stia2023 hashing

8/12/2019 STIA2023 Hashing

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Data Structures & Algorithm

AnalysisWeek 13: Hashing


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Objectives:

At the end of this lesson, the student will be able to: Describe the basic idea of hashing,

Describe the purpose of a hash table, and a hash function,

Describe how a hash function compresses a hash code into an index to

hash table,

Explain what collisions are and why they occur,

Describe open addressing as a method to resolve collisions,

Describe linear probing, and quadratic probing as particular open

addressing schemes,

Describe separate chaining as method to resolve collisions, and

Describe the relative efficiencies of various collisions resolution

techniques.


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Chapter Contents

What is Hashing? Hash Functions

Computing Hash Codes

Compression a Hash Code into an Index for the Hash Table

Resolving Collisions Open Addressing with Linear Probing

Open Addressing with Quadratic Probing

Separate Chaining


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Chapter Contents (ctd.)

Efficiency The Load Factor

The Cost of Open Addressing

The Cost of Separate Chaining


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What is Hashing?

A technique that determines an index or location forstorage of an item in a data structure

The hash function receives the search key

Returns the index of an element in an array called the hashtable

The index is known as the hash index

A perfect hash function maps each search key into adifferent integer suitable as an index to the hash table


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What is Hashing?

Fig. 1: A hash function indexes its hash table.


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What is Hashing?

Two steps of the hash function Convert the search key into an integer called the hash code

Compress the hash code into the range of indices for the hash

table

Typical hash functions are not perfect They can allow more than one search key to map into a single

index

This is known as a collision


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What is Hashing?

Fig. 2: A collision caused by the hash function h


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Hash Functions

General characteristics of a good hash function

Minimize collisions

Distribute entries uniformly throughout the hash

table

Be fast to compute


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We will override the hashCodemethod of Object

Guidelines

If a class overrides the method equals, it should override

hashCode

If the method equalsconsiders two objects equal, hashCodemust

return the same value for both objects

If an object invokes hashCodemore than once during execution

of program on the same data, it must return the same hash code

If an object's hash code during one execution of a program can

differ from its hash code during another execution of the sameprogram


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The hash code for a string, s

Hash code for a primitive type

Use the primitive typed key itself Manipulate internal binary representations

Use folding

int hash = 0;

int n = s.length();

for (int i = 0; i < n; i++)

hash = g * hash + s.charAt(i);

// g is a positive constant


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Compressing a Hash Code

Must compress the hash code so it fits into the indexrange

Typical method for a code c is to compute c modulo n

nis a prime number (the size of the table)

Index will then be between 0 and n 1

private int getHashIndex(Object key)

{ int hashIndex = key.hashCode() % hashTable.length;

if (hashIndex < 0)

hashIndex = hashIndex + hashTable.length;

return hashIndex;

} // end getHashIndex


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Resolving Collisions

Options when hash functions returns location

already used in the table

Use another location in the table (open addressing)

Change the structure of the hash table so that each arraylocation can represent multiple values (separate

chaining)


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Open Addressing with Linear Probing

Open addressing scheme locates alternate location

New location must be open, available

Linear probing

If collision occurs at hashTable[k], look successively atlocation k + 1, k + 2,


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Fig. 3 : The effect of linear probing after adding four

entries whose search keys hash to the same index.



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Fig. 4: A revision of the hash table shown in 19-3 when

linear probing resolves collisions; each entry contains a

search key and its associated value



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Removals

Fig. 5: A hash table if removeused null

to remove entries.


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Fig. 6: A linear probe sequence (a) after adding an entry;

(b) after removing two entries;


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Fig. 6: A linear probe sequence (c) after a search; (d)

during the search while adding an entry; (e) after an

addition to a formerly occupied location.



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Searches that Dictionary Operations Require

To retrieve an entry

Search the probe sequence for the key

Examine entries that are present, ignore locations in available state

Stop search when key is found or null reached

To remove an entry

Search the probe sequence same as for retrieval If key is found, mark location as available

To add an entry

Search probe sequence same as for retrieval

Note first available slot

Use available slot if the key is not found


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Open Addressing, Quadratic Probing

Change the probe sequence

Given search key k

Probe to k + 1, k + 22, k + 32, k + n2

Reaches every location in the hash table if table size

is a prime number

For avoiding primary clustering But can lead to secondary clustering


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Separate Chaining

Alter the structure of the hash table

Each location can represent multiple values

Each location called a bucket

Bucket can be a(n)

List

Sorted list

Chain of linked nodes

Array

Vector


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Separate Chaining

Fig. 9: A hash table for use with separate chaining; each

bucket is a chain of linked nodes.


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Separate Chaining

Fig. 10: Where new entry is inserted into linked bucket

when integer search keys are (a) duplicate and unsorted;


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Separate Chaining


when integer search keys are (b) distinct and unsorted;


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Separate Chaining


when integer search keys are (c) distinct and sorted


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Efficiency Observations

Successful retrieval or removal Same efficiency as successful search

Unsuccessful retrieval or removal

Same efficiency as unsuccessful search

Successful addition

Same efficiency as unsuccessful search

Unsuccessful addition

Same efficiency as successful search


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Load Factor

Perfect hash function not always possible or practical

Thus, collisions likely to occur

As hash table fills

Collisions occur more often

Measure for table fullness, the load factor


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Cost of Open Addressing

Fig. 11: The average number of comparisons required by

a search of the hash table for given values of the load

factor when using linear probing.


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Cost of Open Addressing

Fig. 12: The average number of comparisons

required by a search of the hash table for given

values of the load factor when using either

quadratic probing or double hashing.

Note: for quadraticprobing or double

hashing, should

have < 0.5


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Cost of Separate Chaining

Fig. 13: Average number of comparisons required by

search of hash table for given values of load factor

when using separate chaining.

Note: Reasonable

efficiency requires

only < 1


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Conclusion

Q & A Session


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Question 1

The property that is not expected from good hashing

technique should ______________.

A) produce keys uniformly distributed over the range

B)easy to program

C)produce no collisions

D)minimize collisions


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Question 2

Assume that a hash function has the following

characteristics:

Keys 77, 355, and 276 hash to 3.

Keys 945 and 579 hash to 5.

Key 517 hashes to 0.Key 155 hashes to 2.

Perform insertions in order the 945, 77, 276, 355, 517, 155, 579

A) Using linear probing technique, indicates the position of the data.

B) Which element requires the largest number of probes to be found in thetable (if more than one, give the element with the smallest index in the

hash table)?

C) Which elements(s) can we access with a single probe?

D) What is the load factor?

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