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Lecture 9
Topics
 Hashing
Reference: Introduction to Algorithm by Cormen
Chapter 12: Hash Tables
Data Structure and Algorithm
1.1
Introduction
 A hash table is a generalization of the
simpler notion of an ordinary array
 Searching for an element in a hash table
can take as long as searching for an
element in an array/linked list i.e. O(N)
time in the worst case.
 But under reasonable assumptions,
hashing takes O(1) time to search an
element in a hash table.
Data Structure and Algorithm
1.2
Dictionary/Table
Keys
Given a student ID find the record (entry)
Data Structure and Algorithm
1.3
Direct Addressing
 Direct Addressing is a simple technique that works well
when the universe U of keys is reasonably small.
 Let Universe U= (0,1,…….m-1} where m is not too large
 We may assume that no two elements have the same
key.
 To represent the dynamic set, we use an array or direct
address table T[0..m-1] in which each position or slot
correspond to a key in the universe U
Data Structure and Algorithm
1.4
Direct Addressing
Slot k points to an element in the set with key k
If the set contains no element with key k, then T[k]=nil
0
1
7
9
2
2
3
3
4
2
1
key data
0
U
(universe of keys)
4
T
k4 5
3
5
8k5
6
5
7
8
6
9
actual
keys
Data Structure and Algorithm
1.5
8
Direct-address Table
 Direct addressing is a simple technique that works well when the
universe of keys is small.
Assuming each key corresponds to a unique slot.
Direct-Address-Search(T,k)
return T[k]
Direct-Address-Insert(T,x)
return T[key[x]]  x
Direct-Address-Delete(T,x)
return T[key[x]]  Nil
5
2
3
entry
1
/
/
/
5
6
7
O(1) time for all operations
Data Structure and Algorithm
/
1
4
7
1
0
1.6
5
/
7
The Problem With Direct Addressing
 If the universe U is large, storing a table T of size |U|
may be impractical or even impossible.
 Furthermore, the set K of keys actually stored may be
so small relative to U that most of the space allocated
for T would be wasted.
 Solution: map keys to smaller range 0..m-1
 This mapping is called a hash function
Data Structure and Algorithm
1.7
Hash function
 Hash function h maps the universe U of keys into slots of a hash
table T [0..m-1]:
h:U
{0,1,….m-1}
 But two keys may hash to the same slot – a collision
T
U
(universe of keys)
0
h(k1)
h(k4)
k1
k4
K
(actual
keys)
k2
k5
h(k2) = h(k5)
h(k3)
k3
m-1
Data Structure and Algorithm
1.8
Next Problem
 But two keys may hash to the same slot – a
collision
T
0
U
(universe of keys)
h(k1)
h(k4)
k1
k4
K
(actual
keys)
k2
k5
h(k2) = h(k5)
h(k3)
k3
m-1
Data Structure and Algorithm
1.9
Resolving Collisions
 How can we solve the problem of
collisions?
 Solution 1: chaining
 Solution 2: open addressing
Data Structure and Algorithm
1.10
Chaining
 Chaining puts elements that hash to the same slot in a linked
list:
U
(universe of keys)
k6
k2
k4
k1 ——
k7
k5
——
——
——
k1
k4
K
(actual
k7
keys)
T
——
k5
——
k8
k3
k3 ——
k8
——
Data Structure and Algorithm
1.11
k6 ——
k2 ——
Chaining (insert at the head)
U
(universe of keys)
k1
k4
K
(actual
k7
keys)
k6
k2
k5
k8
k3
Data Structure and Algorithm
T
——
k1 ——
——
——
——
——
——
——
——
——
1.12
Chaining (insert at the head)
U
(universe of keys)
k6
k2
k1 ——
——
——
——
k1
k4
K
(actual
k7
keys)
T
——
k5
k2 ——
——
k8
k3
Data Structure and Algorithm
k3 ——
——
——
1.13
Chaining (insert at the head)
U
(universe of keys)
k6
k2
k41 ——
——
——
——
k1
k4
K
(actual
k7
keys)
T
——
k5
k2 ——
——
k8
k3
Data Structure and Algorithm
k3 ——
——
——
1.14
k1 ——
Chaining (insert at the head)
U
(universe of keys)
k6
k2
k41 ——
k1 ——
k52 ——
k2 ——
——
——
——
k1
k4
K
(actual
k7
keys)
T
——
k5
——
k8
k3
k3 ——
——
k6 ——
Data Structure and Algorithm
1.15
Chaining (Insert to the head)
U
(universe of keys)
k6
k2
k4
k1 ——
k7
k5
——
——
——
k1
k4
K
(actual
k7
keys)
T
——
k5
——
k8
k3
k3 ——
k8
——
Data Structure and Algorithm
1.16
k6 ——
k2 ——
Operations
Direct-Hash-Search(T,k)
Search for an element with key k in list T[h(k)]
(running time is proportional to length of the list)
Direct-Hash-Insert(T,x)
(worst case O(1))
Insert x at the head of the list T[h(key[x])]
Direct-Hash-Delete(T,x)
Delete x from the list T[h(key[x])]
(same as searching)
Data Structure and Algorithm
1.17
Open Addressing
 Basic idea (details in Section 12.4):
 To insert: if slot is full, try another slot, …,
until an open slot is found (probing)
 To search, follow same sequence of probes
as would be used when inserting the
element
 If reach element with correct key, return it
 If reach a NULL pointer, element is not in
table
 Good for fixed sets (adding but no deletion)
 Table needn’t be much bigger than n
Data Structure and Algorithm
1.18
Choosing A Hash Function
 Choosing the hash function well is crucial
 Bad hash function puts all elements in same
slot
 A good hash function:
 Should distribute keys uniformly into slots
 Should not depend on patterns in the data
 Three popular methods:
 Division method
 Multiplication method
 Universal hashing
Data Structure and Algorithm
1.19
The Division Method
 h(k) = k mod m
 hash k into a table with m slots using the slot
given by the remainder of k divided by m
 Elements with adjacent keys hashed to different
slots: good
 If keys bear relation to m: bad
 In Practice: pick table size m = prime number
not too close to a power of 2 (or 10)
Data Structure and Algorithm
1.20