Storage by Hashing

1
Storage by Hashing

• Regio Florea
• COT4810
• Spring 2005
• 1/24/05

2
Introduction

• Familiarity with and prior knowledge of hashing
  is assumed.
• Also, it is assumed that a good portion of
  hashing knowledge is lost or fuzzy.
• This presentation is dual-purpose, serving as
  both an introduction to hashing and as a review,
  depending on experience.
• After the presentation you should understand the
  fundamental concepts of hashing, and have
  knowledge of various hashing techniques.

3
Flow of Discussion

• Background and history of hashing
• Fundamentals of hashing
• Hash functions
• Collision resolution
• Uses of storage by hashing

4
What is Hashing?

• Hashing is a technique of storing data by means
  of a dynamic data structure known as a hash
  table.
• A hash table can be visualized by means of an
  array.
• Insertion, deletion, and searching of data items
  are allowed through the use of a hash table.

5
Background

• hash
• . . .
• tr.v.
• 1. To chop into pieces mince.
• . . .
• hash browns
• pl.n.
• Chopped cooked potatoes, fried until brown.
  Also called hash brown potatoes.

6
Background

• The word hash used in CS comes from the obvious
  actual definition of the term.
• It is believed that Hans Peter Luhn from IBM was
  the first to use the concept, in 1953.
• 10 years later the term hash then began to be
  used.

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Hash Example
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Why Hashing?

• To store and find records or data elements in
  large files one may use one of the following
  three methods
• Sequentially searching through each of the n
  files, requiring O(n) steps to find a given
  record.
• Using a data structure that needs to be ordered,
  or a tree, which would require O(logn) steps to
  find a particular record.
• Or using certain data from a record to generate
  an address of a location in memory, which would
  require O(1) steps to find a specific record.
• Which would you choose?

9
Designations

• k the key, normally an alphanumeric string, we
  hash the key to generate a memory address.
• h the hash function, h(k), which takes a key as
  an input and produces the hash code or index as
  output.
• M the max. integer address used, e.g.- 0 to M
  can be memory address or simply array indices.
• m number of addresses in the hash table that
  contain keys.

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Hash Functions
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Division

• A good, common hash function involves modulus.
• e.g- h(k) k mod M
• We want to select an M that is PRIME so that the
  generated addresses are distributed well.
• Also, M should not take the form rh a, where r
  is the radix and a is a small integer (to ensure
  proper distribution).

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Division

• Example If keys with the hash codes 200, 205,
  210, 215,, 600 are inserted in an array of size
  100, each hash code will collide with three
  others.
• But if we select a size of 101, no collisions
  will occur!

13
Multiplication

• The following steps produce a good hash function,
  provided x is an irrational number
• Multiply k by x.
• Take the fractional part of the previous result.
• Multiply by M.
• Take the integer part of the previous result
  (floor function).

14
Interesting Note

• Selecting the golden mean (designated g) to be
  the value of the irrational number x will produce
  a better hashing than any other rational number.
• Golden ratio 1 / 1.6180339 golden mean
  0.61803399

15
Interesting Note

• Evidence of the use and existence of golden
  mean/ratio can be found throughout time and
  history.
• Present in architecture, art, music, the human
  body, and nature.
• Related to Fibonacci sequence (also present in
  nature, flowers, etc.).
• Most aesthetic.

16
Collisions

• Collisions occur when two or more records hash to
  the same address.
• Requires collision resolution to appropriately
  handle the collision.
• Two techniques
• 1.) Chaining
• 2.) Open addressing

17
Chaining

• More common, simpler collision resolution
  technique.
• Hash table can be viewed as an array of linked
  lists.
• When a collision occurs the data record becomes
  linked to that hash indexs linked list.

18
Chaining

• See the following website at the University of
  Michigan for a view of chaining among other
  animated hashing visualizations
• http//www.engin.umd.umich.edu/CIS/course.des/cis3
  50/hashing/WEB/HashApplet.htm

19
Open Addressing

• More complex, but no auxiliary structures are
  employed.
• All records in a hash table are stored in the
  table itself one element per bin.
• When searching for a record we go through slots
  until its found or its determined that its not
  in the table.
• Table is successively examined (probed) when
  inserting/searching.

