interval > 0; interval /= 2) {
for (i = interval; i < n; i += 1) {
int temp = a[i];
int j;
for (j = i; j >= interval && a[j - interval] > temp; j -= interval) {
a[j] = a[j - interval];
}
a[j] = temp;
}
}
printf("After performed the bubble sort the sorted array is : ");
for(i=0;i<n;i++)
printf("%dt",a[i]);
}

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Time Complexity
 Worst Case Complexity: less than or equal to O(n2
)
Worst case complexity for shell sort is always less than or equal to O(n2
).
According to Poonen Theorem, worst case complexity for shell sort
is Θ(Nlog N)2
/(log log N)2
) or Θ(Nlog N)2
/log log N) or Θ(N(log N)2
) or
something in between.
 Best Case Complexity: O(n*log n)
When the array is already sorted, the total number of comparisons for each
interval (or increment) is equal to the size of the array.
 Average Case Complexity: O(n*log n)
It is around O(n1.25
).
Example 2:

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5. Heap Sort
Heap sort is a comparison based sorting technique based on Binary Heap data structure. It
is similar to selection sort where we first find the maximum element and place the
maximum element at the end. We repeat the same process for the remaining elements.
What is Binary Heap?
Let us first define a Complete Binary Tree. A complete binary tree is a binary tree in which
every level, except possibly the last, is completely filled, and all nodes are as far left as
possible (Source Wikipedia)
A Binary Heap is a Complete Binary Tree where items are stored in a special order such
that value in a parent node is greater(or smaller) than the values in its two children nodes.
The former is called as max heap and the latter is called min-heap. The heap can be
represented by a binary tree or array.
Why array based representation for Binary Heap?
Since a Binary Heap is a Complete Binary Tree, it can be easily represented as an array and
the array-based representation is space-efficient. If the parent node is stored at index I, the
left child can be calculated by 2 * I + 1 and right child by 2 * I + 2 (assuming the indexing
starts at 0).
Heap Sort Algorithm for sorting in increasing order:
1. Build a max heap from the input data.
2. At this point, the largest item is stored at the root of the heap. Replace it with the last
item of the heap followed by reducing the size of heap by 1. Finally, heapify the root of the
tree.
3. Repeat step 2 while size of heap is greater than 1.
How to build the heap?
Heapify procedure can be applied to a node only if its children nodes are heapified. So the
heapification must be performed in the bottom-up order.
Lets understand with the help of an example:
Input data: 4, 10, 3, 5, 1
4(0)
/
10(1) 3(2)
/
5(3) 1(4)
The numbers in bracket represent the indices in the array
representation of data.

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Applying heapify procedure to index 1:
4(0)
/
10(1) 3(2)
/
5(3) 1(4)
Applying heapify procedure to index 0:
10(0)
/
5(1) 3(2)
/
4(3) 1(4)
The heapify procedure calls itself recursively to build heap
in top down manner.
Source Code:
#include <stdio.h>
// Function to swap the the position of two elements
void swap(int *a, int *b) {
int temp = *a;
*a = *b;
*b = temp;
}
void heapify(int arr[], int n, int i) {
// Find largest among root, left child and right child
int largest = i;
int left = 2 * i + 1;
int right = 2 * i + 2;

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if (left < n && arr[left] > arr[largest])
largest = left;
if (right < n && arr[right] > arr[largest])
largest = right;
// Swap and continue heapifying if root is not largest
if (largest != i) {
swap(&arr[i], &arr[largest]);
heapify(arr, n, largest);
}
}
// Main function to do heap sort
void heapSort(int arr[], int n) {
// Build max heap
int i;
for (i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
// Heap sort
for (i = n - 1; i >= 0; i--) {
swap(&arr[0], &arr[i]);
// Heapify root element to get highest element at root again
heapify(arr, i, 0);
}
}
// Print an array
void printArray(int arr[], int n) {
int i;

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for (i = 0; i < n; ++i)
printf("%d ", arr[i]);
printf("n");
}
// Driver code
int main() {
// int arr[] = {1, 12, 9, 5, 6, 10};
//int n = sizeof(arr) / sizeof(arr[0]);
int n,i,a[100];
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&a[i]);
heapSort(a, n);
printf("Sorted array is n");
printArray(a, n);
}

