A hash table is an unordered collection of key-value pairs, where each key is unique.

for each key value, hash function will generate the unique index then that key value will stored in that unique index as look like storing in normal array.





so next what is Direct-address tables

We shall assume that no two elements have the same key , to represent the dynamic set, we use an array, or direct-address table

Each of these operations is fast: only O(1) time is required.









Let have a look in HASH TABLE Collision resolution by chaining

if two elements have the same key we put all the elements that hash to the same slot in a linked list









CHAINED -HASH -INSERT (T, x)

insert x at the head of list T [h(key[x])]



CHAINED -HASH -SEARCH (T, k)

search for an element with key k in list T [h(k)]



CHAINED -HASH -DELETE (T, x)

delete x from the list T [h(key[x])]





The division method : h(k) = k mod m

The multiplication method : h(k) = m (k A mod 1)

Universal hashing

HASH -SEARCH (T, k)

i ← 0

repeat j ← h(k, i)

if T [ j ] = k

then return j

i ← i + 1

until T [ j ] = NIL or i = m

return NIL

hereis the hash tableis the key valueDIRECT -ADDRESS -SEARCH (T, k)return T [k]DIRECT -ADDRESS -INSERT (T, x)T [key[x]] ← xDIRECT -ADDRESS -DELETE (T, x)T [key[x]] ← NILLet check Hash functions, in which are the methods to Interpreting keys as natural numbersk is the keyshort description about hash table have a lookall elements are stored in the hash table itself. That is, each table entry contains either an element of the dynamic set or NIL . When searching for an element, we systematically examine table slots until the desired element is found or it is clear that the element is not in the table. There are no lists and no elements stored outside the table, as there are in chaining. Thus, in open addressing, the hash table can “fill up” so that no further insertions can be madeHASH -INSERT (T, k)i ← 0repeat j ← h(k, i)if T [ j ] = NILthen T [ j ] ← kreturn jelse i ← i + 1until i = merror “hash table overflow”