B Trees

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B Trees

• 14332351
• Programming Methodology II

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Motivation for B-Trees

• So far we have assumed that we can store an
  entire data structure in main memory
• What if we have so much data that it wont fit?
• We will have to use disk storage but when this
  happens our time complexity fails
• The problem is that Big-Oh analysis assumes that
  all operations take roughly equal time
• This is not the case when disk access is involved

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Motivation (cont.)

• Assume that a disk spins at 3600 RPM
• In 1 minute it makes 3600 revolutions, hence one
  revolution occurs in 1/60 of a second, or 16.7ms
• On average what we want is half way round this
  disk it will take 8ms
• This sounds good until you realize that we get
  120 disk accesses a second the same time as 25
  million instructions
• In other words, one disk access takes about the
  same time as 200,000 instructions
• It is worth executing lots of instructions to
  avoid a disk access

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Motivation (cont.)

• Assume that we use an AVL tree to store all car
  driver details (say about 20 million records)
• We still end up with a very deep tree with lots
  of different disk accesses log2 20,000,000 is
  about 24, so this takes about 0.2 seconds
  (248ms, if there is only one user of the
  program)
• We know we cant improve on the log n for a
  binary tree
• But, the solution is to use more branches and
  thus less height!
• As branching increases, depth decreases

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Indexing

• once the records are stored in a file, how do you
  search efficiently? (eg., ssn123?)

SELECT Name FROM STUDENT WHERE ssn123
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Indexing

• once the records are stored in a file, how do you
  search efficiently?
• brute force retrieve all records, report the
  qualifying ones
• better use indices (pointers) to locate the
  records directly

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Indexing main idea
SELECT Name FROM STUDENT WHERE ssn123
Ssn is the search key
Index the SSN attribute index has entries of
the form (value, ptr) sorted on the value To
search for ssn123 1) Do a binary search on
index file 2) Follow the pointer(s) of index
record(s)
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Main concepts and Index choices

• search keys are sorted in the index file and
  point to the actual records
• primary vs. secondary indices
• sparse vs. dense indices

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Index Choices

• Primary index search key physical order search
  key
• Secondary all other indexes
• Dense index entry for every search key value
• Sparse some search key values not in the index
• Single level vs Multilevel (index on the indices)

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Measuring goodness

• On what basis do we compare different indices?
• Access type what type of queries can be
  answered
• selection queries (ssn 123)?
• range queries ( 100 lt ssn lt 200)?
• Access time what is the cost of evaluating
  queries
• Measured in of block operations
• Maintenance overhead cost of insertion /
  deletion?
• Space overhead?
• Reorganization?

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B-Trees

• The B-tree is THE standard file organization for
  applications requiring insertion, deletion and
  key range searches.
• B-trees are always balanced.
• B-trees keep related records on a disk page,
  which takes advantage of locality of reference.
• B-trees guarantee that every node in the tree
  will be full at least to a certain minimum
  percentage. This improves space efficiency while
  reducing the typical number of disk fetches
  necessary during a search or update operation.

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Definition of a B-tree

• A B-tree of order m is an m-way tree (i.e., a
  tree where each node may have up to m children)
  in which
• 1. the number of keys in each non-leaf node is
  one less than the number of its children and
  these keys partition the keys in the children in
  the fashion of a search tree
• 2. all leaves are on the same level
• 3. all non-leaf nodes except the root have at
  least ?m / 2? children and a max of m children
• 4. the root is either a leaf node, or it has from
  two to m children
• 5. a leaf node contains no more than m 1 keys
• The number m should always be odd

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An example B-Tree of order 5 containing 26 items
1. The number of keys in each non-leaf node is
one less than the number of its children and
these keys partition the keys in the children in
the fashion of a search tree

• all non-leaf nodes except the root have at least
  ?m / 2? children and a max of m children
• m 5

2. all leaves are on the same level
5. a leaf node contains no more than m 1 keys
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Note when printed this slide is animated
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An example B-Tree of order 5 containing 26 items
1. The number of keys in each non-leaf node is
one less than the number of its children and
these keys partition the keys in the children in
the fashion of a search tree
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An example B-Tree of order 5 containing 26 items
2. all leaves are on the same level
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An example B-Tree of order 5 containing 26 items

• all non-leaf nodes except the root have at least
  ?m / 2? children and a max of m children
• m 5

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An example B-Tree of order 5 containing 26 items
4. the root is either a leaf node, or it has from
two to m children
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An example B-Tree of order 5 containing 26 items
5. a leaf node contains no more than m 1 keys
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Constructing a B-tree

• Suppose we start with an empty B-tree and keys
  arrive in the following order1 12 8 2 25 5
  14 28 17 7 52 16 48 68 3 26 29 53 55
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• We want to construct a B-tree of order 5
• The first four items go into the root
• To put the fifth item in the root would violate
  condition 5
• Therefore, when 25 arrives, pick the middle key
  to make a new root

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Constructing a B-tree (contd.)
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Constructing a B-tree (contd.)
Adding 17 to the right leaf node would over-fill
it, so we take the middle key, promote it (to the
root) and split the leaf
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Constructing a B-tree (contd.)
Adding 68 causes us to split the right most leaf,
promoting 48 to the root, and adding 3 causes us
to split the left most leaf, promoting 3 to the
root 26, 29, 53, 55 then go into the leaves
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Constructing a B-tree (contd.)
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Inserting into a B-Tree