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Relationship between Array Indexes and Tree
Elements
A complete binary tree has an interesting property that we can use to find
the children and parents of any node.
If the index of any element in the array is i, the element in the
index 2i+1 will become the left child and element in 2i+2 index will become
the right child. Also, the parent of any element at index i is given by the
lower bound of (i-1)/2.
Relationship
betweenarrayandheapindices
Let's test it out,
Left child of 1 (index 0)
= element in (2*0+1) index
= element in 1 index
= 12
Right child of 1
= 9

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Similarly,
Left child of 12 (index 1)
= 5
Right child of 12
= 6
Let us also confirm that the rules hold for finding parent of any node
Parent of 9 (position 2)
= (2-1)/2
= ½
= 0.5
~ 0 index
= 1
Parent of 12 (position 1)
= (1-1)/2
= 0 index
= 1
Now let's think of another scenario in which there is more than one level.

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How to heapify root element when its subtrees are
already max heaps
The top element isn't a max-heap but all the sub-trees are max-heaps.
To maintain the max-heap property for the entire tree, we will have to keep
pushing 2 downwards until it reaches its correct position.
How to heapify
root element when its subtrees are max-heaps

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Thus, to maintain the max-heap property in a tree where both sub-trees are
max-heaps, we need to run heapify on the root element repeatedly until it is
larger than its children or it becomes a leaf node.
We can combine both these conditions in one heapify function as
void heapify(int arr[], int n, int i) {
// Find largest among root, left child and right child
int largest = i;
int left = 2 * i + 1;
int right = 2 * i + 2;
if (left < n && arr[left] > arr[largest])
largest = left;
if (right < n && arr[right] > arr[largest])
largest = right;
// Swap and continue heapifying if root is not largest
if (largest != i) {
swap(&arr[i], &arr[largest]);
heapify(arr, n, largest);
}
}
This function works for both the base case and for a tree of any size. We
can thus move the root element to the correct position to maintain the max-
heap status for any tree size as long as the sub-trees are max-heaps.
Build max-heap
To build a max-heap from any tree, we can thus start heapifying each sub-
tree from the bottom up and end up with a max-heap after the function is
applied to all the elements including the root element.

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In the case of a complete tree, the first index of a non-leaf node is given
by n/2 - 1. All other nodes after that are leaf-nodes and thus don't need to
be heapified.
So, we can build a maximum heap as
// Build heap (rearrange array)
for (int i = n / 2 - 1; i >= 0; i--)
heapify(arr, n, i);
Create array and calculate i

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Steps to build max heap for heap sort

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As shown in the above diagram, we start by heapifying the lowest smallest
trees and gradually move up until we reach the root element.
If you've understood everything till here, congratulations, you are on your
way to mastering the Heap sort.

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How Heap Sort Works?
1. Since the tree satisfies Max-Heap property, then the largest item is stored
at the root node.
2. Swap: Remove the root element and put at the end of the array (nth
position) Put the last item of the tree (heap) at the vacant place.
3. Remove: Reduce the size of the heap by 1.
4. Heapify: Heapify the root element again so that we have the highest
element at root.
5. The process is repeated until all the items of the list are sorted.

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Swap,

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Remove, and Heapify
The code below shows the operation.
// Heap sort
for (int i = n - 1; i >= 0; i--) {
swap(&arr[0], &arr[i]);
// Heapify root element to get highest element at root again
heapify(arr, i, 0);
}
Heap Sort Complexity
Heap Sort has O(nlog n) time complexities for all the cases ( best case, average case, and
worst case).
Let us understand the reason why. The height of a complete binary tree containing n elements
is log n
6. Merge Sort
Merge sort is a sorting technique based on divide and conquer technique. With worst-
case time complexity being Ο(n log n), it is one of the most respected algorithms.
Merge sort first divides the array into equal halves and then combines them in a sorted
manner.
How Merge Sort Works?
To understand merge sort, we take an unsorted array as the following −
We know that merge sort first divides the whole array iteratively into equal halves unless
the atomic values are achieved. We see here that an array of 8 items is divided into two
arrays of size 4.
This does not change the sequence of appearance of items in the original. Now we divide
these two arrays into halves.

