hash tables dr. li jiang school of computer science, the university of adelaide 1

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Hash Tables

Dr. Li JiangSchool of Computer Science,

The University of Adelaide

Overview Hash Table ADT

Table ADT Direct addressing and its problem

Hash table and hash table ADT operations Hash Function

Example of using a hash function Benefit of using a hash function Problem of using a hash function

Collision and collision resolution Collisions Resolution

An example of using chaining

Learning Objectives

By the end of this lecture, you should be able to: Understand and interpret the concepts of hash

table and hash function. Define hash table function and hash table

operations Understand the collision and one of the collision

resolution approaches – chaining approach Use chain approach to solve collision problem

An Example of A Table

BHM Birmingham International Airport

LGB Long Beach

LAX Los Angeles International Airport

OAK Oakland

IAD Washington, Dulles International Airport

HNL Honolulu International Airport

BOS Boston, Logan International Airport

ACY Atlantic City International Airport

CLE Cleveland

PDX Portland International Airport

(Key, Value)

An Example of A Table

BHM Birmingham International Airport

LGB Long Beach

LAX Los Angeles International Airport

OAK Oakland

IAD Washington, Dulles International Airport

HNL Honolulu International Airport

BOS Boston, Logan International Airport

ACY Atlantic City International Airport

CLE Cleveland

PDX Portland International Airport

Key Associated Information (Airports name, or related information )

(cont.)

An Example of A Table

BHM Birmingham International Airport

LGB Long Beach

LAX Los Angeles International Airport

OAK Oakland

IAD Washington, Dulles International Airport

HNL Honolulu International Airport

BOS Boston, Logan International Airport

ACY Atlantic City International Airport

CLE Cleveland

PDX Portland International Airport

Key Associated Information

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(cont.)

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Direct Addressing• Suppose there are n objects required to store in the table:

– The range of keys is 0..n-1 – Keys are distinct

• The idea of the direct addressing:– Table is represented with an array, e.g. airportInfo[0..n-1]– Insert an object to the airport information table

• airportInfo[i] = x if x airportInfo and key[x] = i• airportInfo[i] = NULL otherwise

– Efficiency of the algorithms implementing the operations of Table ADT• Insert operation takes O(1) time• Search operation takes O(n) time• Delete operation takes O(n) time

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Advantages of Direct Addressing

If number of objects and size of table is reasonably small:

• Direct Addressing is an efficient way to access the data

• It takes less time for any operation on direct addressing table.

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Problems with Direct Addressing

When the size of table is very large:• Using a table T of size N and N is a large

number (e.g. >10000), using direct addressing may be impractical, given the memory available on a typical computer.– The number of the objects actually stored

may be so small relative to large space created. Thus, most of the space allocated for T would be wasted.

An Example of A Table (1)

BHM Birmingham International Airport

LGB Long Beach

LAX Los Angeles International Airport

OAK Oakland

IAD Washington, Dulles International Airport

HNL Honolulu International Airport

BOS Boston, Logan International Airport

ACY Atlantic City International Airport

CLE Cleveland

PDX Portland International Airport

Key Associated Information

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An Example of Table

• Assume that– Data items of 400 airports needs to be processed.– The key: Airport code with three letters, used to

identify each airport.

If direct addressing approach is used,

• Number of different three letter combinations will be 26 × 26 × 26 =17576 (possible number of airports)

• The fraction of actual keys (Buckets) needed: 400/17576=2.2%• Percent of the memory allocated for table wasted,

97.8%• Again, the operations on the table will take: O(1) to O(n) time

BucketsThe data item of one airport

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Another ExampleAssume that:• A table is needed to store 50 students in a class.• The key is defined as 9 digit Student Identification Number,

used to identify each student.

If direct addressing approach is used, we will find that

• Number of different 9 digit number will be 109

• The fraction of actual keys needed. 50/109, 0.000005%

• Percent of the memory allocated for table wasted, 99.999995%

A better way is necessary Hash Table

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Hash Table ADT The hash table is a table of elements that have keys,

usually represented as (Key, Element) pair which is actually a table.

A hash function is used for locating a position in the table

Dictionary ADT

Can be any type of object

h( key ) Location of the object containing the key

A hash table maps a huge set of possible keys into index of N buckets by applying a hash function to each hash code

Key S, where S is usually a huge set of possible keys

Notice : Ns = Card |S|, Ns is much larger than N, n is the actual number of objects that are processed

Ideally, n =N or n=a× N +b where a and b is small number

Hash Functions The input into a hash function is a key value

The output from a hash function is an index of an array (hash table) where the object containing the key is located

The most commonly used hash function is:

h( hashCode ) = hashCode mod N

Where the hashCode is the key of an element, N is the number of buckets that is actually used

Notice that the hashCode is not often obvious, building a model to compute it is often required.

