1. COSC 2007 Data Structures II
April 20, 2017
COSC Data Structures II
Chapter 13
Advanced Implementation of Tables III
Advanced Impelmentation of Tables

2. Topics Hashing Open 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 : or

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. 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 Xu
•••
Sub

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. 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. What is Hash Table? [ 4 ] Each record has a special
Number
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. What is Hash Table? The number is person's phone number,
[ 4 ]
Number
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. 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]
Number
Number
Number
. . .
Number
An array of records

13. Inserting a New Record? In order to insert a new record,
Number
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 ]
[100]
Number
Number
Number
. . .
Number
An array of records

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

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

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

17. Inserting a New Record?
The hash value is used for the location of the new record.
[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[100]
Number
Number
Number
Number
. . .
Number
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 function is a function h(K) which transforms a key K into an address.

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

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 Retrieval pseudocode table[i]=newItem
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 Mapping Main problems
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. How to choose the Table Size?1 2 3 4 5 6 7 8 9
110
210
320
460
520
600 10
210,320
TSIZE = 11
H (Key) = Key mod TSIZE
TSIZE = 10 20 22 54 15 26 49 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
110
210
320
460
520
600

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) = % 100 = 21
Names (char strings)
h(name) = (sum over the ascii values) % B
eg. h(“Bill”) = ( ) % 101 = 86

29. Collision Here is another new record to
Number
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]
Number
Number
Number
Number
. . .
Number
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. Open hashing (separate chaining)
Keep a list of all elements that hash to the same value. 1 2 3 4 5 6 7 8 9 81 64 16 36 49 25 1 4 9 16 25 36 49 64 81

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

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
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