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We further divide these arrays and we achieve atomic value which can no more be
divided.
Now, we combine them in exactly the same manner as they were broken down. Please
note the color codes given to these lists.
We first compare the element for each list and then combine them into another list in a
sorted manner. We see that 14 and 33 are in sorted positions. We compare 27 and 10 and
in the target list of 2 values we put 10 first, followed by 27. We change the order of 19
and 35 whereas 42 and 44 are placed sequentially.
In the next iteration of the combining phase, we compare lists of two data values, and
merge them into a list of found data values placing all in a sorted order.
After the final merging, the list should look like this −
Now we should learn some programming aspects of merge sorting.
Algorithm
Merge sort keeps on dividing the list into equal halves until it can no more be divided. By
definition, if it is only one element in the list, it is sorted. Then, merge sort combines the
smaller sorted lists keeping the new list sorted too.
Step 1 − if it is only one element in the list it is already sorted, return.
Step 2 − divide the list recursively into two halves until it can no more be divided.
Step 3 − merge the smaller lists into new list in sorted order.
Pseudocode
MergeSort(arr[], l, r)
If r > l
1. Find the middle point to divide the array into two halves:
middle m = l+ (r-l)/2
2. Call mergeSort for first half:
Call mergeSort(arr, l, m)

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3. Call mergeSort for second half:
Call mergeSort(arr, m+1, r)
4. Merge the two halves sorted in step 2 and 3:
Call merge(arr, l, m, r)
void mergeSort(int a[], int start, int end)
{
int mid;
if(start < end)
{
mid = (start + end) / 2;
mergeSort(a, start, mid);
mergeSort(a, mid+1, end);
merge(a, start, mid, end);
}
}

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Source Code:
#include <stdio.h>
// lets take a[5] = {32, 45, 67, 2, 7} as the array to be sorted.
// merge sort function
void mergeSort(int a[], int start, int end)
{
int mid;
if(start < end)
{
mid = (start + end) / 2;
mergeSort(a, start, mid);
mergeSort(a, mid+1, end);
merge(a, start, mid, end);
}
}
// function to merge the subarrays
void merge(int a[], int start, int mid, int end)
{
int b[100]; //same size of a[]
int i, j, k;
k = 0;
i = start;
j = mid + 1;
while(i <= mid && j <= end)
{
if(a[i] < a[j])
{
b[k++] = a[i++]; // same as b[k]=a[i]; k++; i++;
}
else
{
b[k++] = a[j++];
}
}
while(i <= mid)
{

36
b[k++] = a[i++];
}
while(j <= end)
{
b[k++] = a[j++];
}
for(i=end; i >= start; i--)
{
a[i] = b[--k]; // copying back the sorted list to a[]
}
}
// function to print the array
void printArray(int a[], int size)
{
int i;
for (i=0; i < size; i++)
{
printf("%d ", a[i]);
}
printf("n");
}
int main()
{
int n,i,j,a[100],t;
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&a[i]);
// calling merge sort
mergeSort(a, 0, n - 1);
printf("nSorted array: n");
printArray(a, n);
return 0;
}

37
The merge function works as follows:
1. Create copies of the subarrays L ← A[p..q] and M ← A[q+1..r].
2. Create three pointers i, j and k
a. i maintains current index of L, starting at 1
b. j maintains current index of M, starting at 1
c. k maintains the current index of A[p..q], starting at p.
3. Until we reach the end of either L or M, pick the larger among the elements
from L and M and place them in the correct position at A[p..q]
4. When we run out of elements in either L or M, pick up the remaining elements and put
in A[p..q]
Important Characteristics of Merge Sort:
 Merge Sort is useful for sorting linked lists.
 Merge Sort is a stable sort which means that the same element in an array maintain their original
positions with respect to each other.
 Overall time complexity of Merge sort is O(nLogn). It is more efficient as it is in worst case also
the runtime is O(nlogn)
 The space complexity of Merge sort is O(n). This means that this algorithm takes a lot of space
and may slower down operations for the last data sets.
Recurrence Relation - Merge Sort
T(n) = 2T(n/2) + n

