Hashing, Hashing Tables

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Hashing, Hashing Tables

• Chapter 8

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Class Hierarchy
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Introduction

• Definition
• Key a key is a field or composite of fields that
  uniquely identifies an entry in a table.

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Example

• Table of students in a course sorted by name
• --------------------------------------------------
  ------------
• Name Year Mark
• --------------------------------------------------
  ------------
• Adams, Keith 3 94
• Davis, Susan 1 75
• Jordan, Ann 1 86
• Patterson, Lynn 4 73
• Williams, George 1 65

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Insert Function of ListAsArray
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Find Function of ListAsArray
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Insert Function of ListAsLinkedList
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Find Function of ListAsLinkedList
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Insert Function of SortedListAsArray
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Binary Search
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Hashing

• The implementation of hash tables is called
  Hashing.
• Hashing is a technique used for performing
  insertions and finds in constant average time.
• Efficient removal of items not required

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The General Idea

• Array of some fixed size, containing items.

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Example
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Keys and Hash Functions

• Each key is mapped into some number in the range
  0 to TableSize-1 and placed in the appropriate
  cell.
• The mapping is called a hash function

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Keys and Hash Functions

• Characteristics of a good hash function
• Avoids collisions
• Spread keys evenly in the array
• Easy to compute

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Avoid Collisions

• Ideal situation
• Given a set of nltM distinct keys k1,k2,,kn,
  the set of hash values h(k1),h(k2),,h(kn)
  contains no duplicates
• We can only try to reduce the likelihood of a
  collision using knowledge about the keys
• E.g. if we know the telephone numbers are all
  from the same district, so the district number
  will have little use in our hash function

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Spreading Keys Evenly

• We need to know the distribution of the keys
• An equal number of keys should map into each
  array position

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Ease of Computation

• The running time of the hash function should be
  O(1) (Jumping immediately to the desired record
  is a direct access approach, much like direct
  access of data on a disk)

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Hashing Methods

• We are dealing with integer values first, KZ
• The value of the hash function falls between 0
  and M-1

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Division Method

• The simplest method of hashing an integer
• The division method of hashing
• h(x) x mod M.

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Choice of M

• Generally, any M is good
• we often choose M to be a prime number

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Implementation

• Unsigned int const M 1031 // a prime
• Unsigned int h(unsigned int x)
• return xM

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Middle Square Method

• Avoid division
• Making use of the fact that computer does
  finite-precision integer arithmetic
• All arithmetic is done modulo W, where W2w, w is
  the word size of the computer
• M2k, W2w
• Meaning
• Multiply x by itself, then shift to the right k
  bits.

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Implementation

• unsigned int const k 10 // M1024
• unsigned int const w bitsizeof (unsigned int)
• unsigned int h (unsigned int x)
• return (x x) gtgt (w - k)

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Multiplication Method

• We multiply the key by a

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Implementation

• unsigned int const k 10 // M1024
• unsigned int const w bitsizeof (unsigned int)
• unsigned int const a 2654435769U
• unsigned int h (unsigned int x)
• return (x a) gtgt (w - k)

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Hash Tables
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HashTable Class Definition
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Separate Chaining