Hashing

~~~~~~~~~~~~~~~~~~~~ HASHING ~~~~~~~~~~~~~~~~~~~~

The searching methods Linear Search and Binary Search are based on compariosion of datas. Therefore we need search techniques whose search time is independent of number of elements. Using Hashing, the postion of particular element is determined by value of hash key of that element. 

Hash Table: Hash table is a structure arranged in a form of an array where we store a key value after applying the hash function. Each structure is capable of storing one data only. Therefore time required to locate any element in the hash table is O(1). Each position on a table is known as bucket. H(k) is home bucket for the hash table.

Hash Function: The function that transform the original data into hash data is known as hash function. 
For example: let A is the orginal data having key K and H is the hash function that gives value B, then A is stored in postion H(K); i.e. position B of hash table. 

 The various ways to find hash functions are

1. Division Method: Choose a number m larger than the number n  of elements, may be last prime number in the range 1 to n . The hash function H is defined by

H(K)=K mod m

For Example: Consider a file  and n=100. Let us suppose m=97 then hash function can be applied to some of the elements as follows

H(3205)=3205 mod 97 =4

H(7148)= 7148 mod 97=67

H(2345)=2345 mod 97=17

Hash Table

K     :   3205     7148     2345 ............

H(K) :     4          67         17  ..............

2. Mid Square Method: In this method, the square of elements is taken and then number of digit extracted from the square as hash data.

For Example: Consider a file of 100 elements then we can apply this method as follows

Hash Table

K      :  3205            7148               2345

K²    : 10272025     51093904      5499025    
H(K) :  72                  93                   99  

3. Folding Method: In this method elements are partitioned into number of parts  each of which has a same length except the last part. These parts are added together and hash value is obtained.
For Example:  Consider the file of 100 elements and each element is to be divided into single digit. This method can be applied as follows

Hash Table

K              :    3205          7148         2345
Parts         :   3,2,0,5      7,1,4,8      2,3,4,5
Sum H(K)  :     10             20             14
 

4. Digit Analysis Method: In this method, hash data is obtained by selecting and shifting digits or bits of the original data. Digits position having most uniform distribution is selected.

5. Length Dependent Method: In this method, the length of the data is used along with some original data position to produce hash key.

Following points must be remembered while using any of the above Hash Function

i. Function should be easy and quick to compute.

ii. There should be no or minimum collision.

To search a data with any key k, we compute first H(K) and see whether that data exists at position H(K) of the tabe. 


 

Merge Sort

 First divide the list into the smallest unit (1 element), then compare each element with the adjacent list to sort and merge the two adjacent lists. Finally all the elements are sorted and merged.


Efficiency

• Worst case performance     O(n log n)
• Best case performance   O(n log n)
• Average case performance     O(n log n)

Insertion Sort Method

This method sorts the data by inserting value into its proper poistion.

The steps used in this method are

Step 1: A1 itself is sorted.

Step 2: Compare A2 with A1

• In case of ascending order ,  

  A2 <= A1 if yes then x = A2 ; A2 = A1 ; A1 = x

                             if no 'No Change'

• In case of descending order

    A2 >= A1 if yes then x = A2 ; A2 = A1 ; A1 = x

                             if no 'No Change'

Step 3: Compare A3 with A1 and A2

• In case of ascending order ,  

  A3 <= A1  or A3<= A2  

  then insert A3 in that postion where the condition holds true (first) , 

          otherwise 'No Change'

• In case of descending order

   A3 >= A1  or A3>= A2  

  then insert A3 in that postion where the condition holds true (first) , 

          otherwise 'No Change'

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Step n : Compare An with A1, A2, A3 ... An-1 

• In case of ascending order ,  

  An <= A1  or An<= A2  or ... An <= An-1 

  then insert An in that postion where the condition holds true (first) , 

          otherwise 'No Change'

• In case of descending order

    An >= A1  or An>= A2  or ... An >= An-1 

  then insert An in that postion where the condition holds true (first) , 

          otherwise 'No Change'

Complexity 

• Worst Case : n(n-1)/2 = o(n²)
• Average Case : n(n-1)/4 = o(n²)

Algorithm

1. For step= 1 to n
    Set temp=a[step]
    Set i=step-1
    while temp=0)
            Set a[i+]=a[i]
    end while
    Set a[i+1]=temp
   End For
2. Print Sorted Data
3. Exit

Quick Sort Method

Quick Sort Method

Description: 
Also known as Partition Exchange Method because it sorts data by partitioning the the array / list into two sub arrays / lists. Each partition is in turn sorted recursively. In each step, the goal is
to place a particular data known as key in its final position. In general, key element is the first data of an array / list and rest of the data grouped into  such that

i. One partition contains data smaller than the key value.