Examples of Hash Functions

(1) h( k ) = k % 101 if k is an integer and it is the key for the associated element

(2) What the hash function of the Airport Code will be for processing data items of up to 400 airports?

One of the answers will be: h(Ariport_code) =p(fitstChar) × p(secondChar) × p(thirdChar)%400

p is a position function which maps a character to its position value

Divisor is usually the size of the table, it is set to a prime when

the keys contains a lot of 0s

A B C D E F G H I J K L ……1 2 3 4 5 6 7 8 9 10 11 12 ……

h(CLE) =3 × 12 × 5%400=180

h(CLE)=?

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Hash Table ADT Operations

Insert: to insert an element into a table Retrieve: to retrieve an element from the table Remove: to remove an element from the table Update: to update an element in the table Empty: to empty out the hash table

Inserting an Object in A HashTable

The following pseudo-code for the insert operation:

public: bool insert( key, object) {

1. Compute the key's hash code. 2. Compute the hash function to determine the index of bucket. 3. Insert the object into the bucket's chain with the index of the

bucket obtained from 2.

}

Insertion is done in O( 1 ) time

Notice that here is bucket’s chain, instead of bucket.

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Inserting an Object in A HashTable

An example of insert operation An element (Cleveland) is inserted into a

hash table.(suppose we only need to deal with 101 big airports)

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Buckets

Clevelandh(CLE)=h(180)=180%101=79

What the hash function will be?

h( k ) = k % 101

– To find where an element is to be inserted, use the hash function on its key

– If the key value is 180, the element is to be stored in index 79 of the array

– Insertion is done in O( 1 ) time

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Benefit of Using a Hash Function

Using a hash table, we simply have a function which provides us with the index of the array where the object containing the key is located Other alternative is expensive

If we have millions of objects with (key, values) structure, it may take a long time to search a regular array or a linked list for a specific part number (on average, we might compare 500,000 key values)

Problem of Using a Hash Function Consider the hash function

h( k ) = k % 100

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• a key value of 114 is used for a second object; the result of the hash function is 14, but index 14 is already occupied,

– This is called a collision

How shall we solve this problem?

Collision is the circumstance where several keys hash to the same bucket. This happens when: h( hashCode1 ) == h( hashCode2 )

Suppose that • a key value of 214 is used for an object, and the

object is stored at index 14

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How are Collisions Resolved?

The most popular way to resolve collisions is by chaining Instead of having an array of objects, we

have an array of linked lists, each node of which contains an object

An element is inserted by using the hash function -- the hash function provides an index of a linked list, and the element is inserted at the front of that (usually short) linked list

When searching for an element, the hash function is used to get the correct linked list, then the linked list is searched for the key (and the element) If we had 500,000 keys, this approach

is still much faster than comparing 500,000 keys with other approaches)

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Note: The whole object is stored but only the key value is shown

Value

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Searching an Object in A HashTable

Pseudo-code for the retrieve (search, find) operation

• A search for an element can be done in O( 1 ) time.

The following pseudo-code for the retrieve (find) operation:public: bool retrieve( DataType & key) { 1. Hash the key

• find the hash code and compute hash function with the given key to obtain the index of the bucket.

2. Search through the linked list specified by the bucket index number.

3. If you find the entry with the right key you return it; otherwise return null.

}

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Searching an Object in A HashTable

An example of search operation Suppose our hash function is:

h( k ) = k % 100 We wish to search for the object containing

key value 214 If k is set to 214 in the hash function above, the

result is 14 The object containing key 214 is stored at

index 14 of the array (hash table)

An Example of HashTable Class

template <class DataType> class HashTable { public:

HashTable( int (*hf)(const DataType &), int s ); bool insert( const DataType & newObject ); // returns true if successful;

// returns false if invalid index was returned from hash function bool retrieve( DataType & retrieved ); // retrieve the item for the given key bool remove( DataType & removed ); // remove the item for the given key bool update( DataType & updateObject ); // update the item for the given key void makeEmpty( ); // empty out the hash table

private: Array< LinkedList<DataType> > table; int (*hashfunc)(const DataType &); // pointer to hash function

};

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An Example of Using Chaining

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A hash table which is initially empty.

