Quick Sort Algorithm

By -

Quick Sort:

Quick Sort algorithm is based on divide-and-conquer approach, recursively processing small and small parts of array list at each level. Here is the three-step divide-and-conquer process for sorting a typical subarray A[p....r].

Divide:  Partition(rearrange) the array A[p ...r] into two sub-arrays A[p ...q-1] and A[q+1 ...r] such that each element of A[p...q-1] is less than and or equal to A[q], which is, in turn less than and or equal to A[q+1 ..r]. Compute the index q as part of this partitioning procedure.

 Conquer: Sort the two subarray A[p ...q-1] and A[q+1 ..r] by recursive calls of quicksort.

 Combine: Subarrays are sorted in place, no work is needed to combine them.

  Quick Sort algorithm chooses an element as a pivot. And partitions the given array around choose the pivot element. There are many ways to choose a pivot element for partition.
Always choose the first element as a pivot.
Always choose the last element as a pivot. Choose a random element as a pivot. Choose median as a pivot.

The key process in quickSort is partition(). The target of partitions is, given an array and an element x of the array as a pivot, put x at its correct position in the 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.
Ex: choose the last element as a pivot.

techie delight




 Algorithm:

QUICKSORT(A, p, r)
  1. if  p < r
  2.   then q <-- PARTITION(A, p, r)
  3.         QUICKSORT(A, p, q-1)
  4.          QUICKSORT(A, q+1, r)       
 To sort an entire array A, the initial call is QUICKSORT(A, 1, length[A]).

Partition the array
the Key to the algorithm is PARTITION procedure, which rearranges the subarray A[p ...r] in place.

PARTITION(A, p, r)
  1. x <-- A[r]
  2. i <-- p-1
  3. for j <-- p to r -1
  4.    do if  A[j] <= x
  5.          then i <-- i+1
  6.                exchange A[i] <--> A[j]
  7. exchange A[i+1] <--> A[r]
  8. return i+1
T(n) = f(n) + T(q) + T(n-q-1)

T(n): Time complexity of Quicksort
T(n-q-1): Time complexity of left partition algorithm

T(q): Time complexity of right partition algorithm
f(n): Time complexity of partition algorithm

Best case: O( nlogn ) 
Worse case: O( n2 )


case 1: If element of array, already sorted in increasing order.
 ex:  1 < 2 <3 <4 <5          5 is pivot

 n-q-1 = 0      q = n-1
                                     T(n) = T(n-1) + n ==> O( n2 )
case 2: If element of array, already sorted in decreasing order.
 ex:  5< 4< 3< 2< 1         1 is pivot

 n-q-1 = n-1      q = 0
                                     T(n) = T(n-1) + n ==> O( n2 )

case 3: If Partition algorithm divides the array into two equal halves.

q=n-q-1       ==>>       q=(n-1)/2


           T(n) = T((n-1)/2) + T((n-1)/2) + n

        T(n) = 2 T((n-1)/2) + n

        T(n) = O( nlogn)



case 4: General case











      
                              

Comments

Post a Comment

Popular posts from this blog

C program that contains a string XOR each character in this string with 0 ,127

By -
Write a C program that contains a string (char pointer) with a value \Hello World’. The program should XOR each character in this string with 0 and displays the result. Program: #include<stdlib.h> main() { char str[]="Hello World"; char str1[11]; int i,len; len=strlen(str); for(i=0;i<len;i++) { str1[i]=str[i]^0; printf("%c",str1[i]); } printf("\n"); } Q.2 Write a C program that contains a string (char pointer) with a value \Hello World’. The program should AND or and XOR each character in this string with 127 and display the result. PROGRAM: #include <stdio.h> #include<stdlib.h> void main() { char str[]="Hello World"; char str1[11]; char str2[11]=str[]; int i,len; len = strlen(str); for(i=0;i<len;i++) { str1[i] = str[i]&127; printf("%c",str1[i]); } printf("\n"); for(i=0;i<len;i++) { str3[i] = str2[i]^127; printf("%c",str3[i]); } pri
Read more

Queue in Data Structure(DS)

By -
Image
Queue Queues are data structures that follow the First In First Out (FIFO) i.e. the first element that is added to the queue is the first one to be removed. Elements are always added to the back(rear) and removed from the front. Think of it as  a line of people waiting for a bus. The person who is at the beginning of the line is the first one to enter the bus.   Basic features of Queue Like a stack, the queue is also an ordered list of elements of similar data types. A queue is a FIFO( First in First Out ) structure. Once a new element is inserted into the Queue, all the elements inserted before the new element in the queue  must be removed, to remove the new element. In a simple queue, there is a disadvantage after deleting element we cannot insert the element in the queue because of insertion  is possible from the rear but deletion is performed from the front.                                           insertion                                     ------------------
Read more

Implementation of stack Using Array

By -
Implementation  of stack Using Array #include<stdio.h> void pop(); void push(); void display(); int i,n,ch=0,top,item,stack[50]; int main() {     printf("Enter the number of element in Stack:\t");     scanf("%d",&n);     top=-1;     printf("Stack Operations");     printf("\n1.Push\n2.Pop\n3.Display\n4.Exit\n");    do     {         printf("Enter the choice: ");         scanf("%d",&ch);         switch(ch)         {             case 1:push();                    break;             case 2:pop();                    break;             case 3:display();                    break;             case 4:exit(0);                    break;             default:printf("!!Wrong choice!!\n");         }     }     while(ch!=4);     return 0; } void push(int item) {     if(top==n-1)         printf("Overflow\n");     else     {        
Read more