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**Hash Tables Dr. Li Jiang School of Computer Science, The University of Adelaide**

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**Overview Hash Table Hash Function Hash table ADT operations**

Direct addressing and its problem Hash Table Concept Hash Function Example of using a hash function Benefit and problem of using a hash function Hash table ADT operations Collision and collision resolution Examples: ADT operations and Using chaining approach to resolve collision Hash Table ADT For Hash Table ADT, I will start with using direct-addressing approach for Table related operation and reveal the problem of this approach, and illustrate the necessity of using Hash Table. Then we will talk about - Hash table and hash table ADT operations We will talk about Hash Function Example of using a hash function Benefit of using a hash function Problem of using a hash function From the problem of using a hash function, we will also introduce the concept of collision, and introduce one of collision resolution approaches. Collisions Resolution Finally, through Examples, we will illustrate: ADT operations and Using chaining approach to resolve collision

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**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 for solving simple problem Understand the collision and one of the collision resolution approaches – chaining approach Use chaining approach to solve collision problem

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**An Example of A Table (Key, Value) 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 We have deal with a lot programming issues with Table. The data structure of the table is simple. (Key, Value) There are different approaches to address the elements in the table. For the operations , insert information about new airports, delete an airport name, query about the name of an airport. It can be done in a reasonable efficient manner when N, the number of items is small. Table is an abstract storage device that contains table entries Each table entry contains a unique key k. Each table entry may also contain some information, I, associated with its key. A table entry is an ordered pair (K, I)

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**Associated Information (Airports name, or related information )**

An Example of A Table (cont.) Key Associated Information (Airports name, or related information ) 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 One approach is using a string as a key, Such as airport code Suppose that we want you to develop a program to save the information, insert information about new airports, delete an airport name, query about the name of an airport. Table is an abstract storage device that contains table entries Each table entry contains a unique key k. Each table entry may also contain some information, I, associated with its key. A table entry is an ordered pair (K, I)

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**Associated Information**

An Example of A Table (cont.) Key Associated Information 1 2 3 4 5 6 7 8 9 10 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 Or using numerical values as key. To represent and implement the Table ADT Suppose that we want you to develop a program to save the information, insert information about new airports, delete an airport name, query about the name of an airport. Table is an abstract storage device that contains table entries Each table entry contains a unique key k. Each table entry may also contain some information, I, associated with its key. A table entry is an ordered pair (K, I)

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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 , each number is (uniquely mapped to) the address of a bucket in the table Keys are distinct The idea of the direct addressing: Table is represented with an array, e.g. airportInfo[0..n-1] Efficiency of the algorithms implementing the operations of Table ADT with direct addressing approach: Insert an object to the airport information table airportInfo[i] = x if x airportInfo and key[x] = i airportInfo[i] = NULL otherwise Insert operation takes O(1) time (no order requirement) Search operation takes O(n) time Delete operation takes O(n) time Direct Addressing approach is probably the easiest way to implement a table. The idea of direct addressing is:

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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. When you are not sure how many data items are and you know the number of data items will be big. People tend to create a much bigger table.

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**Associated Information (Airports name, or related information )**

An Example of A Table (cont.) Key Associated Information (Airports name, or related information ) 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 Table is an abstract storage device that contains table entries Each table entry contains a unique key k. Each table entry may also contain some information, I, associated with its key. A table entry is an ordered pair (K, I)

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**The data item of one airport**

An Example of Table (cont.) The data item of one airport Buckets 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, % Again, the operations on the table will take: O(1) to O(n) time How to represent and store the table such that all the operations related to the table can be executed efficiently. Pneumonoultramicroscopicsilicovolcanoconiosis : lung related disease = 4.7e+63 Honorificabilitudinitatibus: 27 : Longest word in Shakespeare's works = 1.6e + 38

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**Another Example Assume that: A better way is necessary Hash Table**

The information about 50 students in a class is to be stored in a table. 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 possible keys with 9 digit number will be 109 The fraction of actual keys needed. 50/109, % Percent of the memory allocated for table wasted, % Apparently, this is not a good way to do it. A better way is necessary Hash Table

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**Can be any type of object**

Hash Table Dictionary The hash table is a table of elements that have keys, usually represented as (Key, Element) pair A hash function is used for locating a position in the table Can be any type of object h( key ) Location of the object containing the key Key S, where S is usually a huge set of possible keys A hash table maps a huge set of possible keys into index of N buckets by applying a hash function to each hash code The elements of the table is the set of data acted upon by the hash table operations The term "hash" comes by way of analogy with its non-technical meaning, to "chop and mix". Indeed, typical hash functions, like the mod operation, "chop" the input domain into many sub-domains that get "mixed" into the output range to improve the uniformity of the key distribution. Notice that uniformity does not mean injective mapping, it might be bijective mapping as well. hash function thoroughly 'mixes up' the values of its domain. because you chop your input that you put in pieces in different places or buckets (your table entries). 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

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**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 How to define a hash function is not trivial for some applications. 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 required.

