11. Hash Tables

1
11. Hash Tables

• Heejin Park
• College of Information and Communications
• Hanyang University

2
Contents

• Direct-address tables
• Hash tables
• Hash functions
• Open addressing

3
Hash functions

• What makes a good hash function?
• Assumption of simple uniform hashing
• Each key is equally likely to hash to any of the
  m slots.
• Each key hashes independently of where any other
  key has hashed to.

4
Hash functions

• Interpreting keys as natural numbers
• Hash functions assume that the universe of keys
  is the set N 0, 1, 2, ... of natural numbers.
• If the keys are not natural numbers, a way is
  found to interpret them as natural numbers.
• For example pt
• p 112 and t 116 in the ASCII character set
• Expressed as a radix-128 integer
• pt becomes (112128)116 14452

5
The division method

• The division method
• Map a key k into one of m slots by taking the
  remainder of k divided by m.
• h(k) k mod m.
• Example
• m 12, k 100
• h(k) 100 mod 12 4

6
The division method

• Certain values of m are avoided.
• m 2p
• h(k) is just the p lowest-order bits of k.
• m 24, h(k) k mod 24
• k 10110100, m 00010000, h(k) 0100
• Thus m should not be a power of 2.
• m 2p 1
• When k is a character string interpreted in radix
  2p, permuting the characters of k does not change
  its hash value.

7
The division method

• A good choice
• A prime not too close to an exact power of 2.
• If the number of keys are about 2000 and we don't
  mind examining an average of 3 elements in an
  unsuccessful search, we can allocate a hash table
  of size m 701.
• 701 is a prime near 2000/3 but not near any power
  of 2.
• h(k) k mod 701.

8
The multiplication method

• The multiplication method
• h(k) ?m(kA mod 1)?
• Multiply the key k by a constant A in the range
  (0 lt A lt 1) and extract the fractional part of
  kA.
• Multiply this value by m and take the floor of
  the result.

9
The multiplication method

• An advantage of the multiplication method is that
  the value of m is not critical.
• We typically choose m to be a power of 2 (m
  2p).
• It makes the implementation of the function easy.
• Suppose that the word size of the machine is w
  bits and that k fits into a single word.
• We restrict A to be a fraction of the form s/2w.
• where s is an integer in the range 0 lt s lt 2w

10
The multiplication method
w bits
k
x
s A2W
r0
r1
h(k)
p bits
11
Open addressing

• Open addressing
• All elements are stored in the hash table itself.
• The advantage of open addressing
• It avoids pointers altogether.
• The extra memory provides the hash table with a
  larger number of slots for the same amount of
  memory.
• Yielding fewer collisions and faster retrieval.

12
Open addressing

• Insertion
• Examine the hash table (probe) until it finds an
  empty slot.
• The sequence of positions probed depends upon the
  key being inserted.
• The probe sequence for every key k
• be a permutation of lt0, 1, , m-1gt.

lt h(k, 0), h(k, 1), . . . , h(k,m-1) gt
13
Open addressing

• HASH-INSERT
• It takes as input a hash table T and a key k.

14
Open addressing

• HASH-SEARCH
• It takes as input a hash table T and a key k.

15
Open addressing

• Deletion
• Can you remove the key physically?
• Mark the slot by the special value DELETED.

16
Open addressing

• Three common techniques for open addressing.
• Linear probing
• Quadratic probing
• Double hashing

17
Linear probing

• Given an ordinary hash function h U ? 0, 1,
  ... , m-1, which we refer to as an auxiliary
  hash function, the method of linear probing uses
  the hash function
• for i 0, 1, , m-1

h(k, i) (h(k) i) mod m
18
Linear probing
T

• m 13
• k 5, 14, 29, 25, 17, 21, 18, 32, 20, 9, 15, 27

0
1
2
3
4
5
6
7
8
9
10
11
12
14
27
15
29
17
h(k, i) (k i) mod 13
5
18
32
20
21
9
25
19
Linear probing

• Linear probing is easy to implement, but it
  suffers from a problem known as primary
  clustering.
• Long runs of occupied slots build up, increasing
  the average search time.
• Clusters arise since an empty slot preceded by i
  full slots gets filled next with probability (i
  1) / m.
• Long runs of occupied slots tend to get longer,
  and the average search time increases.

20
Quadratic probing

• Quadratic probing use a hash function of the form
• where h is an auxiliary hash function, c1 and c2
  ? 0 are auxiliary constants, and i 0, 1, , m-1.

h(k, i) (h(k) c1 i c2 i²) mod m
21
Quadratic probing
T

• m 13
• k 5, 14, 29, 25, 17, 21, 18, 32, 20, 9, 15, 27

0
1
2
3
4
5
6
7
8
9
10
11
12
14
27
15
29
17
h(k, i) (k i 3i²) mod 13
5
18
32
20
21
9
25
22
Quadratic probing

• If two keys have the same initial probe position,
  then their probe sequences are the same, since
  h(k1, 0) h(k2, 0) implies h(k1, i) h(k2, i).
• This property leads to a milder form of
  clustering, called secondary clustering.

23
Double hashing

• Double hashing uses a hash function of the form
• The initial probe is to position Th1(k).
• Successive probe positions are offset from
  previous positions by the amount h2(k), modulo m.

h(k, i) (h1(k) i h2(k)) mod m
24
Double hashing

• The value h2(k) must be relatively prime to the
  hash-table size m for the entire hash table to be
  searched.
• A way to ensure this condition is to let m be a
  power of 2 and to design h2 so that it always
  produces an odd number.
• Another way is to let m be prime and to design h2
  so that it always returns a positive integer less
  than m.

25
Double hashing
T

• m 13
• k 5, 14, 29, 25, 17, 21, 18, 32, 20, 9, 15, 27

0
1
2
3
4
5
6
7
8
9
10
11
12
14
27
15
29
h1(k) k mod 13
17
5
18
h2(k) 1 (k mod 11)
32
h(k, i) (h1(k) i h2(k)) mod 13
20
21
9
25