COSC 2007 Data Structures II

1. COSC 2007 Data Structures II Chapter 13 Advanced Implementation of Tables III

2. Topics • Hashing • Definition • Hash function • Key • Hash value • collision • Open hashing

3. Common Problem • A common pattern in many programs is to store and look up data • Find student record, given ID# • Find person address, given phone # • Because it is so common, many data structures for it have been investigated • How?

4. Phone Number Problem • Problem: phone company wants to implement caller ID. • given a phone number (the key), look up person’s name or address(the data) • lots of phone numbers (P=107-1) in a given area code • only a small fraction of them are in use • Nobody has a phone number :0000000 or 0000001

5. Comparison of Time Complexity (average) Operation Insertion Deletion Search Unsorted Array O(1) O(n) O(n) Unsorted reference O(1) O(n) O(n) Sorted Array O(n) O(n) O(logn) Sorted reference O(n) O(n) O(n) BST O(logn) O(logn) O(logn) Can we do better than O(logn)?

6. Can we do better than O(log N)? • All previous searching techniques require a specified amount of time (O(logn) or O(n)) • Time usually depends on number of elements (n) stored in the table • In some situations searching should be almost instantaneous -- how? • Examples • 911 emergency system • Air-traffic control system

7. ••• Null Sub Null Null Null Null Xu Null ••• ••• ••• 000-0000 000-0001 000-0002 259-1623 263-3049 Can we do better than O(log N)? • Answer: Yes … sort of, if we're lucky. • General idea: take the key of the data record you’re inserting, and use that number directly as the item number in a list (array). • Search is O(1), but huge amount of space wasted. – how to solve this?

8. Hashing • Basic idea: • Don't use the data value directly. • Given an array of size B, use a hash function, h(x), which maps the given data record x to some (hopefully) unique index (“bucket”) in the array. 0 1 h x h(x) B-1

9. What is Hash Table? • The simplest kind of hash table is an array of records. • This example has 101 records. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

10. [ 4 ] What is Hash Table? Number 256-2879 8888 Queen St. Linda Kim • Each record has a special field, called its key. • In this example, the key is a long integer field called Number. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

11. [ 4 ] What is Hash Table? Number 256-2879 • The number is person's phone number, and the rest is person name or address. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

12. Number 281942902 Number 233667136 Number 506643548 Number 155778322 What is Hash Table? • When a hash table is in use, some spots contain valid records, and other spots are "empty". [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

13. Number 281942902 Number 233667136 Number 506643548 Number 155778322 Inserting a New Record? Number 265-1556 • In order to insert a new record, the key must somehow be converted toan array index. • The index is called the hash valueof the key. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

14. Number 281942902 Number 233667136 Number 506643548 Number 155778322 Inserting a New Record? Number 265-1556 • Typical way to create a hash value: (Number mod 101) What is (265-1556 mod 101) ? [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

15. Number 281942902 Number 233667136 Number 506643548 Number 155778322 Inserting a New Record? Number 265-1556 • Typical way to create a hash value: (Number mod 101) 3 What is (2651556 mod 101) ? [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

16. [3] Number 281942902 Number 233667136 Number 506643548 Number 155778322 Number 265-1556 Inserting a New Record? • The hash value is used for the location of the new record. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

17. Number 281942902 Number 233667136 Number 580625685 Number 506643548 Number 155778322 Inserting a New Record? • The hash value is used for the location of the new record. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

18. What is Hashing? • What is hashing? • Each item has a unique key. • Use a large array called a Hash Table. • Use a Hash Function. • Hashing is like indexing in that it involves associating a key with a relative record address. • Hashing, however, is different from indexing in two important ways: • With hashing, there is no obvious connection between the key and the location. • With hashing two different keys may be transformed to the same address. • A Hash functionis a function h(K) which transforms a key K into an address.

19. 0 Array (Hash table) Search key Address Calculator (Hash function) N-1 What is Hashing? • An address calculator (hashing function) is used to determine the location of the item

20. What Can Be Hashed? • Anything! • Can hash on numbers, strings, structures, etc. • Java defines a hashing method for general objects which returns an integer value.

21. Where do we use Hashing? • Databases (phone book, student name list). • Spell checkers. • Computer chess games. • Compilers.

22. Hashing and Tables • Hashing gives us another implementation of Table ADT • Hashing operations • Initialize • all locations in Hash Table are empty. • Insert • Search • Delete • Hash the key; this gives an index; use it to find the value stored in the table in O(1) • Great improvement over Log N.

