Csci 2111: Data and File Structures Week 10, Lectures 1

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Csci 2111 Data and File StructuresWeek 10,
Lectures 1 2

Hashing
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Motivation

• Sequential Searching can be done in O(N) access
  time, meaning that the number of seeks grows in
  proportion to the size of the file.
• B-Trees improve on this greatly, providing O(Logk
  N) access where k is a measure of the leaf size
  (i.e., the number of records that can be stored
  in a leaf).
• What we would like to achieve, however, is an
  O(1) access, which means that no matter how big a
  file grows, access to a record always takes the
  same small number of seeks.
• Static Hashing techniques can achieve such
  performance provided that the file does not
  increase in time.

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What is Hashing?

• A Hash function is a function h(K) which
  transforms a key K into an address.
• Hashing is like indexing in that it involves
  associating a key with a relative record address.
• Hashing, however, is different from indexing in
  two important ways
• With hashing, there is no obvious connection
  between the key and the location.
• With hashing two different keys may be
  transformed to the same address.

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Collisions

• When two different keys produce the same address,
  there is a collision. The keys involved are
  called synonyms.
• Coming up with a hashing function that avoids
  collision is extremely difficult. It is best to
  simply find ways to deal with them.
• Possible Solutions
• Spread out the records
• Use extra memory
• Put more than one record at a single address.

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A Simple Hashing Algorithm

• Step 1 Represent the key in numerical form
• Step 2 Fold and Add
• Step 3 Divide by a prime number and use the
  remainder as the address.

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Hashing Functions and Record Distributions

• Records can be distributed among addresses in
  different ways there may be (a) no synonyms
  (uniform distribution) (b) only synonyms (worst
  case) (c) a few synonyms (happens with random
  distributions).
• Purely uniform distributions are difficult to
  obtain and may not be worth searching for.
• Random distributions can be easily derived, but
  they are not perfect since they may generate a
  fair number of synonyms.
• We want better hashing methods.

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Some Other Hashing Methods

• Though there is no hash function that guarantees
  better-than-random distributions in all cases, by
  taking into considerations the keys that are
  being hashed, certain improvements are possible.
• Here are some methods that are potentially better
  than random
• Examine keys for a pattern
• Fold parts of the key
• Divide the key by a number
• Square the key and take the middle
• Radix transformation

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Predicting the Distribution of Records

• When using a random distribution, we can use a
  number of mathematical tools to obtain
  conservative estimates of how our hashing
  function is likely to behave
• Using the Poisson Function p(x)(r/N)xe-(r/N)/x!
  applied to Hashing, we can conclude that
• In general, if there are N addresses, then the
  expected number of addresses with x records
  assigned to them is Np(x)

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Predicting Collisions for a Full File

• Suppose you have a hashing function that you
  believe will distribute records randomly and you
  want to store 10,000 records in 10,000 addresses.
• How many addresses do you expect to have no
  records assigned to them?
• How many addresses should have one, two, and
  three records assigned respectively?
• How can we reduce the number of overflow records?

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Increasing Memory Space I

• Reducing collisions can be done by choosing a
  good hashing function or using extra memory.
• The question asked here is how much extra memory
  should be use to obtain a given rate of collision
  reduction?
• Definition Packing density refers to the ratio
  of the number of records to be stored (r) to the
  number of available spaces (N).
• The packing density gives a measure of the amount
  of space in a file that is used.

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Increasing Memory Space II

• The Poisson Distribution allows us to predict the
  number of collisions that are likely to
  occur given a certain packing density. We use the
  Poisson Distribution to answer the following
  questions
• How many addresses should have no records
  assigned to them?
• How many addresses should have exactly one record
  assigned (no synonym)?
• How many addresses should have one record plus
  one or more synonyms?
• Assuming that only one record can be assigned to
  each home address, how many overflow records can
  be expected?
• What percentage of records should be overflow
  records?

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Collision Resolution by Progressive Overflow

• How do we deal with records that cannot fit into
  their home address? A simple approach
  Progressive Overflow or Linear Probing.
• If a key, k1, hashes into the same address, a1,
  as another key, k2, then look for the first
  available address, a2, following a1 and place k1
  in a2. If the end of the address space is
  reached, then wrap around it.
• When searching for a key that is not in, if the
  address space is not full, then an empty address
  will be reached or the search will come back to
  where it began.

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Search Length when using Progressive Overflow

• Progressive Overflow causes extra searches and
  thus extra disk accesses.
• If there are many collisions, then many records
  will be far from home.
• Definitions Search length refers to the number
  of accesses required to retrieve a record from
  secondary memory. The average search length is
  the average number of times you can expect to
  have to access the disk to retrieve a record.
• Average search length (Total search
  length)/(Total number of records)

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Storing More than One Record per Address Buckets

• Definition A bucket describes a block of records
  sharing the same address that is retrieved in one
  disk access.
• When a record is to be stored or retrieved, its
  home bucket address is determined by hashing.
  When a bucket is filled, we still have to worry
  about the record overflow problem, but this
  occurs much less often than when each address can
  hold only one record.

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Effect of Buckets on Performance

• To compute how densely packed a file is, we need
  to consider 1) the number of addresses, N,
  (buckets) 2) the number of records we can put at
  each address, b, (bucket size) and 3) the number
  of records, r. Then, Packing Density r/bN.
• Though the packing density does not change when
  halving the number of addresses and doubling the
  size of the buckets, the expected number of
  overflows decreases dramatically.

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Making Deletions

• Deleting a record from a hashed file is more
  complicated than adding a record for two reasons
• The slot freed by the deletion must not be
  allowed to hinder later searches
• It should be possible to reuse the freed slot for
  later additions.
• In order to deal with deletions we use
  tombstones, i.e., a marker indicating that a
  record once lived there but no longer does.
  Tombstones solve both the problems caused by
  deletion.
• Insertion of records is slightly different when
  using tombstones.

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Effects of Deletions and Additions on Performance

• After a large number of deletions and additions
  have taken places, one can expect to find many
  tombstones occupying places that could be
  occupied by records whose home address precedes
  them but that are stored after them.
• This deteriorates average search lengths.
• There are 3 types of solutions for dealing with
  this problem a) local reorganization during
  deletions b) global reorganization when the
  average search length is too large c) use of a
  different collision resolution algorithm.

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Other Collision Resolution Techniques

• There are a few variations on random hashing that
  may improve performance
• Double Hashing When an overflow occurs, use a
  second hashing function to map the record to its
  overflow location.
• Chained Progressive Overflow Like Progressive
  overflow except that synonyms are linked together
  with pointers.
• Chaining with a Separate Overflow Area Like
  chained progressive overflow except that overflow
  addresses do not occupy home addresses.
• Scatter Tables The Hash file contains no
  records, but only pointers to records. I.e., it
  is an index.

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Pattern of Record Access

• If we have some information about what records
  get accessed most often, we can optimize their
  location so that these records will have short
  search lengths.
• By doing this, we try to decrease the effective
  average search length even if the nominal average
  search length remains the same.
• This principle is related to the one used in
  Huffman encoding.