Asst. Prof. Dr. İlker Kocabaş Hash Tables. 2 Overview Information Retrieval Binary Search Trees Hashing. Applications. Example. Hash Functions. Hash Tables

Asst. Prof. Dr. İlker Kocabaş

Hash Tables

2

Overview

Information RetrievalBinary Search TreesHashing.Applications.Example.Hash Functions.

Hash TablesCollisionsLinear ProbingProblems with Linear ProbChaining

3

R. Kruse, C. Tondo, B. Leung, “Data Structures and Program Design in C”, 1991, Prentice Hall.

E. Horowitz, S. Salini, S. Anderson-Freed, “Fundamentals of Data Structures in C”, 1993, Computer Science Press.

R. Sedgewick, “Algorithms in C”, 1990, Addison-Wesley.

A. Aho, J. Hopcroft, J. Ullman, “Data Structures and Algorithms”, 1983, Addison-Wesley.

T.A. Standish, “Data Structures, Algorithms & Software Principles in C”, 1995, Addison-Wesley.

D. Knuth, “The Art of Computer Programming”, 1975, Addison-Wesley.

Y. Langsam, M. Augenstein, M. Fenenbaum, “Data Structures using C and C++”, 1996, Prentice Hall.

Example: Bibliography

4

Insert the information into a Binary Search Tree,

using the first author’s surname as the key

5

Kruse

Horowitz Sedgewick

Aho Knuth

Langsam

Standish

Insert the information into a Binary Search Tree,

using the first author’s surname as the key

Kruse Horowitz Sedgewick Aho Knuth Langsam Standish

6

Complexity

Inserting Balanced Trees O(log(n)) Unbalanced Trees O(n)

Searching Balanced Trees O(log(n)) Unbalanced Trees O(n)

7

Hashing

key

hash function

0

1

2

3

TABLESIZE - 1

:

:

hash table

pos

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“Kruse”

0

1

2

3

6

4

5

hash table

Example:

5

Kruse

hash function

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Hashing

Each item has a unique key.Use a large array called a Hash Table.Use a Hash Function.

10

Applications

Databases.Spell checkers.Computer chess games.Compilers.

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Operations

Initialize all locations in Hash Table are empty.

InsertSearchDelete

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Hash Function

Maps keys to positions in the Hash Table.Be easy to calculate.Use all of the key.Spread the keys uniformly.

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unsigned hash(char* s){ int i = 0; unsigned value = 0; while (s[i] != ‘\0’) { value = (s[i] + 31*value) % 101; i++; } return value;}

Example: Hash Function #1

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A. Aho, J. Hopcroft, J. Ullman, “Data Structures and Algorithms”, 1983, Addison-Wesley.

‘A’ = 65

‘h’ = 104

‘o’ = 111

value = (65 + 31 * 0) % 101 = 65

value = (104 + 31 * 65) % 101 = 99

value = (111 + 31 * 99) % 101 = 49


value = (s[i] + 31*value) % 101;

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resultingtable is “sparse”


value = (s[i] + 31*value) % 101;Hash

Key ValueAho 49Kruse 95Standish 60Horowitz 28Langsam 21Sedgewick 24Knuth 44

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value = (s[i] + 1024*value) % 128;


likely toresult in

“clustering”

Hash Key ValueAho 111Kruse 101Standish 104Horowitz 122Langsam 109Sedgewick 107Knuth 104

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“collisions”

value = (s[i] + 3*value) % 7;

Hash Key Value

Aho 0Kruse 5Standish 1Horowitz 5Langsam 5Sedgewick 2Knuth 1

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Insert

Apply hash function to get a position.Try to insert key at this position.Deal with collision.

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Aho Hash Function

0

1

2

3

6

4

5

hash table

Aho, Kruse, Standish, Horowiz, Langsam, Sedgewick, Knuth

0

Example: Insert

Aho

20

Kruse

0

1

2

3

6

4

5

hash table


5

Example: Insert

Aho

Kruse

Hash Function

21

Standish

0

1

2

3

6

4

5

hash table


1

Example: Insert

Aho

Kruse

StandishHash

Function

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Search

Apply hash function to get a position.Look in that position.Deal with collision.

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Kruse

Kruse

0

1

2

3

6

4

5

hash table


5

Example: Search

Aho

StandishHash

Function

found.

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Kruse

Sedgwick

0

1

2

3

6

4

5

hash table


2

Example: Search

Aho

StandishHash

Function

Not found.

Hash Tables:Collision Resolution

26

Hashing

key

hash function

0

1

2

3

TABLESIZE - 1

:

:

hash table

pos

27

“Kruse”

0

1

2

3

6

4

5

hash table

Example:

5

Kruse

hash function

28

Hashing

Each item has a unique key.Uses a large array called a Hash Table.Uses a Hash Function.

Hash Function• Maps keys to positions in the Hash Table.• Be easy to calculate.• Use all of the key.• Spread the keys uniformly.

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Hash Table Operations

Initialize all locations in Hash Table are empty.

InsertSearchDelete

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“collisions”

value = (s[i] + 3*value) % 7;

Hash Key Value

Aho 0Kruse 5Standish 1Horowitz 5Langsam 5Sedgewick 2Knuth 1

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Collision

When two keys are mapped to the same position.Very likely.

10

20

30

40

50

60

70

0.1169

0.4114

0.7063

0.8912

0.9704

0.9941

0.9992

Number of People ProbabilityBirthdays

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Collision Resolution

Two methods are commonly used: Linear Probing. Chaining.

