Hashing Algorithm

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Hashing Algorithm

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Introduction

• Hashing , a ubiquitous information retrieval
  strategy for providing efficient access to
  information based on a key
• Information can usually be accessed in constant
  time
• Hashings drawbacks

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Concept of hashing

• The problem at hand is to define and implement a
  mapping from a domain of keys to a domain of
  locations
• From the performance standpoint, the goal is to
  avoid collisions (A collision occurs when two or
  more keys map to the same location)
• From the compactness standpoint, no application
  ever stores all keys in a domain simultaneously
  unless the size of the domain is small

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Concept of hashing (cont)

• The information to be retrieved is stored in a
  hash table which is best thought of as an array
  of m locations, called buckets
• The mapping between a key and a bucket is called
  the hash function
• The time to store and retrieve data is
  proportional to the time to compute the hash
  function

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Hashing function

• The ideal function, termed a perfect hash
  function, would distribute all elements across
  the buckets such that no collisions ever occurred
  
• h(v) f(v) mod m
• Knuth(1973) suggests using as the value for m a
  prime number

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Hashing function(cont)

• It is usually better to treat v as a sequence of
  bytes and do one of the following for f(v)
• (1) Sum or multiply all the bytes. Overflow can
  be ignored
• (2) Use the last (or middle) byte instead of
  the first
• (3) Use the square of a few of the middle bytes

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Implementing hashing

• The following operations are usually provided by
  an implementation of hashing
• (1) Initialization
• (2) Insertion
• (3) Retrieval
• (4) Deletion

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Chained hashing
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Chained hashing(cont)

• In the worst case (where all n keys map to a
  single location), the average time to locate an
  element will be proportional to n/2.
• In the best case (where all chains are of equal
  length), the time will be proportional to n/m.

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Open addressing
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Minimal perfect hash functions

• Minimal perfect hash function (MPHF) is a perfect
  hash function with the property that is hashed m
  keys to m buckets with no collisions
• Cichelli(1980) and of Cercone et al.(1983)
  proposed two important concepts
• (1)using tables of values as the parameters
• (2)using a mapping, ordering, and searching
  (MOS) approach

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Minimal perfect hash functions(cont)

• Mappingtransform the key set from an original to
  a new universe
• Orderingplace the keys in a sequence that
  determines the order in which hash values are
  assigned to keys
• Searchingassign hash values to the keys of each
  level

Mapping ? Ordering ? Searching
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Sagers method and improvement

• Sager(1984,1985) formalizes and extends
  Cichellis approach
• In the mapping step, three auxiliary(hash)
  functions are defined on the original universe of
  keys U
• h0U? 0 , , m - 1
• h1U? 0 , , r - 1
• h2U? r , , 2r 1

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Sagers method and improvement

• The class of functions searched is
• h(k) ( h0(k) g(h1(k)) g(h2(k)) (mod m)
• Sager uses a graph that represents the
  constraints among keys
• The mapping step goes from keys to triples to a
  special bipartite graph, the dependency graph,
  whose vertices are the h1(k) and h2(k) values and
  whose edges represent the words

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Sagers method and improvement
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The algorithm

• The mapping step

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The algorithm (cont)

• The ordering step

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The algorithm (cont)

• The searching step

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Discussion

• Hashing algorithm is a constant-time algorithm,
  and there are always advantages to being able to
  predict the time needed to locate a key
• The MPHF uses a large amount of space