Hash Type Algorithm – GeeksforGeeks

There have all the time been arguments about how could be a sorting algorithm of linear time complexity be achieved, as all the standard sorting algorithms are no less than of the order of O(N*log(N)) in worst and instances. 

The rationale for the issue in reaching linearly proportional time complexity in a sorting algorithm is that almost all of those conventional sorting algorithms are comparability primarily based which can’t produce a sorted output of a dataset in linear time within the worst instances. To resolve this downside, the hash kind algorithm got here into the image and is even sooner than the quickest conventional sorting algorithm, i.e. fast kind.

Assumptions of Hash Type:

The assumptions taken whereas implementing the Hash kind are: 

1. The info values are inside a identified vary.
2. The values are numeric in nature.

Working of the Algorithm:

Because the title says, the algorithm combines the idea of hashing with sorting. 

However how could be hashing used for sorting? The reply to that is through the use of a super-hash perform. 

The super-hash perform is a composite perform of two sub-functions: 

1. Hash perform 
2. Mash perform, which helps the super-hash perform to allocate the information values to distinctive addresses (i.e. no collision occurs) in a constant manner (means no scattering of the information).

Now, these mappings of information values obtained from super-hash features are utilized by the principle hash kind strategies. The info buildings used for storing and sorting the information are matrices. There are just a few variations of hash kind strategies, primarily there are two :  

1. In-situ hash kind –  On this technique, each the storage and sorting of the values happen in the identical information construction
2. Direct hash kind – On this technique, a separate information listing is used to retailer the information, after which the mapping is finished into the multidimensional information construction from that listing.

Tremendous-Hash Perform:

Tremendous-Hash perform is a mixture of two sub-functions named hash perform, and mash perform. These 2 features used collectively to realize the excellence of the information values from one another, as a easy hash perform cannot do that because of the situation of collision. The 2 sub-functions work within the following methods to realize the aim of super-hash perform: 

Think about a generic kind for a quantity: cx + r, the place c is a magnitude worth, x is the bottom (normally 10) and r is the rest obtained.

• Hash Perform: This works upon the remainder-based distinction of values for mapping, i.e. mod operator is utilized on the information values to get the remainders and people remainders are used to allocate the information to a specific location in mapping. So, the essential hash perform is: (X mod N). However utilizing this technique alone would trigger collisions as a result of for a given the rest worth there could be a number of information values therefore they can’t be distinguished.

Instance: Taking N=10 and r=1, an equal set of values could be like { 1, 11, 21, 31, 41, 51, 61, 71, 81, 91. . . . . . . c_nx + 1 }, which all can be allotted to similar location.

• Mash perform: The mash perform is known as so as a result of it’s a magnitude hash perform, which implies it calculates the magnitudes to map the values of the information set. 
  For a quantity cx + r, the place c is the magnitude and x is the bottom (normally taken 10), c is calculated through the use of the div operator (X div N) within the mash as a substitute of the mod operator as used within the hash perform. However once more the values usually are not totally distinguished from one another utilizing the mash perform alone and a number of values are mapped to a specific location.

Instance: Taking N=10, c=5, we get the next set: { 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 }. This set comprises all of the values of the cx + r the place c  is fixed for all values (c=5) and r is variable. That’s why for a single worth of c we get a number of values mapping to the identical slot.

Now, to resolve this, we use each hash and mash features collectively to get a singular ordinal pair (c, r), the place c is the magnitude obtained utilizing the div operator within the mash perform and r is the rest worth obtained by the mod operator within the hash perform.

Development of Tremendous-Hash Perform:
Allow us to think about the vary of values [i, j], the place i is the decrease sure and j is the higher sure of the vary. Now,

1. Compute the size of the vary,  
       L = (j – i) + 1.
2. Decide the closest sq. integer ( θ ) to L by :       
       θ = ceil( √L ) 
3. Now the ordinal pair values could be calculated utilizing :    
       worth(dx, mx)  =  dxθ + mx    

Instance: 

Think about the vary of values within the interval [ 20, 144 ], making use of the steps of development of the super-hash perform:

1. Calculating the worth of L ⇒ (144 – 20) + 1 = 125 
2. Calculating nearest sq. worth to L ⇒  θ = ceil( √L ) = ceil( √125 ) = 12
3. Now for mapping, we additionally subtract the decrease sure (right here 20), so that each one the values are mapped within the vary 0. . . .(j – i).
   The resultant super-hash perform: F(x)  =  { d = (x –  20) div 12  ,  m = (x – 20) mod 12 }
4. From the above super-hash perform, we are able to discover that for x = 20, we get the ordinal pair (0, 0) and for x = 144, we get (10, 4).

In-situ Hash Type :

In-situ means “in-site”, named because of the in-place functioning of this variation. In in-situ hash kind, the super-hash perform is used iteratively over the information values to kind the information. However earlier than this, an initialization step happens through which a supply worth is taken, and is utilized by the super-hash perform to map it into one other location, the place the supply worth swaps with the worth current at that vacation spot location and therefore creating a brand new supply worth (After swapping). This course of retains on repeating till the entire information just isn’t sorted, or on the finish of the information listing.

