Hashing

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Hashing

• Chapter 5

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• Hashing is used for
• Insertion, Deletion, Search in constant time
• Hashing is not used for
• Finding minimum, maximum
• Sorting tthe elements
• Appropriate for databases

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What we will learn

• Hash Table
• Hash Function
• Key
• Linear robing
• Collusion
• Double Hashing

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Now Starts

• hash table
• Tables which can be searched for an item
  in O(1) time using a hash function to form an
  address from the key.
• hash function
• Function which, when applied to the key, produces
  a integer which can be used as an address in a
  hash table.
• collision
• When a hash function maps two different keys to
  the same table address, a collision is said to
  occur.
• linear probing
• A simple re-hashing scheme in which the next slot
  in the table is checked on a collision.
• quadratic probing
• A re-hashing scheme in which a higher (usually
  2nd) order function of the hash index is used to
  calculate the address.
• clustering.
• Tendency for clusters of adjacent slots to be
  filled when linear probing is used.
• secondary clustering.
• Collision sequences generated by addresses
  calculated with quadratic probing.
• perfect hash function
• Function which, when applied to all the members
  of the set of items to be stored in a hash table,
  produces a unique set of integers within some
  suitable range.

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An Example

• In this example john maps 3
• Phil maps 4
• Problem
• How mapping will be done?
• If two items maps the same place what haappens?

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A Plan For Hashing

• Save items in a key-indexed table. Index is a
  function of the key.
• Hash function.
• Method for computing table index from key.
• Collision resolution strategy.
• Algorithm and data structure to handle two keys
  that hash to the same index.
• If there is no space limitation
• Trivial hash function with key as address.
• If there is no time limitation
• Trivial collision resolution sequential search.
• Limitations on both time and space hashing (the
  real world)

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Finding A Hash Function

• Goal scramble the keys.
• Each table position equally likely for each key.
• Ex Vatandaslik Numarasi for 10000 person
• Bad The Whole Number Since 10000 will not be
  used forever
• Better last three digits. But every number is
  even
• The Best Use 2,3,4,5 digits
• Ex date of birth.
• Bad first three digits of birth year.
• Better birthday.
• Ex phone numbers.
• Bad first three digits.
• Better last three digits.

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Example for Hash Function
Question What should be the tableSize?
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Example for Hash Function
Question Is it suitable for table size 10007
Answer No English is not random. Although
there are 26317576 possible combinations of
three character, only 2851 different combinations
occur. So 100-28 72 waste of table
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New Hash Example
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Problem.

• Collision two keys hashing to same value.
• Essentially unavoidable.
• In probability theory, the birthday problem or
  birthday paradox pertains to the probability that
  in a set of randomly chosen people some pair of
  them will have the same birthday. In a group of
  23 (or more) randomly chosen people, there is
  more than 50 probability that some pair of them
  will both have been born on the same day. For 57
  or more people, the probability is more than 99,
  reaching 100 as the number of people reaches
  366. The mathematics behind this problem leads to
  a well-known cryptographic attack called the
  birthday attack.
• With M hash values, expect a collision after
  sqrt( M/2) insertions.
• Conclusion can't avoid collisions unlessyou have
  a ridiculous amount of memory.
• Challenge efficiently cope with collisions.

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Collision
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When collusion occurs

• Seperate Chaning
• Keep a list of elements that hash to the same
  value.

• Separate chaining array of M linked lists.
• Hash map key to integer i between 0 and M-1.
• Insert put at front of ith chain.
• constant time
• Search only need to search ith chain.
• proportional to length of chain

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Seperate Chain Performance

• Search cost is proportional to length of chain.
• Trivial average length N / M.
• Worst case all keys hash to same chain.
• Theorem. Let ? N / M gt 1 be average length of
  list which is called loading factor.
• Average search cost 1 ?/2
• What is the choice of M
• M too large too many empty chains.
• M too small chains too long.
• Typical choice N / M 10 constant-time
  search/insert.

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Hash Table without Link List

• Linear probing array of size M.
• Hash map key to integer i between 0 and M-1.
• Insert put in slot i if free, if not try i1,
  i2, etc.
• Search search slot i, if occupied but no match,
  try i1, i2, etc.
• Cluster.
• Contiguous block of items.
• Search through cluster using elementary
  algorithm for arrays.

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Lineer Probing
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Lineer Probing Performance

• Insert and search cost depend on length of
  cluster.
• Trivial average length of cluster ? N / M.
• Worst case all keys hash to same cluster.
• Theorem (Knuth, 1962). Let ? N / M lt 1 be
  average length of list.
• insert search
  
• Parameters.
• M too large too many empty array entries.
• M too small clusters coalesce.
• Typical choice M 2N constant-time
  search/insert.

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Lineer Probing Analyze
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Avoid Clustering

• Use Double Hash
• When collusion occurs use second hash function
• Avoid to get 0 at the second hash

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Rehashing

• After 70 of table is full, double the size of
  the hash table.
• Dont forget to have prime number

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Rehashing
Rehashing