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COSC 2007 Data Structures II

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COSC 2007 Data Structures II

Chapter 13

Advanced Implementation of Tables III

- Hashing
- Definition
- Hash function
- Key
- Hash value
- collision

- Definition
- Open hashing

- 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?

- 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

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)?

- 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

- Examples

•••

Null

Sub

Null

Null

Null

Null

Xu

Null

•••

•••

•••

000-0000

000-0001

000-0002

259-1623

263-3049

- 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?

- 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

- 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

[ 4 ]

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

[ 4 ]

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

Number 281942902

Number 233667136

Number 506643548

Number 155778322

- 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

Number 281942902

Number 233667136

Number 506643548

Number 155778322

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

Number 281942902

Number 233667136

Number 506643548

Number 155778322

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

Number 281942902

Number 233667136

Number 506643548

Number 155778322

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

[3]

Number 281942902

Number 233667136

Number 506643548

Number 155778322

Number 265-1556

- The hash value is used for
the location of the

new record.

[ 0 ]

[ 1 ]

[ 2 ]

[ 3 ]

[ 4 ]

[ 5 ]

[100]

. . .

An array of records

Number 281942902

Number 233667136

Number 580625685

Number 506643548

Number 155778322

- The hash value is used for the location of the new record.

[ 0 ]

[ 1 ]

[ 2 ]

[ 3 ]

[ 4 ]

[ 5 ]

[100]

. . .

An array of records

- What is hashing?
- Each item has a unique key.
- Use a large array called a Hash Table.
- Use a Hash Function.

- 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.

0

Array

(Hash table)

Search key

Address Calculator

(Hash function)

N-1

- An address calculator (hashing function) is used to determine the location of the item

- Anything!
- Can hash on numbers, strings, structures, etc.
- Java defines a hashing method for general objects which returns an integer value.

- Databases (phone book, student name list).
- Spell checkers.
- Computer chess games.
- Compilers.

- Hashing gives us another implementation of Table ADT
- Hashing operations
- Initialize
- all locations in Hash Table are empty.

- Insert
- Search
- Delete

- Initialize
- 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.

- 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

- 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

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

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

H (Key) = Key mod TSIZE

TSIZE = 10

- 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.

- 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

- Student Ids (integers)

Number 281942902

Number 233667136

Number 580625685

Number 506643548

Number 155778322

Number 2641455

- 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

Open hashing (separate chaining)

Close hashing (open address)

Linear Probing

Quadratic Probing

Double hashing

0

49

4

81

1

0

25

9

16

36

64

1

2

3

4

5

6

7

8

9

- Keep a list of all elements that hash to the same value.

0

1

4

9

16

25

36

49

64

81

0

9

16

49

36

64

1

81

4

25

0

1

2

3

4

5

6

7

8

9

- Secondary Data Structure
- List
- Search tree
- another hash table

- We expect small collision
- List
- Simple
- Small overhead

- List

- 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.

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

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

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 ()

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

- 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

- 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

- A hash table is a(n) ______.
- stack
- queue
- array
- list

- 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

- ______ 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

- ______ 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

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

- 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