Hash Tables

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Hash Tables

• Computer Science and Engineering
• Chapter 8 of Goodrich and Tomassias Text

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Topics

• Hashing
• Hash functions
• Hash Tables
• Collision
• Collision resolution
• Chaining, linear probing, quadratic probing,
  double hashing
• Java Hash table
• Example

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Hashing Concept

• Another approach for storing and searching
  elements.
• Used in applications where add and delete,
  besides search, are used.
• Worst case linear but can get O(1) best case.

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Hash Function

• A hash function h maps keys of a given type into
  an integer interval 0, N-1
• A simple function
• h(x) x mod N
• A good hash function will uniformly disperse the
  keys in the range 0, N-1

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Hash Table

• Hash table ADT
• Has a hash function h
• Array of size N
• Collision occurs when two keys map to the same
  array index.
• Two major collision resolution schemes are
  chaining and open addressing

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Example

• Item(name, ssn) where ssn is 9 digit positive
  integer.
• Hash table N 10000.
• Hash function last four digits of of x
• Use chaining to handle collision.

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Hash Table Example
0 1 2 3
045-34-0002
. . . .
9996 9997 9998 9999
567-34-9996
045-35-9996
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Hash Functions

• Usually specified as two components
• Component1 called the hash code map, collects the
  parts of the data and maps them to integers
  (numeric data)
• H1 keys ? integers
• Component2 called the compression map, takes the
  integers and maps them to 0 to N-1.
• H2 integers? 0, N-1

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Hash Code Maps

• Memory address of data
• Integer cast of non-numeric data bit/byte
  pattern of data
• Sum of all components add the ascii values of
  your last name
• Polynomials of various parts of data.

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Compression maps

• Reminder of division (mod)
• h(y) y mod N
• Multiple, Add and Divide (MAD)
• h(y) (ay b) mod N where
• a mod N ltgt 0 (otherwise everything will map to
  b!)

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Linear Probing

• Linear probing resolves collision by placing the
  colliding item in the next available empty cell.
• Each entry inspected is referred to as the
  probe
• Example h(x) x mod 13
• Insert keys 18, 41, 22, 44, 59, 32, 31, 73, in
  that order

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Search with linear probing find(k)

• j h(k) probe 0
• while (p lt N)
• 2.1 c Aj
• 2.2 if (c null) return NOT_FOUND
• else if (c.key() k)
• return c.element()
• else j (j1) mod N
• p p 1
• 3. return NOT_FOUND

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Double Hashing

• h1 primary hash function
• If it results in collision, resolve by applying
  another hash function, secondary hash function
  here is an example of such a function.
• d(k) q k mod q
• Where q lt N,
• Possible values of d(k) are 1,2,3,..q
• It cannot be 0
• When collision occurs (h(k)jd(k)) mod N
• j 0, 1, ..N-1

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Example

• h(k) k mod 13
• d(k) 7 k mod 7
• Insert 18, 41, 22, 44, 59, 32, 31, 73.