A sorting technique that sequences a list by continuously dividing the list into two parts and moving the lower items to one side and the higher items to the other. It starts by picking one item in the entire list to serve as a pivot point.

**QUICK SORT**

A sorting technique that sequences a
list by continuously dividing the list into two parts and moving the lower
items to one side and the higher items to the other. It starts by picking one
item in the entire list to serve as a pivot point.

The pivot could be the first item or a
randomly chosen one. All items that compare lower than the pivot are moved to
the left of the pivot; all equal or higher items are moved to the right.

It then picks a pivot for the left side
and moves those items to left and right of the pivot and continues the pivot
picking and dividing until there is only one item left in the group. It then
proceeds to the right side and performs the same operation again. Also known as
**Partition**

**Exchange Sort.**

Quick
sort algorithm applies a special designing technique called Divide and Conquer,

The
*divide-and-conquer* strategy solves a problem by:

1. Breaking
it into *sub problems* that are themselves smaller instances of the same
type of problem

2. Recursively
solving these sub problems

3. Appropriately
combining their answers

Divide
and Conquer in Quick sort

**Divide**: If the
sequence S has 2 or more elements, select an element x from S to be your**
pivot**.** **Any arbitrary element, like the last, will do. Remove all the
elements of S and divide them into 3 sequences:

L, holds S’s elements less than x E, holds S’s
elements equal to x

G, holds
S’s elements greater
than x

2) **Recurse**:
Recursively sort L and G** **

3) **Conquer**:
Finally, to put elements back into S in order, first inserts the elements of L,
then** **those of E, and those of G.

**Step 1: Select**:
pick an element** **

**Step 2: Divide**:
rearrange elements so that x goes to its final position E

**Step 3: Recurse and conquer**:
recursively sort

**Quick Sort Algorithm:**

**#include<stdio.h>**

**#include<conio.h>**

**void qsort(int arr[20],
int left, int right); int main()**

**{**

**int arr[30]; int i,n;**

**printf("Enter total no. of the
elements : ");**

**scanf("%d",&n);**

**printf("Enter total %d elements :
\n",size); for(i=0; i<n; i++)**

**scanf("%d",&arr[i]);
qsort(arr,0,n-1);**

**printf("Quick sorted elements are
as : \n"); for(i=0; i<n; i++)**

**printf("%d\t",arr[i]);**

**getch(); return 0;**

**}**

**void qsort(int arr[20], int left, int
right)**

**{**

**int i,j,pivot,tmp; if(left<right)**

**{**

**pivot=left;**

**i=left+1;**

**j=right;**

**while(i<j)**

**{**

**while(arr[i]<=arr[pivot] &&
i<right) i++;**

**while(arr[j]>arr[pivot]) j--;**

**if(i<j)**

**{**

**tmp=arr[i];**

**arr[i]=arr[j];**

**arr[j]=tmp;**

**}**

**}**

**tmp=arr[pivot];**

**arr[pivot]=arr[j];**

**arr[j]=tmp; qsort(arr,left,j-1);
qsort(arr,j+1,right);**

**}**

**}**

The partition method
uses two while statements one is for incrementing the variable to arrange the
sequence in left side with least values. Another while statement for
decrementing the variable to arrange the sequence in right side with greatest
values when compared with the pivot element.

**Execution for Partition method:**

There are a number of ways to pick the
pivot element. In this example, we will use the first element in the array

Compare 20 and pivot element 40 the 20
is least value so down alone get incremented.

Again compare pivot element and element
pointing down, 10 is least value again so down value get incremented.

But Here the situation
is 80 value pointed by down index is high when compared with the pivot element.
So 1^{st} while statement get terminated.

Now, up value will start to get
decremented. The array will be as follows

Up value will get decrement if the value
pointing up is greater than pivot element, but now the situation is up pointing
element is least when compared with the pivot element. So now second while also
get terminated.

Now after the end of two while
statements if down is less than up swap these two values to arrange left sub
array with least values in the left side of the pivot element and greatest
values at the right side of the pivot element.

Continue the two whiles again and find
two values to get swapped until down < up

So now stop the iteration and swap the
pivot element towards the up position like below.

Current pivot index has been updated
with the value 4 and the pivot element also swapped to the pivot index.

These Sub arrays are
now again introduced into the partition with the above steps. The two sub
arrays are from 0 to 3 and 4 to 8. So the first sub array again partitioned
into two, depending upon pivot element position. The quick sort procedure with
the partition method is as follows.

void quicksort ( int a[ ], int left, int right )

{

int
mid;

if
( right > left )

{

mid = quicksort (a, left, right) ; quicksort ( a,
left, mid - 1 ) ; quicksort ( a, mid+ 1, right ) ;

}

}

Here the variable i is
the partition index. And accordingly the quick sort performs two partitions
each time depending upon the pivot index.

**Complexity Analysis:**

o
Recursion:

^{§ }Partition
splits array in two sub-arrays of size n/2 ^{}

^{ }

^{§ }Quick
sort each sub-array ^{}

o Depth of recursion tree
O(log_{2}n)

o Number of accesses in partition
O(n)

Assume that
keys are random, uniformly distributed.

Best case
running time: O(n log_{2}n)

The Master theorem can be applied to
calculate the efficiency. The theorem is as follows,

For the Quick sort the standard format
is T(n) = aT(n/b)+F(n)

Where a and b are Number of recursions and number of
partitions respectively.

For quick sort a=2 and b=2, so a and b are equal so
the efficiency is **θ (n
log n)**

__Selection sorts__

Selection sort
is a sorting algorithm, specifically an in-place comparison sort. It has O (n2)
time complexity, making it inefficient on large lists, and generally performs
worse than the similar insertion sort.

Selection sort
is noted for its simplicity, and also has performance advantages over more
complicated algorithms in certain situations, particularly where auxiliary
memory is limited.

Three types of
selection sorts are discussed here,

i)
Straight Selection sort

ii)
Binary tree Sort

iii)
Heap sort

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