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Linear Probing

Linear search in the array from the position where collision occurred.

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Insert with Linear Probing

Apply hash function to get a position.Try to insert key at this position.Deal with collision. Must also deal with a full table!

35

Aho Hash Function

0

1

2

3

6

4

5

hash table


0

Example: Insert with Linear Probing

Aho

36

Kruse

0

1

2

3

6

4

5

hash table


5


Aho

Kruse

Hash Function

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Standish

0

1

2

3

6

4

5

hash table


1


Aho

Kruse

Standish

Hash Function

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Horowitz

0

1

2

3

6

4

5

hash table


5


Aho

Kruse

Standish

Horowitz

Hash Function

39

Langsam

0

1

2

3

6

4

5

hash table


5


Aho

Kruse

Standish

Horowitz

Hash Function

Langsam

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module linearProbe(item){ position = hash(key of item) count = 0 loop { if (count == hashTableSize) then { output “Table is full” exit loop } if (hashTable[position] is empty) then { hashTable[position] = item exit loop } position = (position + 1) % hashTableSize count++ }}

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Search with Linear Probing

Apply hash function to get a position.Look in that position.Deal with collision.

42

Kruse

Langsam

0

1

2

3

6

4

5

hash table


5

Example: Search with Linear Probing

Aho

Standish

Horowitz

Hash Function

Langsam

found.

43

Kruse

Knuth

0

1

2

3

6

4

5

hash table

1

Example: Search with Linear Probing

Aho

Standish

Horowitz

Hash Function

Langsam

not found.


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module search(target){ count = 0 position = hash(key of target) loop { if (count == hashTableSize) then { output “Target is not in Hash Table” return -1. } else if (hashTable[position] is empty) then { output “Item is not in Hash Table” return -1. } else if (hashTable[position].key == target) then { return position. } position = (position + 1) % hashTableSize count++ }}

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Delete with Linear Probing

Use the search function to find the itemIf found check that items after that also don’t hash to the item’s positionIf items after do hash to that position, move them back in the hash table and delete the item.

Very difficult and time/resource consuming!Very difficult and time/resource consuming!

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Linear Probing: Problems

Speed.Tendency for clustering to occur as the table becomes half full.Deletion of records is very difficult.If implemented in arrays – table may become full fairly quickly, resizing is time and resource consuming

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Chaining

Uses a Linked List at each position in the Hash Table. Linked list at a position contains all the items that ‘hash’

to that position. May keep linked lists sorted or not.

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hash table0

1

2

3

:

:

49

0

1

2

3

6

4

5

Aho, Kruse, Standish, Horowiz, Langsam, Sedgwick, Knuth

Example: Chaining

0, 5, 1, 5, 5, 2, 1

3

1

2

Kruse Horowitz

Knuth

1

Standish

Aho

Sedgewick

Langsam

0

0

0

50

Hashtable with ChainingAt each position in the array you have a list:

List hashTable[MAXTABLE];

You must initialise each list in the table.0

1

2

1

2

1

:

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Insert with Chaining

Apply hash function to get a position in the array.Insert key into the Linked List at this position in the array.

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module InsertChaining(item){ posHash = hash(key of item)

insert (hashTable[posHash], item); }

0

1

2

1

2 Knuth

1

Standish

Aho

Sedgewick:

53

Search with Chaining

Apply hash function to get a position in the array.Search the Linked List at this position in the array.

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/* module returns NULL if not found, or the address of the * node if found */

module SearchChaining(item){ posHash = hash(key of item) Node* found;

found = searchList (hashTable[posHash], item);

return found;

}

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0

1

2

1

2 Knuth

1

Standish

Aho

Sedgewick:

55

Delete with Chaining

Apply hash function to get a position in the array.Delete the node in the Linked List at this position in the array.

/* module uses the Linked list delete function to delete an item *inside that list, it does nothing if that item isn’t there. */

module DeleteChaining(item){ posHash = hash(key of item) deleteList (hashTable[posHash], item);}

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0

1

2

1

2 Knuth

1

Standish

Aho

Sedgewick:

57

Disadvantages of Chaining

Uses more space.More complex to implement. Contains a linked list at every element in

the array. Requires linear searching. May be time consuming.

58

Advantages of Chaining

Insertions and Deletions are easy and quick.Allows more records to be stored.Naturally resizable, allows a varying number of records to be stored.

Dictionaries and Hash Tables 59

Double HashingDouble hashing uses a secondary hash function d(k) and handles collisions by placing an item in the first available cell of the series

(i jd(k)) mod N for j 0, 1, … , N 1The secondary hash function d(k) cannot have zero valuesThe table size N must be a prime to allow probing of all the cells

Common choice of compression map for the secondary hash function: d2(k) q k mod q

where q N q is a prime

The possible values for d2(k) are

1, 2, … , q

Dictionaries and Hash Tables 60

Performance of Hashing

In the worst case, searches, insertions and removals on a hash table take O(n) timeThe worst case occurs when all the keys inserted into the dictionary collideThe load factor nN affects the performance of a hash tableAssuming that the hash values are like random numbers, it can be shown that the expected number of probes for an insertion with open addressing is

1 (1 )

The expected running time of all the dictionary ADT operations in a hash table is O(1) In practice, hashing is very fast provided the load factor is not close to 100%Applications of hash tables:

small databases compilers browser caches

Documents

Asst. Prof. Dr. İlker Kocabaş Hash Tables. 2 Overview Information Retrieval Binary Search Trees Hashing. Applications. Example. Hash Functions. Hash Tables