Pseudo-code of the Algorithm:

(v1, v2, v3, . . . . . .vn) <– initialize         // initialization for retrieving the supply code
Whereas Not(Finish of Checklist) Do
    Temp <– get(v1, v2, v3. . . . vn);       // swapping of supply and vacation spot worth
    Worth –> put(v1, v2, v3. . . ..vn);
    Worth = temp;
    (v1, v2, v3. . . . . .vn) –> super_Hash_Function(temp);      // utilizing super-hash perform to  retrieve the following supply worth
Finish of Loop                                                              

Observe the under illustration for a greater understanding of the algorithm:

Illustration:

Allow us to think about a 2-dimensional 3 x 3 matrix with values starting from 1-9 as an example the in-situ hash kind: 

Discovering the vary size, L  = (9 – 1) + 1 = 9 

Computing the closest sq. integer to L = 9 ⇒ θ = ceil ( √9 ) = 3, Decrease sure, i = 1
Subsequently, the resultant super-hash perform comes out to be: d = (x – 1) / 3, m = (x – 1) % 3
Let the preliminary matrix configuration be like this: 

• Beginning initially at worth(0, 0) = 5,  
  • Calculate the worth of d and m for the worth 5: d = (5- 1) div 3 = 1,  m = (5 -1 ) mod 3 = 1.
  • Therefore, the brand new vacation spot is at (1, 1) with worth = 7.
  • Now, 5 and seven will swap their respective positions. The resultant matrix will probably be: 

• Once more utilizing the (0, 0) place with worth = 7. 
  • Discover the values of d and m: d = (7 – 1) div 3 = 2,  m = (7 – 1) mod 3 = 0. 
  • Therefore the obtained location is at (2, 0) the place the worth 4 is current. 
  • So, 4 and seven will swap.

• On place (0, 0) worth = 4. 
  • Calculating d and m for x = 4: d = (4 – 1) div 3 = 1, m = (4 – 1) mod 3 = 0.
  • The obtained place is (1, 0) with vacation spot worth = 9. 
  • Subsequently, 9 and 4 will swap their positions. 

• Now, at place (0, 0), we have now worth = 9. 
  • So, d = (9 – 1) div 3 = 2,  m = (9 – 1) mod 3 = 2
  • The obtained place is (2, 2) with worth  = 3. 
  • Subsequently, 9 and three will swap their positions.

• At place (0, 0) worth =  3.
  • Calculate d and m, d = (3 – 1) div 3 = 0, m = (3 – 1) mod 3 = 2.
  • Resultant place is (0, 2) with worth = 1. 
  • Subsequently, 1 and three will swap their positions.

• At place (0, 0), worth = 1.
  • d = (1 – 1) div 3 = 0,  m = (1 – 1) mod 3 = 0.
  • The obtained place: (0, 0). That is the case of hysteresis, the place the supply worth maps to its personal location. This may trigger an infinite loop as 1 on (0, 0) will carry on mapping place (0, 0) itself. 
  • So, to resolve this, we increment the place by 1 and get (0, 1) as our new supply location. 
  • Now, computing the values of d and m for worth = 8 at place (0, 1): d = (8 – 1) div 3 = 2, m  = (8 -1) mod 3 = 1. 
  • The brand new obtained location is (2, 1) with worth = 6. 
  • Subsequently, we swap the positions of 8 and 6.

• At place (0, 1) worth = 6.
  • d = (6 – 1) div 3 = 1, m = (6 – 1) mod 3 = 2.
  • The brand new obtained location is at (1, 2) with worth = 2. 
  • Subsequently, 6 and a pair of will swap their positions. The resultant matrix is:

The matrix is totally sorted now, so we cease our implementation right here. 

Beneath is the code to implement In-Situ hash kind.

The resultant sorted array is: 
1 2 3 
4 5 6 
7 8 9 

Time and House Complexity Evaluation:

Time Complexity: O(N)

• Hash kind mapping features have a number of attainable variety of implementations because of the extendible nature of the hash kind, so we are able to take a relentless c, the place c >=1, denoting that no less than one mapping is required. 
• Now, because the super-hash perform is a composite perform of two sub-functions, so the general time for mappings will probably be twice the variety of sub-mappings (denoted by c), So we multiply c by 2 to get the time taken for the mapping perform for one information worth. 
• The mapping perform will probably be utilized iteratively on each ingredient of the information, which is n on this case, so the full time complexity comes out to be:T(n) = 2.c.N

Subsequently, T(N) = O(N)

Auxiliary House: O(N)

Auxiliary House relies upon upon the variation of the hash kind used. 

• For in-situ hash kind, the sorting occurs in place within the information construction itself. 
• Subsequently, we require (N + 1) area for implementing the in-situ algorithm, the place N is the variety of parts within the information construction and one further area is required for holding the short-term values whereas doing the swapping course of.

Subsequently, S(N) = O(N)

Purposes of Hash Type:

• Knowledge mining: Hash kind could be helpful for organizing and looking of information in information mining the place there are large portions of information.
• Database Techniques: For an environment friendly retrieval of the information.
• Working Techniques: File group.