Quick Review of material covered Apr 8

1
Quick Review of material covered Apr 8

• B-Tree Overview and some definitions
• balanced tree
• multi-level
• reorganizes itself on insertion and deletion
• built so each node fits on a single disk page
• Examined mechanics of B-tree Insertion and
  Deletion
• looked at several examples
• Well finish up B-trees with two more concepts
• B-tree File Organization
• B-tree index files

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B-tree File Organization

• B-Tree Indices solve the problem of index file
  degradation. The original data file will still
  degrade upon a stream of insert/delete
  operations.
• Solve data-file degradation by using a B-tree
  file organization
• Leaf nodes in a B-tree file organization store
  records, not pointers into a separate original
  datafile
• since records are larger than pointers, the
  maximum number of recrods that can be stored in a
  leaf node is less than the number of pointers in
  a non-leaf node
• leaf nodes must still be maintained at least half
  full
• insert and delete are handled in the same was as
  insert and delete for entries in a B-tree index

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B-tree File Organization Example

• Records are much bigger than pointers, so good
  space usage is important
• To improve space usage, involve more sibling
  nodes in redistribution during splits and merges
  (to avoid split/merge when possible)
• involving one sibling guarantees 50 space use
• involving two guarantees at least 2/3 space use,
  etc.

4
B-tree Index Files

• B-trees are similar to B-trees, but search-key
  values appear only once in the index (eliminates
  redundant storage of key values)
• search keys in non-leaf nodes dont appear in the
  leaf nodes, so an additional pointer field for
  each search key in a non-leaf node must be stored
  to point to the bucket or record for that key
  value
• leaf nodes look like B-tree leaf nodes
• (P1, K1, P2, K2, , Pn)
• non-leaf nodes look like so
• (P1, B1, K1, P2, B2, K2, , Pn)
• where the Bi are pointers to buckets or file
  records.

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B-tree Index File Example

• B-tree
• and
• B-tree

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B-tree Index Files (cont.)

• Advantages of B-tree Indices (vs. B-trees)
• May use less tree nodes than a B-tree on the
  same data
• Sometimes possible to find a specific key value
  before reaching a leaf node
• Disadvantages of B-tree Indices
• Only a small fraction of key values are found
  early
• Non-leaf nodes are larger, so fanout is reduced,
  and B-trees may be slightly taller than B-trees
  on the same data
• Insertion and deletion are more complicated than
  on B-trees
• Implementation is more difficult than B-trees
• In general, advantages dont outweigh
  disadvantages

7
Hashing

• Weve examined Ordered Indices (design based upon
  sorting or ordering search key values) the other
  type of major indexing technique is Hashing
• Underlying concept is very simple
• observation small files dont require indices or
  complicated search methods
• use some clever method, based upon the search
  key, to split a large file into a lot of little
  buckets
• each bucket is sufficiently small
• use the same method to find the bucket for a
  given search key

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Hashing Basics

• A bucket is a unit of storage containing one or
  more records (typically a bucket is one disk
  block in size)
• In a hash file organization we find the bucket
  for a record directly from its search-key value
  using a hash function
• A hash function is a function that maps from the
  set of all search-key values K to the set of all
  bucket addresses B
• The hash function is used to locate records for
  access, insertion, and deletion
• Records with different search-key values may be
  mapped to the same bucket
• the entire bucket must be searched to find a
  record
• buckets are designed to be small, so this task is
  usually not onerous

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Hashed File Example

• So we
• divide the set of disk blocks that make up the
  file into buckets
• devise a hash function that maps each key value
  into a bucket
• V set of key values
• B number of buckets
• H hashing function H V--gt (0, 1, 2, 3, , B-1)
• Example V 9 digit SS B1000 H key modulo
  1000

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Hash Functions

• To search/insert/delete/modify a key do
• compute H(k) to get the bucket number
• search sequentially in the bucket (heap
  organization within each bucket)
• Choosing H almost any function that generates
  random numbers in the range 0, B-1
• try to distribute the keys evenly into the B
  buckets
• one rule of thumb when using MOD -- use a prime
  number

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Hash Functions (2)

• Collision is when two or more key values go to
  the same bucket
• too many collisions increases search time and
  degrades performance
• no or few collisions means that each bucket has
  only one (or very few) key(s)
• Worst-case hash functions map all search keys to
  the same bucket

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Hash Functions (3)

• Ideal hash functions are uniform
• each bucket is assigned the same number of
  search-key values from the set of all possible
  values
• Ideal hash functions are random
• each bucket has approximately the same number of
  records assigned to it irrespective of the actual
  distribution of search-key values in the file
• Finding a good hash function is not always easy

13
Examples of Hash Functions

• Given 26 buckets and a string-valued search key,
  consider the following possible hash functions
• Hash based upon the first letter of the string
• Hash based upon the last letter of the string
• Hash based upon the middle letter of the string
• Hash based upon the most common letter in the
  string
• Hash based upon the average letter in the
  string the sum of the letters (using A0, B1,
  etc) divided by the number of letters
• Hash based upon the length of the string (modulo
  26)
• Typical hash functions perform computation on the
  internal binary representation of the search key
• example searching on a string value, hash based
  upon the binary sum of the characters in the
  string, modulo the number of buckets