Question

In open addressing, why do we need a table size that’s at least the number of keys? State how much larger it should be, to get an efficient time complexity.

Answer 1

Open Addressing
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 the table must be greater than or equal to the total number of keys.

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 the search doesn’t stop at a deleted slot.

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

 m = Number of slots in the hash table
 n = Number of keys to be inserted in the hash table
 
 Load factor α = n/m  ( < 1 )

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

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

--> So, to make it efficient α should be zero, Hence m should be high.