Data Structure and Algorithms Hashing

Hashing Introduction
Hash Functions
Hash Table
Closed hashing(open addressing)
Linear Probing
Quadratic Probing
Double hashing
Open hashing(separate chaining)

Hashing
 The search time of each algorithm discussed so far
depends on number n depends on the number n of
items in the collection S of data.
A searching Technique, called Hashing or Hash
addressing, which is independent of number n.
We assume that
1. there is a file F of n records with a set K of keys which
unlikely determine the record in F.
2. F is maintained in memory by a Table T of m memory
locations and L is the set of memory addresses of
locations in T.
3. For notational convenience, the keys in K and
Address L are Integers.

Example
Suppose A company with 250 employees assign a 5-
digit employee number to each employee which is
used as primary key in company’s employee file.
We can use employee number as a address of record in
memory.
The search will require no comparisons at all.
Unfortunately, this technique will require space for
1,00,000 memory locations, where as fewer locations
would actually used.
So, this trade off for time is not worth the expense.

Hashing
 The general idea of using the key to determine the
address of record is an excellent idea, but it must be
modified so that great deal of space is not wasted.
This modification takes the form of a function H from
the set K of keys in to set L of memory address.
 H: K L , Is called a Hash Function or
 Unfortunately, Such a function H may not yield distinct
values: it is possible that two different keys k1 and k2 will
yield the same hash address. This situation is called Collision,
and some method must be used to resolve it.

Hash Functions
 the two principal criteria used in selecting a hash
function H: K L are as follows:
1. The function H should be very easy and quick to
compute.
2.The function H should as far as possible,
uniformly distribute the hash address through out
the set L so that there are minimum number of
collision.

Hash Functions
1. Division method: choose a number m larger than the number n of keys
in K. (m is usually either a prime number or a number without
small divisor) the hash function H is defined by
H(k) = k (mod m) or H(k) = k (mod m) + 1.
here k (mod m) denotes the reminder when k is divided by m. the
second formula is used when we want a hash address to range from
1 to m rather than 0 to m-1.
2. Midsquare method: the key k is squared. Then the hash function H is
defined by H(k) = l. where l is obtained by deleting digits from
both end of k^2.
3. Folding Method: the key k is portioned into a number of parts, k1, k2,
……,kr, where each part is added togather, ignoring the last carry.
H(k) = k1+k2+ ……………+Kr.
Sometimes, for extra “milling”, the even numbered parts, k2, k4, …. Are
each reversed befor addition.

Example of Hash Functions
 consider a company with 68 employees assigns a 4-digit employee
number to each employee. Suppose L consists of 100 two-digit
address: 00, 01, 02 , ……….99. we apply above hash functions to each of
following employee numbers: 3205, 7148,2345.
1. Division Method:
choose a prime number m close to 99, m=97.
H(k)=k(mod m): H(3205)=4, H(7148)=67, H(2345)=17.
2. Midsquare Method:
k= 3205 7148 2345
k^2= 10272025 51093904 5499025
H(k)= 72 93 99
3. Folding Method: chopping the key k into two parts and adding yield
the following hash address:
H(3205)=32+05=37, H(7148)=71+48=19, H(2345)=23+45=68
Or,
H(3205)=32+50=82, H(7148)=71+84=55, H(2345)=23+54=77

Collision Resolution
Suppose we want to add a new record R with key K to our file F, but
suppose the memory location address H(k) is already occupied. This
situation is called Collision.
There are two general ways to resolve collisions :
 Open addressing,(array method)
 Separate Chaining (linked list method)
The particular procedure that one choose depends on many factors.
One important factor is load factor (λ=n/m)i.e. ratio of number n of
keys in K (number of records in F) to m of hash address in L.
e.g. suppose a student class has 24 students and table has space for 365
records.
The efficiency of hash function with a collision resolution procedure is
measured by the average number of probes (key comparison) needed
to find the location of record with a given k. The efficiency mainly
depend on load factor.
Specially we are interested in following two quantities:
 S(λ) = average number of probes for a successful search
 U(λ) = average number of probes for an unsuccessful search

Open Addressing: Liner Probing and Modifications
 Suppose that new record R with key k is added to
memory table T, but that the memory location with hash
address H(k)=h is already filled.
One natural way to resolve the collision is to assign R to
the first variable locating following T[h] (we assume that
the table T with m location is circular i.e. T[1] comes after
T[m]).
With such a collision procedure, we will search for record R in
table T by linearly search the locations T[h], T[h+1], T[h+2],
…………. Until finding R or meeting empty location, which indicates
an unsuccessful search.
The above collision resolution is called Linear probing.
The average number of probes for load factor (λ =n/m) are:

Open Addressing
One main disadvantage of linear probing is that records
tend to cluster, that is, appear next to one another, when
the load factor is greater then 50%.
The two technique that minimize the clustering are as:
1. Quadratic probing: Suppose the record R with key K has
the hash address H(k)=h. Then instead of searching the
locations with address h, h+1, h+2, …….. ., we search the
location with address
h,h+1,h+4,h+9, …………..,h+i^2,……
2. Double hashing: here the second hash function H’ is used
for resolving a collision, as follows. Suppose a record R
with key k has a hash address H(k)=h and H’(k)=h’≠m.
Then we linearly search the locations with address
h, h+h’, h+2h’, h+3h’, …………

Chaining
Chaining involves maintaining two tables in memory.
First of all, as before, there is a table T in memory which
contains the records in F, except that T now has an
additional field LINK which is used so that all record in T
with same hash address h may be linked together to form
a linked list. Second, there is a hash address table LIST
which contain pointers to linked lists in T.
Suppose a new record R with key k is added to the file F.
we place R in the first available location in the table T and
then add R to the linked list with pointer LIST[H(k)].
The average number of probes for load factor (λ =n/m may be greater
than 1) are:
S(λ)≈1+ λ/2 and U(λ)≈e^(- λ)+ λ.

Ex: Chaining
Using chaining, the record will appear in memory as:

Data Structure and Algorithms Hashing

Data Structure and Algorithms Hashing

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