Different ways of Open Addressing

1. Linear Probing: 

In linear probing, the hash table is searched sequentially that starts from the original location of the hash. If in case the location that we get is already occupied, then we check for the next location. 

The function used for rehashing is as follows: rehash(key) = (n+1)%table-size. 

For example, The typical gap between two probes is 1 as seen in the example below:

Let hash(x) be the slot index computed using a hash function and S be the table size 

If slot hash(x) % S is full, then we try (hash(x) + 1) % S
If (hash(x) + 1) % S is also full, then we try (hash(x) + 2) % S
If (hash(x) + 2) % S is also full, then we try (hash(x) + 3) % S 

Example: Let us consider a simple hash function as “key mod 5” and a sequence of keys that are to be inserted are 50, 70, 76, 85, 93. 

2. Quadratic Probing 

If you observe carefully, then you will understand that the interval between probes will increase proportionally to the hash value. Quadratic probing is a method with the help of which we can solve the problem of clustering that was discussed above.  This method is also known as the mid-square method. In this method, we look for the i²‘th slot in the i^th iteration. We always start from the original hash location. If only the location is occupied then we check the other slots.

let hash(x) be the slot index computed using hash function.  

If slot hash(x) % S is full, then we try (hash(x) + 1*1) % S
If (hash(x) + 1*1) % S is also full, then we try (hash(x) + 2*2) % S
If (hash(x) + 2*2) % S is also full, then we try (hash(x) + 3*3) % S

Example: Let us consider table Size = 7, hash function as Hash(x) = x % 7 and collision resolution strategy to be f(i) = i² . Insert = 22, 30, and 50.

3. Double Hashing 

The intervals that lie between probes are computed by another hash function. Double hashing is a technique that reduces clustering in an optimized way. In this technique, the increments for the probing sequence are computed by using another hash function. We use another hash function hash2(x) and look for the i*hash2(x) slot in the i^th rotation. 

let hash(x) be the slot index computed using hash function.  

If slot hash(x) % S is full, then we try (hash(x) + 1*hash2(x)) % S
If (hash(x) + 1*hash2(x)) % S is also full, then we try (hash(x) + 2*hash2(x)) % S
If (hash(x) + 2*hash2(x)) % S is also full, then we try (hash(x) + 3*hash2(x)) % S

Example: Insert the keys 27, 43, 692, 72 into the Hash Table of size 7. where first hash-function is h1​(k) = k mod 7 and second hash-function is h2(k) = 1 + (k mod 5)

Open Addressing is a method for handling collisions. In Open Addressing, all elements are stored in the hash table itself. So at any point, the size of the table must be greater than or equal to the total number of keys (Note that we can increase table size by copying old data if needed). This approach is also known as closed hashing. This entire procedure is based upon probing. We will understand the types of probing ahead:

• Insert(k): Keep probing until an empty slot is found. Once an empty slot is found, insert k. 
• Search(k): Keep probing until the slot’s key doesn’t become equal to k or an empty slot is reached. 
• Delete(k): Delete operation is interesting. If we simply delete a key, then the search may fail. So slots of deleted keys are marked specially as “deleted”. 
  The insert can insert an item in a deleted slot, but the search doesn’t stop at a deleted slot.