38
7. Quick Sort
Like Merge Sort, QuickSort is a Divide and Conquer algorithm. It picks an element as pivot
and partitions the given array around the picked pivot. There are many different versions of
quickSort that pick pivot in different ways.
1. Always pick first element as pivot.
2. Always pick last element as pivot (implemented below)
3. Pick a random element as pivot.
4. Pick median as pivot.
The key process in quickSort is partition(). Target of partitions is, given an array and an
element x of array as pivot, put x at its correct position in sorted array and put all smaller
elements (smaller than x) before x, and put all greater elements (greater than x) after x. All
this should be done in linear time.
Quick Sort Algorithm
Using pivot algorithm recursively, we end up with smaller possible partitions. Each
partition is then processed for quick sort. We define recursive algorithm for quicksort
as follows −
Step 1 − Make the right-most index value pivot
Step 2 − partition the array using pivot value
Step 3 − quicksort left partition recursively
Step 4 − quicksort right partition recursively
Quick Sort Pseudocode
To get more into it, let see the pseudocode for quick sort algorithm −
procedure quickSort(left, right)
if right-left <= 0
return
else

39
pivot = A[right]
partition = partitionFunc(left, right, pivot)
quickSort(left,partition-1)
quickSort(partition+1,right)
end if
end procedure
Source Code:
#include<stdio.h>
#include<conio.h>
int a[10];
void main()
{
int a[];
int i,n,l,u;
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&a[i]);
qsort(a,0,n-1);
printf("Elements After QuickSort:n");
for(i=0;i<n;i++)
printf("%d",a[i]);
getch();
}
qsort(int a[],int l,int u)
{
int i,j,temp,k;
i=l+1;

40
j=u;
k=l;
if(l<u)
{
while(i<=j)
{
while(a[i]<a[k])
i++;
while(a[j]>a[k])
j--;
if(i<j)
{
temp=a[i];
a[i]=a[j];
a[j]=temp;
}
}
temp=a[j];
a[j]=a[k]; //pivot and j element are swapped
a[k]=temp;
qsort(a,l,j-1);
qsort(a,j+1,u);
}
}
Source Code:
#include<stdio.h>
#include<conio.h>
int a[10];

41
void swap(int* a, int* b)
{
int t = *a;
*a = *b;
*b = t;
}
void main()
{
int a[100];
int i,n,l,u;
scanf("%d",&n);
for(i=0;i<n;i++)
scanf("%d",&a[i]);
qsort(a,0,n-1);
printf("Elements After QuickSort:n");
for(i=0;i<n;i++)
printf("%d",a[i]);
getch();
}
qsort(int a[],int l,int u)
{
int i,j,temp,k;
i=l+1;
j=u;
k=l;
if(l<u)
{
while(i<=j)
{
while(a[i]<a[k])
i++;
while(a[j]>a[k])
j--;
if(i<j)
{
swap(&a[i],&a[j]);
}
}
swap(&a[j],&a[k]); //pivot and j element are swapped
qsort(a,l,j-1);

42
qsort(a,j+1,u);
}
}
How QuickSort Works?
1. A pivot element is chosen from the array. You can choose any element
from the array as the pivot element.
Here, we have taken the rightmost (ie. the last element) of the array as the
pivot element.
Select a pivot element
2. The elements smaller than the pivot element are put on the left and the
elements greater than the pivot element are put on the right.
Put all the
smaller elements on the left and greater on the right of pivot element
The above arrangement is achieved by the following steps.
a. A pointer is fixed at the pivot element. The pivot element is compared with
the elements beginning from the first index. If the element greater than the
pivot element is reached, a second pointer is set for that element.
b. Now, the pivot element is compared with the other elements (a third
pointer). If an element smaller than the pivot element is reached, the
smaller element is swapped with the greater element found earlier.

