Multilevel Indexing and B Trees

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Multilevel Indexing and B Trees
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Problems with simple indexes

• If index does not fit in memory
• Seeking the index is slow (binary search)
• We dont want more than 3 or 4 seeks for a
  search.
• Insertions and deletions take O(N) disk accesses.

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Indexed Sequential Files

• Provide a choice between two alternative views of
  a file
• Indexed the file can be seen as a set of records
  that is indexed by key or
• Sequential the file can be accessed sequentially
  (physically contiguous records), returning
  records in order by key.

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Example of applications

• Student record system in a university
• Indexed view access to individual records
• Sequential view batch processing when posting
  grades
• Credit card system
• Indexed view interactive check of accounts
• Sequential view batch processing of payments

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The initial idea

• Maintain a sequence set
• Group the records into blocks in a sorted way.
• Maintain the order in the blocks as records are
  added or deleted through splitting,
  concatenation, and redistribution.
• Construct a simple, single level index for these
  blocks.
• Choose to build an index that contain the key for
  the last record in each block.

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Maintaining a Sequence Set

• Sorting and re-organizing after insertions and
  deletions is out of question. We organize the
  sequence set in the following way
• Records are grouped in blocks.
• Blocks should be at least half full.
• Link fields are used to point to the preceding
  block and the following block (similar to doubly
  linked lists)
• Changes (insertion/deletion) are localized into
  blocks by performing
• Block splitting when insertion causes overflow
• Block merging or redistribution when deletion
  causes underflow.

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Example insertion

• Block size 4
• Key Last name

Block 1

• Insert BAIRD

Block 1
Block 2
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Example deletion
Block 1
Block 2
Block 3
Block 4

• Delete DAVIS, BYNUM, CARTER,

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Add an Index set

• Key Block
• BERNE 1
• CAGE 2
• DUTTON 3
• EVANS 4
• FOLK 5
• GADDIS 6

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Tree indexes

• This simple scheme is nice if the index fits in
  memory.
• If index doesnt fit in memory
• Divide the index structure into blocks,
• Organize these blocks similarly building a tree
  structure.
• Tree indexes
• B Trees
• B Trees
• Simple prefix B Trees

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Separators

• Block Range of Keys Separator
• 1 ADAMS-BERNE
• BOLEN
• 2 BOLEN-CAGE
• CAMP
• 3 CAMP-DUTTON
• EMBRY
• 4 EMBRY-EVANS
• FABER
• 5 FABER-FOLK
• FOLKS
• 6 FOLKS-GADDIS

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root
EMBRY
Index set
BOLEN CAMP
FABER FOLKS
ADAMS-BERNE
FOLKS-GADDIS
CAMP-DUTTON
EMBRY-EVANS
1
3
4
6
FABER-FOLK
BOLEN-CAGE
2
5
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B Trees

• B-tree is one of the most important data
  structures in computer science.
• What does B stand for? (Not binary!)
• B-tree is a multiway search tree.
• Several versions of B-trees have been proposed,
  but only B Trees has been used with large files.
• A Btree is a B-tree in which data records are in
  leaf nodes, and faster sequential access is
  possible.

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Formal definition of B Tree Properties

• Properties of a B Tree of order v
• All internal nodes (except root) has at least v
  keys and at most 2v keys .
• The root has at least 2 children unless its a
  leaf..
• All leaves are on the same level.
• An internal node with k keys has k1 children

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B tree Internal/root node structure
P0 K1 P1 K2
Pn-1 Kn Pn
Each Pi is a pointer to a child node each Ki is
a search key value of search key values n,
of pointers n1

• Requirements
• K1 lt K2 lt lt Kn
• For any search key value K in the subtree pointed
  by Pi,
• If Pi P0, we require K lt K1
• If Pi Pn, Kn ? K
• If Pi P1, , Pn-1, Ki lt K ? Ki1

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B tree leaf node structure
L K1 r1 K2
Kn rn R

• Pointer L points to the left neighbor R points
  to the right neighbor
• K1 lt K2 lt lt Kn
• v ? n ? 2v (v is the order of this B tree)
• We will use Ki for the pair ltKi, rigt and omit L
  and R for simplicity

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Example B tree with order of 1

• Each node must hold at least 1 entry, and at most
  2 entries

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Example Search in a B tree order 2

• Search how to find the records with a given
  search key value?
• Begin at root, and use key comparisons to go to
  leaf
• Examples search for 5, 16, all data entries gt
  24 ...
• The last one is a range search, we need to do the
  sequential scan, starting from the first leaf
  containing a value gt 24.

