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COSC 2007 Data Structures II Chapter 12 Advanced Implementation of Tables III

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

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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

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4 Phone Number Problem Problem: phone company wants to implement caller ID. given a phone number (the key), look up persons name or address(the data) lots of phone numbers (P=10 7 -1) in a given area code only a small fraction of them are in use Nobody has a phone number :0000000 or 0000001

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5 Comparison of Time Complexity (average) Operation Insertion Deletion Search Unsorted ArrayO(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)?

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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 Examples 911 emergency system Air-traffic control system

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7 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 youre inserting, and use that number directly as the item number in a list (array). Search is O(1), but huge amount of space wasted. Null 259-1623 Xu 000-0000000-0001000-0002 Null NullSub 263-3049

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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) B-1 x h

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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 ] An array of records... [100]

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10 What is Hash Table? 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 ] An array of records... [100] [ 4 ] Number 256-2879 8888 Queen St. Linda Kim

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

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12 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 ] An array of records... [100] Number 506643548 Number 233667136 Number 281942902 Number 155778322

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

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

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

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

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

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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 function is a function h(K) which transforms a key K into an address.

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

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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.

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

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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.

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23 Hashing Insert pseudocode tableInsert (newItem) i = the array index that the address calculator gives you for the new items 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

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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

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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

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

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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.

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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

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

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

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31 Keep a list of all elements that hash to the same value. Open hashing (separate chaining) 0 1 2 3 4 5 6 7 8 9 0 181 464 1636 949 25 0 1 4 9 16 25 36 49 64 81

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32 Open hashing (separate chaining) Secondary Data Structure List Search tree another hash table We expect small collision List Simple Small overhead 0 1 2 3 4 5 6 7 8 9 0 181 464 1636 949 25

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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.

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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

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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

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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 ()

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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

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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

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