COSC 2007 Data Structures II Chapter 13 Advanced Implementation of Tables IV

COSC 2007 Data Structures II

Chapter 13Advanced Implementation of

Tables IV

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Topics

How to choose a Hash Function? Closed hashing

Linear hashing Quadratic hashing Double hashing

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Hash Functions Good hash function:

Easy & fast to compute Has minimal number of clashes Data items are spread uniformly throughout the array

Hashing problems reduce to the following points: Finding a hashing method that minimizes collisions Resolving collisions when they do happen

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Hashing Methods

Integer Type It is sufficient for a hash function to operate on integers Any arbitrary integer can be converted into an integer

within a certain range The index of the hash table lies within a specific range

Solutions Digit Selection Folding Modulo arithmetic

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Hashing Methods

Digit Selection Choose a group of digits from the number Use combination of Mod/div operations on the

search key One of the most effective hashing methods

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Hashing Methods

Digit Selection Example

Assume table size = 1000 Key = 01234567

Choose 2nd, 4th,& last digits H(key) = 147

key = d1 d2 d3 d4 d5 d6 d7 d8 d9 Choose leftmost 3 digits

H(key) = key Div 1000000 = d1 d2 d3 Choose rightmost 3 digits

H(key) = key Mod 1000 = d7 d8 d9

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Hashing Methods

Digit Selection Mid-square Method (Multiplication)

First Variant Key is squared, then some digits of this square are selected to

give the index.

Example k = 54321 H(k) = k2 = 2950771041 Pick up 3 middle digits index = 077

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Hashing Methods

Folding Method Digits are added together instead of just being selected Digits can first be grouped and then add the groups Folding can be done more than once on the search key

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Hashing Methods

Folding Method Example:

Key = 1234567 H(Key) = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

Disadvantage All values will be put in the range

Solution Divide into groups then fold

Key = 1234567 Groups: 12 345 67 Fold: 12 + 345 + 67 = 454 Hash again to fit into table size

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Hashing Methods

Modulo Arithmetic Choose a prime table size Divide the search key using modulo the size of the table

h(x) = x mod TableSize

Items will be distributed over the table Advantages

Simple Reduces collisions

items will be evenly distributed if table size is a prime number

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Hashing Methods What should be done if the search key is a

character? Convert the character string into some integer

before applying the hash function How should we do that?

Use the ASCII code: Can lead to duplication (e.g. NOTE and TONE will result in the

same hash function)

Write a numeric value for each character in binary Concatenate the results

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Hashing Methods

Example: Key = NOTE

ASCII code for each character N = 14 = (01110) // Order of ‘N’ in alphabet O = 15 = (01111) T = 20 = (10100) E = 5 = (00101)

Concatenation Binary result:

y = (01110 01111 10100 00101) Equivalent decimal

X = 474,757 Apply hash function

h(x) = x mod TableSize

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Closed Hashing (Open Addressing)

No secondary data structure All the data goes inside the table. On collision, try alternate cells until an

empty cell is found. How? Bigger table is needed.

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Linear Probing

Linear search from position where collision occurred.

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Linear Probing This is called a collision, because

there is already another valid record at [2].

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .

[100]Number 506643548Number 233667136Number 281942902 Number 155778322Number 580625685

Number 265-7917

My hashvalue is [2].[2] is occupied, how to do[2] is occupied, how to do

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[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917

My hashvalue is [2].

When a collision occurs,move forward until you

find an empty spot.

When a collision occurs,move forward until you

find an empty spot.

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Linear Probing

This is called a collision, because


[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917

My hashvalue is [2].[5] is empty, I can insert it[5] is empty, I can insert it

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[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


The new record goesin the empty spot.

The new record goesin the empty spot.

Number 701466868

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Linear Probing

Find the next index in the array up until the maximum subscript is reached and then it should return to the first index (wrap around)

Try alternate cells Cells h0(x), h1(x), h2(x), … are tried until an free cell

is found hi(x) = ( hash(x) + f(i) ) mod TSIZE f(0) = 0

Linear probing f(i) = i

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Searching for a Key The data that's attached to a key

can be found fairly quickly.

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .

[100]Number 506643548Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868

Number 265-7917

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Searching for a Key Calculate the hash value. Check that location of the array

for the key..

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917

My hashvalue is [2].Not me.

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Searching for a Key Keep moving forward until you find the

key, or you reach an empty spot.

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917


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[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917


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[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917

My hashvalue is [2].Yes!

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Searching for a Key When the item is found, the information

can be copied to the necessary location.

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Number 265-7917

My hashvalue is [2].

Yes!

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Deleting a Record

Records may also be deleted from a hash table

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .


Pleasedelete me.

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Deleting a Record Records may also be deleted from a hash table. But the location must not be left as an ordinary

"empty spot" since that could interfere with searches.

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .

[100]Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868

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Deleting a Record Records may also be deleted from a hash table. But the location must not be left as an ordinary "empty

spot" since that could interfere with searches. The location must be marked in some special way so

that a search can tell that the spot used to have something in it.

[ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]

. . .

[100]Number 233667136Number 281942902 Number 155778322Number 580625685 Number 701466868

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Linear Probing Advantage

Uses less memory than chaining don’t have to store all the links

Disadvantages Can be slower than chaining

may have to walk along the table for a long way Difficult to delete a key and associated record.

has an impact on the search process Clustering

Primary clustering Table contains groups of consecutively occupied locations

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Linear probing: f(i) = i Quadratic probing: f(i) = i2

Insert 10, 40, 60, 20, 30, 70, 80

Quadratic Probing

0123456789

10

02

1040

12

1040

60

22

1040

60

2032

1040

60

30

2042

1040

607030

2052

1040

607030

2062 mod 10 = 6

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Quadratic Probing Advantages

Easy to compute Avoids primary clustering

Disadvantage Not all entries are searched. Might not encounter a free storage

location even when there are locations that are still free Elements that has the same hash value will probe the same set

of alternate cells Secondary clustering Not a big problem in practice

Use a good hash function

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Double Hashing

Use two hash functions one as before that generates the ‘home’ position. second one generates a sequence of offsets from

the home position that define the probe sequence. probe = (probe + offset) mod N

If the size of the table is prime, this method will eventually examine every position in the table.

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Problems with Closed Hashing

Table too full Running time too long Inserts could fail

Must be chosen in advance Don’t know the number of elements

Rehashing Build a new table that is about twice as big Hash the elements into the new table

Need to apply new hash function to every item in the old hash table

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Summary

Hash tables are specialized for dictionary operations: Insert, Delete, Search

Principle: Turn the key field of the record into a number, which we use as an index for locating the item in an array.

O(1) in the ideal case Problems: find a good hash function, collisions,

wasted space, do not support ordering queries Implementations: open hashing, closed hashing,

dynamic hashing

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Reveiw

What is a perfect hash function? What is a collision? What is meant by clustering? How does

clustering affect the overall efficiency of hashing?

What is a bucket? What is the time complexity for insertion,

deletion, and search in Hashing?

Documents

COSC 2007 Data Structures II Chapter 13 Advanced Implementation of Tables IV