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Cost for searching a value in B tree

• Typically, a node is a page (block or cluster)
• Let H be the height of the B tree we need to
  read H1 pages to reach a leaf node
• Let F be the (average) number of pointers in a
  node (for internal node, called fanout )
• Level 1 1 page F0 page
• Level 2 F pages F1 pages
• Level 3 F F pages F2 pages
• Level H1 .. FH pages (i.e., leaf nodes)
• Suppose there are D data entries. So there are
  D/(F-1) leaf nodes
• D/(F-1) FH. That is, H logF( )

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B Trees in Practice

• Typical order 100. Typical fill-factor 67.
• average fanout 133 (i.e, of pointers in
  internal node)
• Can often hold top levels in buffer pool
• Level 1 1 page 8 Kbytes
• Level 2 133 pages 1 Mbyte
• Level 3 17,689 pages 133 MBytes
• Suppose there are 1,000,000,000 data entries.
• H log133(1000000000/132) lt 4
• The cost is 5 pages read

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How to Insert a Data Entry into a B Tree?

• Lets look at several examples first.

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Inserting 16, 8 into Example B tree
Root
30
17
24
13
16
14 15
One new child (leaf node) generated must add one
more pointer to its parent, thus one more key
value as well.
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Inserting 8 (cont.)

• Copy up the middle value (leaf split)

13 17 24 30
Entry to be inserted in parent node.
(Note that 5 is
s copied up and
5
continues to appear in the leaf.)
3
5
2
7
8
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Insertion into B tree (cont.)

• Understand difference between copy-up and push-up
• Observe how minimum occupancy is guaranteed in
  both leaf and index pg splits.

We split this node, redistribute entries evenly,
and push up middle key. ?
(Note that 17 is pushed up and only
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Example B Tree After Inserting 8
Root
17
24
30
13
5
2
3
39
19
20
22
24
27
38
7
5
8
14
15
29
33
34
Notice that root was split, leading to increase
in height.
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Inserting a Data Entry into a B Tree Summary

• Find correct leaf L.
• Put data entry onto L.
• If L has enough space, done!
• Else, must split L (into L and a new node L2)
• Redistribute entries evenly, put middle key in L2
• copy up middle key.
• Insert index entry pointing to L2 into parent of
  L.
• This can happen recursively
• To split index node, redistribute entries evenly,
  but push up middle key. (Contrast with leaf
  splits.)
• Splits grow tree root split increases height.
  
• Tree growth gets wider or one level taller at
  top.

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Deleting a Data Entry from a B Tree

• Examine examples first

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Delete 19 and 20
Root
17
24
30
13
5
2
3
39
19
20
22
24
27
38
7
5
8
14
16
29
33
34
You underflow
Have we still forgot something?
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Deleting 19 and 20 (cont.)
Root
17
27
30
13
5
2
3
39
38
7
5
8
22
24
27
29
14
16
33
34

• Notice how 27 is copied up.
• But can we move it up?
• Now we want to delete 24
• Underflow again! But can we redistribute this
  time?

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Deleting 24
You underflow Merge with sibling!

• Observe the two leaf nodes are merged, and 27 is
  discarded from their parent, but
• Observe pull down of index entry (below).

30
39
22
27
38
29
33
34
New root
13
5
30
17
3
39
2
7
22
38
5
8
27
33
34
14
16
29
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Deleting a Data Entry from a B Tree Summary

• Start at root, find leaf L where entry belongs.
• Remove the entry.
• If L is at least half-full, done!
• If L has only d-1 entries,
• Try to re-distribute, borrowing from sibling
  (adjacent node with same parent as L).
• If re-distribution fails, merge L and sibling.
• If merge occurred, must delete entry (pointing to
  L or sibling) from parent of L.
• Merge could propagate to root, decreasing height.

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Example of Non-leaf Re-distribution

• Tree is shown below during deletion of 24. (What
  could be a possible initial tree?)
• In contrast to previous example, can
  re-distribute entry from left child of root to
  right child.

Root
22
30
13
5
17
20
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After Re-distribution

• Intuitively, entries are re-distributed by
  pushing through the splitting entry in the
  parent node.
• It suffices to re-distribute index entry with key
  20 weve re-distributed 17 as well for
  illustration.

Root
17
30
22
13
5
20
39
7
5
8
2
3
38
17
18
33
34
22
27
29
20
21
14
16
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Terminology

• Bucket Factor the number of records which can
  fit in a leaf node.
• Fan-out the average number of children of an
  internal node.
• A Btree index can be used either as a primary
  index or a secondary index.
• Primary index determines the way the records are
  actually stored (also called a sparse index)
• Secondary index the records in the file are not
  grouped in buckets according to keys of secondary
  indexes (also called a dense index)

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Summary

• Tree-structured indexes are ideal for
  range-searches, also good for equality searches.
• B tree is a dynamic structure.
• Inserts/deletes leave tree height-balanced High
  fanout (F) means depth rarely more than 3 or 4.
• Almost always better than maintaining a sorted
  file.
• Typically, 67 occupancy on average.
• If data entries are data records, splits can
  change rids!
• Most widely used index in database management
  systems because of its versatility. One of the
  most optimized components of a DBMS.