Hash Table

1
Hash Table
SoongSil Univ.
MultiMedia Lab.
Baek Hae Jung
2
Contents

• 1. Direct-address table
• 2. Hash tables
• 3. Hash functions
• Division method
• Multiplication method
• Universal hashings
• 4. Open addressing
• Linear probing
• Quadratic probing
• Double hashing

3
Direct-address table
T
0 1 2 3 4 5 6 7 8 9
/
U (universe of keys)
Key
/
2
3
/
K (actual keys)
5
/
/
8
/
slot
Key K1
Slot K1
4
Direct-address table

• Direct-address table Tm
• The set of actual key determines the slots in the
  table that contain pointers to elements
• The other slots contain NIL(/)

Each key in the universe corresponds to an index
in the table
T
0 1 2 3 4 5 6 7 8 9
/
Key
/
2
3
/
5
/
/
8
/
slot
5
Operations
Direct-address-search(T, k) Return Tk
T
0 1 2 3 4 5 6 7 8 9
/
Key
/
2
Direct-address-insert(T, x) Tkeyx x
3
/
5
/
/
8
/
Direct-address-delete(T, x) Tkeyx NIL
Time Complexity O(1)
6
Hashing
T
U (universe of keys)
/
H(k1)
H(k4)
K (actual keys)
/
H(k2)H(k5)
/
/
H(k3)
/
Slot
Key K1
Function H
Slot H(K1)
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Basic Idea
Key K1
Slot K1
Direct addressing
Key K1
Function h
Slot h(K1)
Hashing
An Element with key k hashes to slot h(k)
H(k) is the hash value of key K
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Collision

• Collision
• Two key hash to the same slot by hash function
• U gt m

9
Collision Resolution Policies

• Two classes
• (1) Open hashing, separate chaining
• (2) Closed hashing, open addressing 12.4
• Difference has to do with
• whether collisions are stored outside the table
  (open hashing)
• whether collisions result in storing one of the
  records
• at another slot in the table
  (closed hashing)

10
Open Hashing

• Collision resolution by chaining
• Chaining
• Put all the elements that hash to the same slot
  in a linked list

11
Example of Collision

• Example
• H(K2) H( K5 )

12
Operations
Chained-hash-inert(T,x) insert x at the head of
list Th(keyx)

• Chained-hash-delete(T,x)
• delete x from the list Th(keyx)

Insert/Delete Time Complexity O(1)

• Chained-hash-search(T,k)
• search for an element with key k in list
  Th(k)

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Analysis of open hashing

• Load factor(?)
• The average number of elements stored in a chain.
• N elements / M slots

Ex) 3 Slots, 6 Elements
/ /
k1
/
k5
k3
/
k4
k6
k5
k3
/
k4
k1
k2
k2
/
k6
Search Time Complexity ?(n) worst-case
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Simple uniform hashing

• Average Performance of hashing depends on
• How well the hash function h distributes the set
  of keys 12.3
• Assumption of Simple uniform hashing
• Any given element is equally likely to hash into
  any of the m slots,
• independently of where any other element has
  hashed to.
• Insertion/Delete/Search Time Complexity
• O(1)

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Analysis of hashing with chaining

• Theorem 12.1

In a hash table in which collisions are resolved
by chaining, Unsuccessful search takes time (
?(1?) ), on the average.
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Analysis of hashing with chaining

• Theorem 12.2

In a hash table in which collisions are resolved
by chaining, Successful search takes time (
?(1?) ), on the average.
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Hash functions

• A Good hash function
• Avoids collisions.
• Minimize the chance that such variants hash to
  the same slot
• Tends to spread keys evenly in the array.
• Satisfies the assumption of simple uniform
  hashing
• Is easy to compute.
• Probability distribution P

for j 0, 1, , m-1.
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Interpreting keys as natural number

• Most hash functions
• The universe of keys
• 0, 1, 2, of natural number

Key 30
Slot
30
Key 14452(pt)
Function h
Slot h(K1)
Pt
P112 128 T116 1 gt 14452
gt sums the ASCII values of the letters in the
string
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Three schemes for Hash function

• Division method
• H(k) k mod m
• Multiplication method
• H(k) ?m(k A mod 1)?
• Universal hashing
• Choose the hash function randomly

