Coding and hashing

1. Hashing

2. Hashing
When implementing a hash table using arrays, the nodes/elements are not stored
consecutively, instead the location of an element/node to be
inserted/deleted/searched is computed using the key and a hash function.

3. Hash Table
• Hash table is a data structure which is designed specifically
with the objective of providing efficient insertion and
searching operations (deletion is not a primary objective).
• Hash Table achieves constant time performance O(1) time
• To achieve constant time performance objective, the
implementation must be based in some way on an array
rather than a linked list.

4. Hash Table
• Hash table is a container which will be used to hold some number
of items of a given set K. We call the elements of the set K keys and
K is called the key space
• In the general case, we expect the size of the set of keys, denoted
as |K|, to be relatively large in comparison with the number of
items stored in the table of size M
• The general approach is to store the keys in an array. The position
(also called location or index) i of a key k in the array is given by a
function hash(k) or f(k), called a hash function
• Hashing – Hash function determines the position of a given key
directly from that key (i.e., i = f(k) is called the hash code of k)
Example :For a Student Record USN is the key
Example USN= 1234
Value = Hash(1234) or value =f(1234) where value is the hash value of
the Student Record with key(USN) 1234.

5. Hashing : Hash Function &
Hash Collision
• Hash Function f
f: K → {0, 1, ..., M-1}.
• Hash function f maps (or transforms) the set of key values(K) to
subscripts (indexes) in an array(Table) of size M.
• In general, since |K| >> M, the mapping defined by a hash
function will be a many-to-one mapping.
• Hash Collision : Hash Collision occurs when hash function
maps distinct keys x and y to the same index, i.e. When x not
equal to y f(x) = f(y) .
• Collisions can be reduced with a selection of a good hash
function

6. Hashing: Hash Function Example
Example : A record with key value 37
f(37) : 37%10
= 7
A record with key value 37 is mapped to index 7
Example : A record with key value 87
f(87) : 87%10
= 7
A record with key value 37 is mapped to index 7
Two records/items with different key values ie
37,87 mapped to same index ie 7 of the hash
table/array results in collision

7. Hashing: Hash Function Example
Example:
•K = {0, 1, ..., 199}, ie Number of different
key values =200
•Hash Table Size (Array Size) M =10
•Hash Function -for each key k in K,
Hash function Defintion
f(k) : k % M
i.e. f(k) : k % Table Size
i.e. f(k) : k % 10
Example : A record with key value 25
f(25) : 25%10
= 5
A record with key value 25 is mapped to
index 5

8. Example :

9. Insert 36,48,90,12 using key%10 hash
function
Index HT[index]
0 90
1 12
2
3
4
5
6 36
7
8 48
9
10

10. Hash Function
The characteristics of a good hash function are
as follows.
• It avoids collisions.
• It tends to spread keys evenly in the array.
• It is easy to compute (i.e., computational time
of a hash function should be O(1)

11. Collision Resolution Techniques
Collision Resolution Techniques
• Open Hashing : In open hashing, keys are stored in linked lists
attached to cells of a hash table
 Chaining( Separate Chaining)
• Closed Hashing(Open Addressing):In closed hashing, all keys
are stored in the hash table itself without the use of linked
lists.
 Linear Probing Method
 Quadratic Probing Method
 Double Hashing Method

12. Separate Chaining
• When an element needs to be searched on the hash table, the hash
function f(k) = k % M will specify the address i within the range [0,
M -1]corresponding the singly linked list i that may contain the
node.
• Searching an element on the hash table turns out the problem of
searching an element on the singly linked list
• The hash table is implemented by using singly linked list.
• Elements on the hash table are hashed into M (e.g., M = 10) singly
linked lists (from list 0 to list M-1).
• Elements conflicted at the address i are directly connected in the
list i, where 0 ≤i ≤M-1.
• When an element with the key k is added into the hash table, the
hash function f(k) = k % M will identify the address i between 0 and
M -1 corresponding the singly linked list i where this node will be
added

13. Separate Chaining(Table size 10 hash
function f(k)= k%10)

14. Linear Probing Method
• All elements are stored in the hash table itself
• The hash table is initialized, all M locations assigned to -1 (i.e., empty or vacant).
• When a node with the key k needs to be added into the hash table, the hash function
f(k) = k % M
will specify the address/index
i = f(k) (i.e., an index of an array)
within the range [0, M-1].
• If there is no conflict (i.e., the cell i is unoccupied), then this node is added into the hash
table at the address i.
• If a collision takes place, then the hash function rehashes first time to consider the
next address (i.e., i + 1).
• If conflict occurs again, then the hash function rehashes second time to examine the
next address (i.e., i + 2).
• This process repeats until the available address found then this node will be added at
this address.

15. 15
Linear probing: Inserting a key
• Idea: when there is a collision, check the next available
position in the table (i.e., probing)
h(k,i) = (h1(k) + i) mod m
i=0,1,2,...
• First slot probed: h1(k)
• Second slot probed: h1(k) + 1
• Third slot probed: h1(k)+2, and so on
• Can generate m probe sequences maximum, why?
probe sequence: < h1(k), h1(k)+1 , h1(k)+2 , ....>
wrap around

16. 16
Linear probing: Deleting a key
• Problems
– Cannot mark the slot as empty
– Impossible to retrieve keys inserted
after that slot was occupied
• Solution
– Mark the slot with a sentinel value
DELETED
• The deleted slot can later be used
for insertion
• Searching will be able to find all the
keys
0
m - 1

17. Linear Probing

18. Linear Probing
• Hash Table Size : 7 , So f(k) = k%7
Probe Sequence is for Insert(47): {5,6,0,1,2,3,4}
Table[4]= 47 //Insert

19. Linear Probing

20. Linear Probing

21. Quadratic Probing

22. Quadratic Probing

23. Quadratic Probing

24. Quadratic Probing

25. Quadratic Probing

26. 26
Double Hashing
(1) Use one hash function to determine the first slot
(2) Use a second hash function to determine the
increment for the probe sequence
h(k,i) = (h1(k) + i h2(k) ) mod m, i=0,1,...
• Initial probe: h1(k)
• Second probe is offset by h2(k) mod m, so on ...
• Advantage: avoids clustering
• Disadvantage: harder to delete an element
• Can generate m2 probe sequences maximum

27. 27
Double Hashing: Example
h1(k) = k mod 13
h2(k) = 1+ (k mod 11)
h(k,i) = (h1(k) + i h2(k) ) mod 13
• Insert key 14:
h1(14,0) = 14 mod 13 = 1
h(14,1) = (h1(14) + h2(14)) mod 13
= (1 + 4) mod 13 = 5
h(14,2) = (h1(14) + 2 h2(14)) mod 13
= (1 + 8) mod 13 = 9
79
69
98
72
50
0
9
4
2
3
1
5
6
7
8
10
11
12
14

28. Example

29. For the input 30, 20, 56, 75, 31, 19 and hash function h(K) = K mod 11
construct the open hash table.
• find the largest number of key comparisons in a successful search in
this table.
• find the average number of key comparisons in a successful search
in this table.
For the input 30, 20, 56, 75, 31, 19 and hash function h(K) = K mod 11
construct the closed hash table.
• find the largest number of key comparisons in a successful search
in this table
• find the average number of key comparisons in a successful search
in this table.