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The University of Texas at Austin
Hashing
Department of Computer Sciences
Professor Vijaya Ramachandran
Lecture 20
CS357: ALGORITHMS, Spring 2006
1
Hashing
Hashing is a widelyused class of data structures that support the operations
of insert, delete and search on a dynamic set of keys
S
. The keys are drawn
from a universe
U
(

U

>>

S

), which for our purposes will be a set of
positive integers. These keys are mapped into a
hash table
using a
hash
function
; the size of the hash table is usually within a constant factor of the
number of elements in the current set
S
.
There are two main methods used to implement hashing:
hashing with chain
ing
and
hashing with open addressing
. We will study hashing with chaining
here.
1.1
Hashing with Chaining
In
hashing with chaining
, the elements in
S
are stored in a
hash table
T
[0
.. m
−
1] of size
m
, where
m
is somewhat larger than
n
, the size of
S
. The hash
table is said to have
m
slots
. Associated with the hashing scheme is a hash
function
h
which is a mapping from
U
to
{
0
,
···
, m
−
1
}
. Each key
k
∈
S
is
stored in location
T
[
h
(
k
)], and we say that key
k
is
hashed into slot
h
(
k
). If
more than one key in
S
hashes into the same slot, then we have a
collision
.
In such a case, all keys that hash into the same slot are placed in a linked
list associated with that slot; this linked list is called the
chain
at that slot.
The
load factor
of a hash table is deFned to be
α
=
n/m
; it represents the
average number of keys per slot. We typically operate in the range
m
= Θ(
n
)
so
α
is usually a constant (usually
α <
1). If a large number of insertions
or deletions destroys this property, a more suitable value of
m
is chosen and
the keys are rehashed into a new table with a new hash function.
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simple uniform hashing
we assume that we have a ‘good’ hash function
h
such that any element in
U
is equally likely to hash into any of the
m
slots
in
T
, independent of the other keys in the table. We also assume that
h
(
k
)
can be computed in constant time.
In hashing with chaining, inserts and deletes can be performed in constant
time with a suitable linked list representation for the chains. In the worst
case, a search operation can take as long as
n
steps if all keys hash into the
same slot. However, if we assume simple uniform hashing then the expected
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 Spring '06
 Ramachandran
 Algorithms, Data Structures

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