ii. Another partition contains data larger the key value.

Example:

Consider following elements, the quick sort
method to sort data in ascending order is

44,33,11,55,77,90,40,60,99,22,88,66

Step 1: Key is 44.
The First step is as follows

i. Scan: Right to Left

• stop when find first number less than key, 44
• 22
• Data after interchange: 22,
  33,11,55,77,90,40,60,99,44,88,66

ii. Left to Right

• stop when find first number greater than key, 44
• 55
• Data after interchange:: 22,33,11,44,77,90,40,60,99,55,88,66

iii. Right to Left

• stop when find first number less than key, 44
• 40
• Data after interchange:: 22, 33,11,40,77,90,44,60,99,55,88,66

iii. Left to Right

• stop when find first number greater than key, 44
• 77
• Data after interchange:: 22,33,11,40,44,90,77,60,99,55,88,66

so the list can be partitioned into two sub list

Sub Part 1: 22,33,11,40. 

Sub Part 2: 90,77,60,99,55,88,66.  

Step 2: Repeat
same steps on Sub Part1 with
key as 22

Step 3: Repeat
same steps on Sub Part2 with
key as 99

and so on

Efficiency:

• Worst Case:: O(n²) 
• Best case performance    O(n log n)  
• Average case performance    O(n log n)

Algorithm

Quicksort (int data[],int left,int right) 

{

int mid,tmp,i,j;

i = left;

j = right;

mid = data[(left + right)/2];

do 

{

     while(data[i] < mid)

         i++;

      while(mid < data[j])

         j--;

      if (i <= j) 

     {

            tmp = data[i];

            data[i] = data[j];

            data[j] = tmp;

            i++;

            j--;

     }

} while (i <= j);

   

if (left < j) Quicksort(data,left,j);

if (i < right) Quicksort(data,i,right);

}

Bubble Sort Method

In bubble sort element, each element is compared with its adjacent element  and if first element is greater than second element  then elements are interchanged otherwise no change. This process is repeated for all the elements in the list.

Compelxity

• Worst Case: n(n-1)/2 = o(n²)
• Best case performance     O(n)
•  Average case performance     o(n²) 

Algorithm:  Given array A of N elements.


1. Repeat thru step 2 for STEP=1 to N
2. [Compare adajanct elements]     Repeat for i=1 to N-1
         if a[i]>a[i+1]
         then swap
3. Print Sorted data
4. Exit

Selection Sort Method

The steps to sort data using selection sort method are

Consider A is an list / array of n elements

Step 1:  Find the postion of smallest element from A1 to An and then interchage the element of that location  with A1. 

Step 2: Find the postion of smallest element from A2 to An and then interchage the element of that location  with  A2.

Step 3: Find the postion of smallest element from A3 to An and then interchage the element of that location  with  A3.

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Step n: Find the postion of smallest element from An to An and then interchage the element of that location  with  An.

Complexity

• Worst Case: n(n-1)/2 = o(n²)
  
• Average Case: n(n-1)/2 = o(n²)
• Best Case: n(n-1)/2 = o(n²) 

Algorithm: Given A is an array of N elements. POS denoted the position of smallest element

1. Repeat thru step 4 for STEP=1 to N
2. POS=STEP
3. [for Asce order find position of smallest element]
    Repeat for i=STEP+1 to n
       if A[i]       then POS=i
4. [Swap]
     if POS<> step
     then  temp=a[setp]
              a[step]=a[POS]
              a[POS]=temp
5. Print Sorted Data
6. Exit