Data Structures

Open Addressing for Collision Handling

We strongly recommend to refer below post as a prerequisite of this.
Hashing Introduction
Separate Chaining for Collision Handling

Open Addressing
Like separate chaining, open addressing is a method for handling collisions. In Open Addressing, all elements are stored in the hash table itself. So at any point, size of table must be greater than or equal to total number of keys (Note that we can increase table size by copying old data if needed).

Insert(k): Keep probing until an empty slot is found. Once an empty slot is found, insert k.

Search(k): Keep probing until 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 search may fail. So slots of deleted keys are marked specially as “deleted”.
Insert can insert an item in a deleted slot, but search doesn’t stop at a deleted slot.

Open Addressing is done following ways:

a) Linear Probing: In linear probing, we linearly probe for next slot. For example, typical gap between two probes is 1 as taken in below example also.
let hash(x) be the slot index computed using 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 
..................................................
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Let us consider a simple hash function as “key mod 7” and sequence of keys as 50, 700, 76, 85, 92, 73, 101.

Clustering: The main problem with linear probing is clustering, many consecutive elements form groups and it starts taking time to find a free slot or to search an element.

b) Quadratic Probing We look for i2‘th slot in i’th iteration.

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
..................................................
..................................................

c) Double Hashing We use another hash function hash2(x) and look for i*hash2(x) slot in 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
..................................................
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See this for step by step diagrams.

Comparison of above three:
Linear probing has the best cache performance, but suffers from clustering. One more advantage of Linear probing is easy to compute.

Quadratic probing lies between the two in terms of cache performance and clustering.

Double hashing has poor cache performance but no clustering. Double hashing requires more computation time as two hash functions need to be computed.

Open Addressing vs. Separate Chaining
Advantages of Chaining:
1) Chaining is Simpler to implement.
2) In chaining, Hash table never fills up, we can always add more elements to chain. In open addressing, table may become full.
3) Chaining is Less sensitive to the hash function or load factors.
4) Chaining is mostly used when it is unknown how many and how frequently keys may be inserted or deleted.
5) Open addressing requires extra care for to avoid clustering and load factor.

Advantages of Open Addressing
1) Cache performance of chaining is not good as keys are stored using linked list. Open addressing provides better cache performance as everything is stored in same table.
2) Wastage of Space (Some Parts of hash table in chaining are never used). In Open addressing, a slot can be used even if an input doesn’t map to it.
3) Chaining uses extra space for links.

Performance of Open Addressing:
Like Chaining, performance of hashing can be evaluated under the assumption that each key is equally likely to be hashed to any slot of table (simple uniform hashing)

 m = Number of slots in hash table
 n = Number of keys to be inserted in has table

 Load factor ? = n/m  ( < 1 )

 Expected time to search/insert/delete < 1/(1 - ?) 

 So Search, Insert and Delete take (1/(1 - ?)) time

References:
http://courses.csail.mit.edu/6.006/fall11/lectures/lecture10.pdf
https://www.cse.cuhk.edu.hk/irwin.king/_media/teaching/csc2100b/tu6.pdf


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