Hashing: Collision Resolution Schemes

1
Hashing Collision Resolution Schemes

• Collision Resolution Techniques
• Introduction to Separate Chaining
• Collision Resolution using Separate Chaining
• Introduction to Collision Resolution using Open
• Addressing

2
Collision Resolution Techniques

• There are three broad ways of collision
  resolution
• 1. Separate Chaining A linked list-based
  implementation.
• 2. Open Addressing Array-based implementation.
• (i) Linear probing (linear search)
• (ii) Quadratic probing (nonlinear search)
• (iii) Random increments/decrements
• (iv) Rehashing (double hashing)
• 3. Buckets methods Usually a combination of (1)
  (2)

3
Introduction to Separate Chaining

• The hash table is implemented as an array of
  linked lists.
• Inserting an item, r, at index i is simply
  insertion into the linked list at position i.
• Synonyms are chained in the same linked list.
• Retrieval of an item, r, with hash address, i, is
  simply retrieval from the linked list at position
  i.
• Deletion of an item, r, with hash address, i, is
  simply deleting r from the linked list at
  position i.

4
Separate Chaining with String Keys

• Recall that search keys can be numbers, strings
  or some other object.
• The following Java method implements such
  technique
• public static int hash(String key, int
  tableSize)
• int hashVal 0
• for (int i 0 i lt key.length() i)
• hashVal key.charAt(i)
• return hashVal tableSize
• The following class which describes commodity
  items
• class CommodityItem
• String name // commodity name
• int quantity // commodity quantity needed
• double price // commodity price

5
Example 1 Separate Chaining

• Devise an appropriate hash function and use it to
  load the information about the following
  commodity items into a hash table of size 13
  using separate chaining.
• onion 1 10.0
• tomato 1 8.50
• cabbage 3 3.50
• carrot 1 5.50
• okra 1 6.50
• mellon 2 10.0
• potato 2 7.50
• Banana 3 4.0
• olive 2 15.0
• salt 2 2.50
• cucumber 3 4.50
• mushroom 3 5.50
• orange 2 3.00

6
Example 1 Separate Chaining (cont'd)
okra
potato













0 1 2 3 4 5 6 7 8 9 10 11 12
onion
carrot

• Item Qty Price h(key)
• onion 1 10.0 1
• tomato 1 8.50 10
• cabbage 3 3.50 4
• carrot 1 5.50 1
• okra 1 6.50 0
• mellon 2 10.0 10
• potato 2 7.50 0
• Banana 3 4.0 11
• olive 2 15.0 10
• salt 2 2.50 7
• cucumber 3 4.50 9
• mushroom 3 5.50 6
• orange 2 3.00 12

cabbage
mushroom
salt
cucumber
mellon
tomato
olive
banana
orange
7
Introduction to Open Addressing

• In this method the entries are placed inside the
  array itself.
• The probe sequence is essentially a sequence of
  functions
• h0, h1, h2, , hn-1
• where,
• hi K -gt 0, 1, , n-1
• To insert item r, we examine array locations
• h0(r), h1(r), h2(r), ...,
• Similarly, to find item r, we examine the same
  sequence of
• locations in the same order.

8
Introduction to Open Addressing (cont'd)

• The most common probe sequences are of the form
• hi(r) (h(r) c(i)) mod n, i 0, 1, ,
  n-1.
• The function c(i) is required to have the
  following two properties
• Property 1
• c(0) 0.
• Property 2 The set of values
• c(0) mod n, c(1) mod n, c(2) mod n, ,
  c(n-1) mod n
• must contain every integer between 0 and n-1
  inclusive.

9
Open Addressing Linear Probing

• Linear Probe Here the function c(i) is a linear
  function
• in i
• c(i) ai b
• Property 1 requires that c(0) 0. Therefore, b
  must be
• zero.
• For c(i) ai to satisfy Property 2, a and n
  must be relatively prime.
• The linear probing sequence that is usually used
  is
• hi (r) (h(r) i) mod n, i0,1,2,, n-1
• Insert record at first empty slot and if no empty
  slot is found then the hash table is full and
  insertion fails.

10
Example 2 Linear Probing

• Use the hash function h(r) r.id 13 to load
  the following records into an array of size 13.
• Al-Otaibi Ziyad 1.73 985926
• Al-Turki, Musab Ahmad Bakeer 1.60 970876
• Al-Saegh, Radha Mahdi 1.58 980962
• Al-Shahrani, Adel Saad 1.80 986074
• Al-Awami, Louai Adnan Muhammad 1.73 970728
• Al-Amer, Yousuf Jauwad 1.66 994593
• Al-Helal, Husain Ali AbdulMohsen 1.70 996321
• Then insert the following records using
  linear probing to
• resolve collisions, if any.
• Al-Najjar, Khaled Ziyad 1.69 987615
• Al-Ali, Amr Ali Zaid 1.79 987630
• Al-Ramadi, Husam Yahya 1.58 987602

11
Example 2 Introduction to Hashing (cont'd)
0 1 2 3 4 5 6
7 8 9 10 11 12

Husain
Yousuf
Khalid
Radha
Amr
Musab
Adel
Husam
Louai
Ziyad
12
Linear Probing Some Notes

• Notice from this table that a large cluster has
  already been formed.
• In general, empty cells following the cluster
  have higher
• chance of being hashed into.
• The probability of taking longer probe sequences
  is much
• higher with clusters.
• This is one disadvantage of linear probing. Other
  methods
• attempt to improve on this.

13
Introduction to Retrieval Deletion

• Retrieval To search for a record we
• Calculate its hash value.
• Check that location of the array for the record.
• If found, return the record.
• If not, keep searching until you find the
  record or you
• reach an empty table location.
• Attempting to retrieve a non-existent record is
  very expensive.
• Deletion
• In open addressing, where a record is stored is
  not ecessarily its home position.
• We cannot just set the location of a deleted
  record to empty.
• A special flag or key value is needed to mark
  deleted records
• locations.

14
Example 3 Retrieval Deletion

• Consider the following hash table constructed in
  Example 2

0 1 2 3 4 5 6
7 8 9 10 11 12

Husain
Yousuf
Khalid
Radha
Amr
Musab
Adel
Husam
Louai
Ziyad
Delete Khalid's record (id 987615) and then
retrieve the records for Amr and then that of
Husam.
15
Example 3 Retrieval Deletion
0 1 2 3 4 5 6
7 8 9 10 11 12

?
Husain
Yousuf
Radha
Amr
Musab
Adel
Husam
Louai
Ziyad
16
Exercises

• 1.Given that,
• c(i) ai,
• for c(i) in linear probing, we discussed that
  this equation
• satisfies Property 2 only when a and n are
  relatively prime. Explain what the
• requirement of being relatively prime means in
  simple plain language.
• 2.Consider the general probe sequence,
• hi (r) (h(r)
  c(i))mod n.
• Are we sure that if c(i) satisfies Property 2,
  then hi(r) will cover all n hash table
• locations, 0,1,...,n-1? Explain.
• 3.Suppose you are given k records to be loaded
  into a hash table of
• size n, with k lt n using linear probing. Does the
  order in which these
• records are loaded matter for retrieval and
  insertion? Explain.
• 4.A prime number is always the best choice of a
  hash table size. Is this
• statement true of false? Justify your answer
  either way.