Every element is a LinkedList object. Only the start pointer of the LinkedList object is shown, which is set to NULL at the beginning.

The hash function is: h( k ) = k % 7

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An Example of Using Chaining (cont.)

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INSERT object with key 31

The hash function is: h( k ) = k % 7

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Example Using Chaining (cont.)

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INSERT object with key 31

The hash function is: h( k ) = k % 7

h(31)=31 % 7= 3

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Example Using Chaining (cont.)

Assumed that the hash function is: h( k ) = k % 7

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6Note: The whole object is stored but only the key value is shown

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Example Using Chaining (cont.)

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The hash function is: h( k ) = k % 7

INSERT object with key 931

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Example Using Chaining (cont.)

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INSERT object with key 9

9 % 7 = 231

The hash function is: h( k ) = k % 7

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Example Using Chaining (cont.)

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9 INSERT object with key 9

9 % 7 is 231

The hash function is: h( k ) = k % 7

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Example Using Chaining (cont.)

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INSERT object with key 36

36 % 7 is 1

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Example Using Chaining (cont.)

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INSERT object with key 36

36 % 7 is 1

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Example Using Chaining (cont.)

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INSERT object with key 42

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Example Using Chaining (cont.)

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INSERT object with key 42

42 % 7 is 0

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Example Using Chaining (cont.)

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INSERT object with key 42

42 % 7 is 0

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Example Using Chaining (cont.)

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INSERT object with key 46

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Example Using Chaining (cont.)

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INSERT object with key 46

46 % 7 is 4

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Example Using Chaining (cont.)

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INSERT object with key 46

46 % 7 is 4

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Example Using Chaining (cont.)

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INSERT object with key 20

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Example Using Chaining (cont.)

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INSERT object with key 20

20 % 7 is 6

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Example Using Chaining (cont.)

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INSERT object with key 20

20 % 7 is 6

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Example Using Chaining (cont.)

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INSERT object with key 2

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Example Using Chaining (cont.)

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INSERT object with key 2

2 % 7 is 2

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Example Using Chaining (cont.)

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COLLISION occurs !!

INSERT object with key 2

2 % 7 is 2

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But an object has been inserted in the location with index 2 of the linked list before

Inserts the new element at the BEGINNING of the list

How to resolve this?

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Example Using Chaining (cont.)

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INSERT object with key 2

2 % 7 is 2

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Example Using Chaining (cont.)

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INSERT object with key 2

2 % 7 is 2

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Example Using Chaining (cont.)

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INSERT object with key 2

2 % 7 is 2

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Example Using Chaining (cont.)

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INSERT object with key 2

2 % 7 is 2

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Example Using Chaining (cont.)

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2 % 7 is 2

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Example Using Chaining (cont.)

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INSERT object with key 24 9

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Example Using Chaining (cont.)

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INSERT object with key 24

24 % 7 is 3

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Example Using Chaining (cont.)

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INSERT object with key 24

24 % 7 is 3

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Example Using Chaining (cont.)

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INSERT object with key 24

24 % 7 is 3

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Example Using Chaining (cont.)

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INSERT object with key 24

24 % 7 is 3

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Example Using Chaining (cont.)

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INSERT object with key 24

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Example Using Chaining (cont.)

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24 % 7 is 3

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Example Using Chaining (cont.)

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e.g. FIND the object with key 9

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Supposed that all objects were stored in the linked list.

How to Find an object?

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Example Using Chaining(cont.)

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FIND the object with key 9

9 % 7 is 2

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Example Using Chaining(cont.)

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We search this linked list for the object with key 9

FIND the object with key 9

9 % 7 is 2

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Example Using Chaining(cont.)

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Remember…the whole object is stored, only the key is shown

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Example Using Chaining(cont.)

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Does this object contain key 9?

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Example Using Chaining(cont.)

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FIND the object with key 9

Does this object contain key 9?

No, so go on to the next object.

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Example Using Chaining(cont.) FIND the object with key 9

Does this object contain key 9?

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Example Using Chaining(cont.)

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Does this object contain key 9? YES, found it! Return the object.

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FIND the object with key 9

Summary Hash Table ADT

Table ADT Direct addressing and its problem

Hash table and hash table ADT operations Hash Function

Example of using a hash function Benefit of using a hash function Problem of using a hash function

Collision and collision resolution Collisions Resolution

An example of using chaining

END

Thank You !

Look Forward To Seeing You Again !