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Division method

• Hash function
• h(k) k mod m
• ex) k 123, m 15
• gt h(123) 123 mod 15 3
• Certain Value of m should not be used
• m is even
• m is a power of 2
• m is decimal numbers
• m 2P -1 and k is character
• ex) abcd 97 . 83 98 . 82 99 .
  8 100 56828 mod 7 2
• badc 98 . 83 97 . 82
  100 . 8 99 57283 mod 7 2
• cf) Good Value of m are primes
• gt primes not too close to exact powers of 2

21
Multiplication method

• Hash Function
• H(k) ?m(kA mod 1)?
• Two steps in the multiplication method
• 1. The key k is multiplied by a constant A in
  the range 0 lt A lt 1
• and the fractional part of kA extracted.
  
• 2. This fractional part is multiplied by m and
  the floor taken.
• Ex)

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Analysis of Universal hashing

• Theorem 12.3

If h is chosen from a universal collection of
hash functions and is used to hash n keys into a
table of size m, where n ?m, The expected number
of collisions involving a particular key x is
less than 1.
n-1 / m
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Analysis of Universal hashing

• Theorem 12.4

The Class H defined by equations (12.3) and
(12.4) is a universal class of hash functions.
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Open addressing

• Collision Resolution Policy
• All elements are stored in hash table itself
• Each table entry contains
• either an element of the dynamic set or NIL
• Hash table fill up so that
• no further insertions can be made
• Strength
• Save Memory
• Fewer collisions
• Faster retrieval

T
/
0
/
1
69
2
98
3
/
4
72
5
14
6
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Probe
T

• Probe
• Hash table until we find an empty slot in which
  to put the key
• The sequence of positions probed depends upon the
  key being inserted

/
0
/
1
69
2
98
3
/
4
72
5
14
6
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Operations

• Hash-Insert(T, k)
• I 0
• Repeat j h(k, I)
• if Tj NIL
• then Tj k
• return j
• else
• I I I
• Until I m
• Error hash table overflow

Hash-Search(T, k) I 0 Repeat j
h(k, I) if Tj k then return
j I I I Utile TjNIL or I
m Return NIL
/
0
/
1
69
2
98
3
/
4
72
5
14
6
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Operations

• Delete
• Using DELETED Value

Ex2) 98 Delete -gt 100 Search
Ex1) 98 Delete -gt 100 Search
/
0
/
0
/
1
/
1
69
2
69
2
Deleted 98
3
/
3
100
4
100
4
72
5
72
5
14
6
14
6
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Three Techniques of probing
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Linear probing

• D8, keys a,b,c,d have hash values h(a)3,
  h(b)0, h(c)4, h(d)3

b
0

• Where do we insert d? 3 already filled
• Probe sequence using linear hashing
• h1(d) (h(d)1)8 48 4
• h2(d) (h(d)2)8 58 5
• h3(d) (h(d)3)8 68 6
• etc.
• 7, 0, 1, 2
• Wraps around the beginning of the table!

1
2
3
a
c
4
d
5
6
7
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Quadratic probing

• Quadratic probing uses a hash function of the
  form
• h(k, p) (h(k) c1p c2p2) mod m
• Of course, the values of c1, c2 and m determine
  whether or not the entire table will be used.

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Analysis of open-address hashing

• Theorem 12.5

Given and open-address hash table with load
factor a n/m lt1, The expected number of probes
in an unsuccessful search is at most 1/(1-a),
assuming uniform hashing
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Analysis of open-address hashing

• Theorem 12.6

Inserting and element into an open-address hash
table with load factor a requires at most 1/(1-a)
probes on average, assuming uniform hashing.
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Analysis of open-address hashing

• Theorem 12.7

Given an open-address hash table with load factor
a lt 1, the expected number of probes in a
successful is at most Assuming uniform hashing
and assuming that each key in the table is
equally likely to be searched for.
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Simulations

• Linear probing
• Quadratic probing

http//swww.ee.uwa.edu.au/plsd210/ds/hash_tables.
html
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Another Scheme Overflow Area

• Divide the pre-allocated table into two sections
  primary area to which keys are mapped
• overflow area which is an area for collisions
• Possible to design systems
• with multiple overflow tables
• which provide flexibility without losing
• the advantages of the overflow sheme.

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Summary

• Hash Table Organization