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Open Addressing

• 3 Types
• Linear probing
• Quadratic probing
• Double hashing

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Linear Probing

• Given an ordinary hash function h(k), the probe
  sequence becomes
• hi(k) (h(k) i) mod M,
• i 0, 1, 2,, M
• Examination sequence is h(k), h(k)1,, M, 0,,
  h(k)-1.
• Simple technique, but primary clustering occurs,
  where large amounts of consecutive slots are
  full.
• Primary clustering increases the average search
  time.

22
Quadratic Probing

• Similar to linear probing, but works better.
• Probe sequence takes the form
• hi(k) (h(k) c1i c2i2) mod M,
• i 0, 1, 2, M
• Clustering occurs as in linear probing, but it is
  milder and known as secondary clustering.

23
Quadratic Probing

• If the table size isnt prime, then its possible
  that an empty slot wont be found, even before
  the table is half full.
• Logical conclusion GO PRIME!
• Prime numbers should always be used with hashing.

24
Double Hashing

• Considered to be one of the best open addressing
  methods.
• Takes the form
• hi(k) (h(k) i h(k)) mod M,
• h(k) and h(k) are hash functions
• The entire probe sequence depends on the key.

25
Practical Applications

• Aside from being relevant in general computing
  areas (e.g- when data items need to be stored),
  there exist more specific areas where hashing
  plays a significant part.
• Three main areas in computer science where
  hashing plays a role is in the following
• Databases
• Cryptography
• Compiler Design

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Databases

• The need or at least possible use of hashing in
  databases should be obvious.
• Namely, the storing and searching of DATA!
• Hashing can be used effectively in a database
  management system (DBMS), for instance.

27
Databases

• Seeing as adding records and searching for
  records is such a common operation, deciding to
  use hashing is a no-brainer.
• As databases come in various sizes, hashing can
  likely be used effectively in all but the most
  large databases (hashing isnt so
  memory-efficient!).

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Cryptography

• One use is in encrypting and decrypting digital
  signatures, which are used to authenticate both
  the sender and receiver of a message.
• The signature is transformed by the hash function
  and then the resulting value (aka message-digest)
  and the signature itself are sent to the receiver
  (in separate transmissions).

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Cryptography

• The receiver then uses the same hash function,
  derives a message-digest from the signature and
  then compares the resulting value to the
  message-digest it received from the sender.
• These two values being the same confirms the
  authentication.

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Compiler Design

• Hashing often used in the design of compilers.
• The common place to find hashing used in a
  compiler is in the symbol table.
• It is considered that good compilers use
  hashing for their symbol tables.
• Symbol tables are simply the data areas where
  symbols and their associated values are held.
• Some common symbols include labels in assembly
  language, and variable names in higher-level
  languages.

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Summary

• Hashing is an effective method for managing data.
• Common hashing functions employ the
  previously-mentioned division and multiplication
  methods.
• Collision resolution needs to be done, either in
  the form of chaining or open addressing.

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Summary

• Its strength lies in its speed, namely in tending
  towards O(1) constant performance when done
  correctly.
• Its main weakness is that it doesnt always use
  memory effectively (e.g.- more data you have,
  more empty slots will exist).
• Hashing is part of the bread and butter of
  computer science, and is used in various related
  areas.

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References

• Baranova, Nadia. Hash Tables. Lecture
  presentation. 3 March 2004.
• Dewdney, A.K. The New Turing Omnibus. New York
  Henry Holt and Company, 2001.
• Goodrich, Michael T., and Roberto Tamassia. Data
  Structures and Algorithms in Java. 3rd ed.
  Hoboken John Wiley Sons, 2004.
• Hashing. searchDatabase.com 20 Jan. 2005
  http//search.database.techtarget.com/sDefinition/
  0,,sid13_gci212230,00.html.
• RSA Security 2.1.6 What is a hash function? 2
  Jan. 2005 http//www.rsasecurity.com/rsalabs/node.
  asp?id2176.