23. Hashing • Insert pseudocode tableInsert (newItem) i = the array index that the address calculator gives you for the new item’s search key table[i]=newItem • Retrieval pseudocode tableRerieve (searchKey) i = array index for searchKey given by the hash function if (table[i].getKey( ) == searchKey) return table[i] else return null

24. Hashing • Deletion pseudocode tableDelete (searchKey) i = array index for searchKey given by the hash function success=(tabke[I].getKey() equals searchKey if (success) Delete the item from table[i] Return success

25. Hash Tables Table size Entries are numbered 0 to TSIZE-1 Mapping Simple to compute Ideally 1-1: not possible Even distribution Main problems Choosing table size Choosing a good hash function What to do on collisions

26. TSIZE = 11 110 0 110 210 320 460 520 600 110 210 320 460 520 600 0 210,320 1 0 15 20 22 26 49 54 20 1 2 1 520 2 3 22 2 4 3 3 5 4 4 54 5 6 5 600 15 6 7 6 26 7 8 7 8 9 8 460 9 10 9 49 How to choose the Table Size? H (Key) = Key mod TSIZE TSIZE = 10

27. How to choose a Hashing Function? • The hash function we choose depends on the type of the key field (the key we use to do our lookup). • Finding a good one can be hard • Rule • Be easy to calculate. • Use all of the key. • Spread the keys uniformly.

28. How to choose a Hashing Function? • Example: • Student Ids (integers) h(idNumber) = idNumber % B eg. h(678921) = 678921 % 100 = 21 • Names (char strings) h(name) = (sum over the ascii values) % B eg. h(“Bill”) = (66+105+108+108) % 101 = 86

29. Number 281942902 Number 233667136 Number 580625685 Number 506643548 Number 155778322 Number 2641455 Collision • Here is another new record to insert, with a hash value of 2. My hash value is [2]. [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [100] . . . An array of records

30. What to do on collisions? Open hashing (separate chaining) Close hashing (open address) Linear Probing Quadratic Probing Double hashing

31. 0 49 4 81 1 0 25 9 16 36 64 1 2 3 4 5 6 7 8 9 Open hashing (separate chaining) • Keep a list of all elements that hash to the same value. 0 1 4 9 16 25 36 49 64 81

32. 0 9 16 49 36 64 1 81 4 25 0 1 2 3 4 5 6 7 8 9 Open hashing (separate chaining) • Secondary Data Structure • List • Search tree • another hash table • We expect small collision • List • Simple • Small overhead

33. Operations with Chaining • Insert with chaining • Apply hash function to get a position. • Insert key into the Linked List at this position. • Search with chaining • Apply hash function to get a position. • Search the Linked List at this position.

34. Open hashing (separate chaining) public class ChainNode { Private KeyedItem item; private ChainNode next; public ChainNode(KeyedItem newItem, ChainNode nextNode) { item = newItem; next= nextNode; // set and get methods } } // end of ChainNode

35. Open hashing (separate chaining) public class HashTable { private final int HASH_TABLE_SIZE = 101; // size of hash table private ChainNode [] table; //hash table private int size; //size of hash table public HashTable() { table = new ChainNode [HASH_TABLE_SIZE]; size =0; } public bool tableIsEmpty() { return size ==0;} public int tableLength() { return size;} public void tableInsert(KeyedItem newItem) throws HashException {} public boolean tableDelete(Comparable searchKey) {} public KeyedIten tableRetrieve(Comparable searchKey) {} } // end of hashtable

36. Open hashing (separate chaining) tableInsert(newItem) if (table is not full) { searchKey= the search key of newItem i = hashIndex (searchKey) node= reference to a new node containing newItem node.setNext (table[I]); table[I] = node } else //table full throw new HashException ()

37. Open hashing (separate chaining) tableRetrieve (searchKey) i = hashIndex (searchKey) node= table [I]; while ((node !=null)&& node.getItem().getKey()!= searchKey ) node=getNext () if (node !=null) return node.getITem() else return null

38. Evaluation of Chaining • Disadvantages of Chaining • More complex to implement. • Search and Delete are harder. We need to know: The number of elements in the table (N); the number of buckets (B); the quality of the hash function • Worse case (O(n)) for searching • Advantage of Chaining • Insertions is easy and quick. • Allows more records to be stored. • The size of table is dynamic

39. Review • A(n) ______ maps the search key of a table item into a location that will contain the item. • hash function • hash table • AVL tree • heap

40. Review • A hash table is a(n) ______. • stack • queue • array • list

41. Review • The condition that occurs when a hash function maps two or more distinct search keys into the same location is called a(n) ______. • disturbance • collision • Rotation • congestion

42. Review • ______ is a collision-resolution scheme that searches the hash table sequentially, starting from the original location specified by the hash function, for an unoccupied location. • Linear probing • Quadratic probing • Double hashing • Separate chaining

43. Review • ______ is a collision-resolution scheme that searches the hash table for an unoccupied location beginning with the original location that the hash function specifies and continuing at increments of 12, 22, 32, and so on. • Linear probing • Double hashing • Quadratic probing • Separate chaining

44. Review • ______ is a collision-resolution scheme that uses an array of linked lists as a hash table. • Linear probing • Double hashing • Quadratic probing • Separate chaining

45. Review • The load factor of a hash table is calculated as ______. • table size + current number of table items • table size – current number of table items • current number of table items * table size • current number of table items / table size