• Attempt to insert the new key into a leaf
• If this would result in that leaf becoming too
  big, split the leaf into two, promoting the
  middle key to the leafs parent
• If this would result in the parent becoming too
  big, split the parent into two, promoting the
  middle key
• This strategy might have to be repeated all the
  way to the top
• If necessary, the root is split in two and the
  middle key is promoted to a new root, making the
  tree one level higher

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Removal from a B-tree

• During insertion, the key always goes into a
  leaf. For deletion we wish to remove from a
  leaf. There are three possible ways we can do
  this
• 1 - If the key is already in a leaf node, and
  removing it doesnt cause that leaf node to have
  too few keys, then simply remove the key to be
  deleted.
• 2 - If the key is not in a leaf then it is
  guaranteed (by the nature of a B-tree) that its
  predecessor or successor will be in a leaf -- in
  this case can we delete the key and promote the
  predecessor or successor key to the non-leaf
  deleted keys position.

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Removal from a B-tree (2)

• If (1) or (2) lead to a leaf node containing less
  than the minimum number of keys then we have to
  look at the siblings immediately adjacent to the
  leaf in question
• 3 if one of them has more than the min number
  of keys then we can promote one of its keys to
  the parent and take the parent key into our
  lacking leaf
• 4 if neither of them has more than the min
  number of keys then the lacking leaf and one of
  its neighbours can be combined with their shared
  parent (the opposite of promoting a key) and the
  new leaf will have the correct number of keys if
  this step leave the parent with too few keys then
  we repeat the process up to the root itself, if
  required

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Type 1 Simple leaf deletion
Assuming a 5-way B-Tree, as before...
Delete 2 Since there are enough keys in the
node, just delete it
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Type 2 Simple non-leaf deletion
Delete 52
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Borrow the predecessor or (in this case) successor
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Type 4 Too few keys in node and its siblings
Too few keys!
Delete 72
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Type 4 Too few keys in node and its siblings
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Type 3 Enough siblings
Delete 22
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Type 3 Enough siblings
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Analysis of B-Trees

• The maximum number of items in a B-tree of order
  m and height h
• root m 1
• level 1 m(m 1)
• level 2 m2(m 1)
• . . .
• level h mh(m 1)
• So, the total number of items is (1 m m2
  m3 mh)(m 1) (mh1 1)/ (m 1) (m
  1) mh1 1
• When m 5 and h 2 this gives 53 1 124

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Reasons for using B-Trees

• When searching tables held on disc, the cost of
  each disc transfer is high but doesn't depend
  much on the amount of data transferred,
  especially if consecutive items are transferred
• If we use a B-tree of order 101, say, we can
  transfer each node in one disc read operation
• A B-tree of order 101 and height 3 can hold 1014
  1 items (approximately 100 million) and any
  item can be accessed with 3 disc reads (assuming
  we hold the root in memory)
• If we take m 3, we get a 2-3 tree, in which
  non-leaf nodes have two or three children (i.e.,
  one or two keys)
• B-Trees are always balanced (since the leaves are
  all at the same level), so 2-3 trees make a good
  type of balanced tree

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Comparing Trees

• Binary trees
• Can become unbalanced and lose their good time
  complexity (big O)
• AVL trees are strict binary trees that overcome
  the balance problem
• Red Black trees are approximately balanced but
  still binary trees, so larger tree depth
• Multi-way trees
• B-Trees can be m-way, they can have any (odd)
  number of children
• One B-Tree, the 2-3 (or 3-way) B-Tree,
  approximates a permanently balanced binary tree

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B trees - Motivation

• B-tree print keys in sorted order

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B trees - Motivation

• B-tree needs back-tracking how to avoid it?

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Solution B - trees

• facilitate sequential ops
• They string all leaf nodes together
• AND
• replicate keys from non-leaf nodes, to make sure
  every key appears at the leaf level

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B-trees

• Eg., B-tree of order 3

root internal node
6
9
lt6
gt9
lt9
gt6
leaf node
4
3
9
6
7
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(3, Joe, 23)
(4, John, 23)
Data File
(3, Bob, 23)



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Reference

• Introduction to B trees
• http//www.comsc.ucok.edu/mcdaniel/mcdaniel/ds/b_
  trees
• Introduction to Algorithms by Cormen, Thomas H.,
  Charles E. Leiserson, and Ronald L. Rivest
• Java Applet
• B trees http//www.seanster.com/BplusTree/BplusT
  ree.html
• B trees http//sky.fit.qut.edu.au/maire/baobab/ba
  obab.html

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Indexing
Primary (or clustering) index on SSN
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Indexing
Secondary (or non-clustering) index duplicates
may exist

• Can have many secondary indices
• but only one primary index

Address-index
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Indexing
secondary key index typically, with postings
lists
Postings lists
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Indexing
Clustering/sparse index on ssn (Primary)
gt123
gt456
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Indexing
Non-clustering / dense index
Secondary on a candidate key No duplicates, no
need for posting lists
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Summary

• All combinations are possible

• at most one sparse/clustering index
• as many as desired dense indices
• usually one primary-key index (maybe
  clustering) and a few secondary-key indices
  (non-clustering)