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COSC 2007 Data Structures II
Chapter 13Advanced Implementation of
Tables III
2
Topics
Hashing Definition
Hash function Key Hash value collision
Open hashing
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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?
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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
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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)?
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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
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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?
Null Null Null Null
259
-162
3
Xu
000
-000
0
000
-000
1
000
-000
2
•••
Null ••• Null Sub
263
-304
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••• •••
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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
xh
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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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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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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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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 506643548Number 233667136Number 281942902 Number 155778322
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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 506643548Number 233667136Number 281942902 Number 155778322
Number 265-1556
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Inserting a New Record? Typical way to create a hash value:
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
An array of records
. . .
[100]Number 506643548Number 233667136Number 281942902 Number 155778322
Number 265-1556
(Number mod 101)
What is (265-1556 mod 101) ?
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Inserting a New Record? Typical way to create a hash value:
[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
An array of records
. . .
[100]Number 506643548Number 233667136Number 281942902 Number 155778322
Number 265-1556
(Number mod 101)
What is (2651556 mod 101) ? 3
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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 506643548Number 233667136Number 281942902 Number 155778322
Number 265-1556
[3]
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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 506643548Number 233667136Number 281942902 Number 155778322Number 580625685
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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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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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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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Where do we use Hashing?
Databases (phone book, student name list). Spell checkers. Computer chess games. Compilers.
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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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Hashing Insert pseudocode
tableInsert (newItem)
i = the array index that the address calculator gives you for the new item’s search keytable[i]=newItem
Retrieval pseudocodetableRerieve (searchKey)
i = array index for searchKey given by the hash functionif (table[i].getKey( ) == searchKey)
return table[i] else
return null
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Hashing
Deletion pseudocodetableDelete (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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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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How to choose the Table Size?
H (Key) = Key mod TSIZETSIZE = 10
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22
541526
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0123456789
152022264954
0123456789
110210320460520600
0123456789
110210320460520600
10
110210,320
520
600
460
TSIZE = 11
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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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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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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 506643548Number 233667136Number 281942902 Number 155778322Number 580625685
Number 2641455
My hashvalue is [2].
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What to do on collisions?
Open hashing (separate chaining) Close hashing (open address)
Linear Probing Quadratic Probing Double hashing
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Keep a list of all elements that hash to the same value.
Open hashing (separate chaining)
0123456789
01 81
4 64
16 36
9 49
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0149162536496481
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Open hashing (separate chaining)
Secondary Data Structure List Search tree another hash table
We expect small collision List
Simple Small overhead
0123456789
01 81
4 64
16 36
9 49
25
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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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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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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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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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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()
elsereturn null
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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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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
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Review A hash table is a(n) ______.
stack queue array list
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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
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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
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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
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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
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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