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**Examples of Hash Functions**

Divisor is usually the size of the table, it is set to a prime when the keys contains a lot of 0s h( k ) = k % 101 if k is an integer and it is the key for the associated element h(CLE)=? A B C D E F G H I J K L …… …… How to define a hash function is not trivial for some applications. 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 h(CLE) =3 × 12 × 5%400=180

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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 Like general Table ADT, and Stack, Hash Table ADT include following functions.

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**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. } Notice that here is bucket’s chain, instead of bucket. Insertion is done in O( 1 ) time

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**Inserting an Object in A HashTable**

1 …… 80 2 79 …… Buckets An example of insert operation An element (Cleveland) is inserted into a hash table. (suppose we only need to deal with 101 big airports) What the hash function will be? h( k ) = k % 101 h(CLE)=h(180)=180%101=79 Cleveland 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 A good hash function and implementation algorithm are essential for good hash table performance, but may be difficult to achieve.

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**The 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)

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**The Problem of Using a Hash Function**

Consider the hash function h( k ) = k % 100 Suppose that a key value of 214 is used for an object, and the object is stored at index 14 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 Collision is the circumstance where several keys hash to the same bucket. This happens when: h( hashCode1 ) == h( hashCode2 ) How do we solve this problem?

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**How are Collisions Resolved?**

Note: The whole object is stored but only the key value is shown 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 (with the element) If we had 500,000 keys, this approach is still much faster than comparing 500,000 keys with other approaches) 9 20 46 42 36 2 31 Value 1 2 3 4 5 6 That linked list is called a chain We handle collisions via chaining. In chaining, each bucket instead of storing a single entry/definition it will store a linked list of entries/definitions. That linked list is called a chain. However, chaining represents a new problem as now we have to lookup the definition in the linked list. To solve this problem we end up not storing just the definition but the word as well so we can ensure that we'll get the right definition from the chain. Thus, each entry will be represented as a pair, i.e. entry = ( key, value )

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**An Example of Searching an Object in A HashTable**

Pseudo-code for the retrieve (search, find) operation 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. } A search for an element can be done in O( 1 ) time.

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**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**

1 2 3 4 5 6 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.)**

1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 31

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**An Example Using Chaining (cont.)**

1 2 3 4 5 6 The hash function is: h( k ) = k % 7 h(31)=31 % 7= 3 INSERT object with key 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 Assumed that the hash function is: h( k ) = k % 7 31 Note: The whole object is stored but only the key value is shown

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 9 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 9 9 % 7 = 2 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 9 9 % 7 is 2 9 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 9 INSERT object with key 36 36 % 7 is 1 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 36 9 INSERT object with key 36 36 % 7 is 1 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 36 9 INSERT object with key 42 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 36 9 INSERT object with key 42 42 % 7 is 0 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 42 42 % 7 is 0 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 46 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 46 46 % 7 is 4 31

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 46 46 % 7 is 4 31 46

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 20 31 46

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 20 20 % 7 is 6 31 46

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 20 20 % 7 is 6 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 INSERT object with key 2 2 % 7 is 2 36 9 But an object has been inserted in the location with index 2 of the linked list before 31 46 COLLISION occurs !! How to resolve this? 20 Inserts the new element at the BEGINNING of the list

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 INSERT object with key 2 2 % 7 is 2 9 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 2 2 % 7 is 2 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 24 % 7 is 3 24 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 Supposed that all objects were stored in the linked list. How to Find an object? 36 2 9 e.g. FIND the object with key 9 24 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 FIND the object with key 9 9 % 7 is 2 36 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

FIND the object with key 9 9 % 7 is 2 1 2 3 4 5 6 42 36 We search this linked list for the object with key 9 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 Remember…the whole object is stored, only the key is shown 36 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

1 2 3 4 5 6 42 Does this object contain key 9? 36 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

FIND the object with key 9 1 2 3 4 5 6 42 Does this object contain key 9? No, so go on to the next object. 36 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

FIND the object with key 9 1 2 3 4 5 6 42 Does this object contain key 9? 36 2 9 24 31 46 20

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**Example Using Chaining (cont.)**

FIND the object with key 9 1 2 3 4 5 6 42 Does this object contain key 9? YES, found it! Return the object. 36 2 9 24 31 46 20

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**Summary Hash Table Hash Function Collision and collision resolution**

Table ADT Direct addressing and its problem Hash Table Concept Hash Function Example of using a hash function Benefit and problem of using a hash function Collision and collision resolution Hash table ADT operations Examples: ADT operations and Using chaining approach to resolve collision

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END Thank You ! Look Forward To Seeing You Again !

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