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

1. Hash TablesDr. Li JiangSchool of Computer Science, The University of Adelaide

2. Overview • Hash Table • 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 • Hash table ADT operations • Collision and collision resolution • Examples: • ADT operations and • Using chainingapproach to resolvecollision

3. 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

4. An Example of A Table (Key, Value)

5. An Example of A Table (cont.)

6. An Example of A Table (cont.)

7. 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

8. 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.

9. 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.

10. An Example of A Table (cont.)

11. An Example of Table (cont.) • Assume that • Data items of 400 airports needs to be processed. • The key: Airport code with three letters, used to identify each airport. The data item of one airport Buckets 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

12. Another Example Assume that: • 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, 0.000005% • Percent of the memory allocated for table wasted, 99.999995% A better way is necessary Hash Table

13. 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 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

14. 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 required.

15. 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 …… 1 2 3 4 5 6 7 8 9 10 11 12 …… • 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

16. 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

17. 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

18. 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) Buckets 1 2 What the hash function will be? …… …… h( k ) = k % 101 h(CLE)=h(180)=180%101=79 Cleveland 79 • 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 80 …… …… …… ……

19. 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)

20. 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?

21. 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) 42 36 Value 0 1 2 3 4 5 6 2 9 31 46 20

22. 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.

23. 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   };

24. An Example of Using Chaining 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. 0 1 2 3 4 5 6 The hash function is: h( k ) = k % 7

25. An Example of Using Chaining (cont.) 0 1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 31

26. An Example Using Chaining (cont.) 0 1 2 3 4 5 6 The hash function is: h( k ) = k % 7 h(31)=31 % 7= 3 INSERT object with key 31

27. Example Using Chaining (cont.) 0 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

28. Example Using Chaining (cont.) 0 1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 9 31

29. Example Using Chaining (cont.) 0 1 2 3 4 5 6 The hash function is: h( k ) = k % 7 INSERT object with key 9 9 % 7 = 2 31

30. Example Using Chaining (cont.) 0 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

31. Example Using Chaining (cont.) 0 1 2 3 4 5 6 9 INSERT object with key 36 36 % 7 is 1 31

32. Example Using Chaining (cont.) 0 1 2 3 4 5 6 36 9 INSERT object with key 36 36 % 7 is 1 31

33. Example Using Chaining (cont.) 0 1 2 3 4 5 6 36 9 INSERT object with key 42 31

34. Example Using Chaining (cont.) 0 1 2 3 4 5 6 36 9 INSERT object with key 42 42 % 7 is 0 31

35. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 42 42 % 7 is 0 31

36. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 46 31

37. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 46 46 % 7 is 4 31

38. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 46 46 % 7 is 4 31 46

39. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 20 31 46

40. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 20 20 % 7 is 6 31 46

41. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 20 20 % 7 is 6 31 46 20

42. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 2 31 46 20

43. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

44. Example Using Chaining (cont.) INSERT object with key 2 2 % 7 is 2 0 1 2 3 4 5 6 42 36 But an object has been inserted in the location with index 2 of the linked list before 9 31 46 COLLISION occurs !! How to resolve this? 20 Inserts the new element at the BEGINNING of the list

45. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 INSERT object with key 2 2 % 7 is 2 9 31 46 20

46. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

47. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

48. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 9 INSERT object with key 2 2 % 7 is 2 31 46 20

49. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 2 9 INSERT object with key 2 2 % 7 is 2 31 46 20

50. Example Using Chaining (cont.) 0 1 2 3 4 5 6 42 36 2 9 INSERT object with key 24 31 46 20