43
Comparison
of pivot element with other elements
c. The process goes on until the second last element is reached.
Finally, the pivot element is swapped with the second pointer.

44
Swap pivot
element with the second pointer
d. Now the left and right subparts of this pivot element are taken for further
processing in the steps below.
3. Pivot elements are again chosen for the left and the right sub-parts
separately. Within these sub-parts, the pivot elements are placed at their
right position. Then, step 2 is repeated.

45
Select pivot
element of in each half and put at correct place using recursion
4. The sub-parts are again divided into smaller sub-parts until each subpart is
formed of a single element.
5. At this point, the array is already sorted.
Quicksort uses recursion for sorting the sub-parts.
On the basis of Divide and conquer approach, quicksort algorithm can be explained
as:
 Divide
The array is divided into subparts taking pivot as the partitioning point. The
elements smaller than the pivot are placed to the left of the pivot and the
elements greater than the pivot are placed to the right.
 Conquer
The left and the right subparts are again partitioned using the by selecting pivot
elements for them. This can be achieved by recursively passing the subparts into
the algorithm.
 Combine
This step does not play a significant role in quicksort. The array is already sorted
at the end of the conquer step.

46
You can understand the working of quicksort with the help of the illustrations
below.
Sorting the elements on the left of pivot using recursion
Recurrence Relation
Assume we constructed a quicksort and the pivot value takes linear time. Find the
recurrence for worst-case running time.
My answer: T(n)= T(n-1) + T(1) + theta(n)
Worst case occurs when the subarrays are completely unbalanced. There is 1 element
in one subarray and (n-1) elements in the other subarray. theta(n) because it takes
running time n to find the pivot.
Best case
The recurrence relation for quicksort is:
T(n)=2T(n/2)+O(n)

47
Merge Two Arrays in C
#include<stdio.h>
#include<conio.h>
int main()
{
int arr1[50], arr2[50], size1, size2, i, k, merge[100];
printf("Enter Array 1 Size: ");
scanf("%d", &size1);
printf("Enter Array 1 Elements: ");
for(i=0; i<size1; i++)
{
scanf("%d", &arr1[i]);
merge[i] = arr1[i];
}
k = i;
printf("nEnter Array 2 Size: ");
scanf("%d", &size2);
printf("Enter Array 2 Elements: ");
for(i=0; i<size2; i++)
{
scanf("%d", &arr2[i]);
merge[k] = arr2[i];
k++;
}
printf("nThe new array after merging is:n");
for(i=0; i<k; i++)
printf("%d ", merge[i]);
getch();
return 0;
}

48
Radix Sort:
#include<stdio.h>
int get_max (int a[], int n){
int i, max = a[0];
for (i = 1; i < n; i++)
if (a[i] > max)
max = a[i];
return max;
}
void radix_sort (int a[], int n){
int bucket[10][10], bucket_cnt[10];
int i, j, k, r, NOP = 0, divisor = 1, lar, pass;
lar = get_max (a, n);
while (lar > 0){
NOP++;
lar /= 10;
}
for (pass = 0; pass < NOP; pass++){
for (i = 0; i < 10; i++){
bucket_cnt[i] = 0;
}
for (i = 0; i < n; i++){
r = (a[i] / divisor) % 10;
bucket[r][bucket_cnt[r]] = a[i];
bucket_cnt[r] += 1;
}
i = 0;
for (k = 0; k < 10; k++){
for (j = 0; j < bucket_cnt[k]; j++){

49
a[i] = bucket[k][j];
i++;
}
}
divisor *= 10;
printf ("After pass %d : ", pass + 1);
for (i = 0; i < n; i++)
printf ("%d ", a[i]);
printf ("n");
}
}
int main (){
int i, n, a[10];
printf ("Enter the number of items to be sorted: ");
scanf ("%d", &n);
printf ("Enter items: ");
for (i = 0; i < n; i++){
scanf ("%d", &a[i]);
}
radix_sort (a, n);
printf ("Sorted items : ");
for (i = 0; i < n; i++)
printf ("%d ", a[i]);
printf ("n");
return 0;
}

50

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