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CS 728
Advanced Database Systems
Chapter 18
Database File Indexing Techniques, BTrees, and B+-Trees
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Indexes as Access Paths
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A single-level index is an auxiliary file that makes
it more efficient to search for a record in the data
file.
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The index is usually specified on one field of the
file (although it could be specified on several
fields)
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One form of an index is a file of entries <field
value, pointer to record>, which is ordered by
field value
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The index is called an access path on the field.
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Indexes as Access Paths (cont.)
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The index file usually occupies considerably less
disk blocks than the data file because its entries are
much smaller
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A binary search on the index yields a pointer to the
file record
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Indexes can also be characterized as dense or
sparse
 dense index
 has index entry for every record in the file.
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sparse (nondense) index
 has index entries for only some of the search-key
values.
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Sparse Vs. Dense Indices
Id
Name
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Dept
Sparse primary
index sorted
on Id
Ordered file sorted on Id
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Dense secondary
index sorted
on Name
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Sparse Vs. Dense Indices
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Ashby, 25, 3000
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Basu, 33, 4003
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Bristow, 30, 2007
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Ashby
33
Cass
Cass, 50, 5004
Smith
Daniels, 22, 6003
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Jones, 40, 6003
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Sparse primary
index on Name
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Smith, 44, 3000
50
Tracy, 44, 5004
Dense secondary
Ordered file on Name index on Age
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Sparse Indices
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Sparse index contains index records for only some
search-key values.
 Some keys in the data file will not have an entry
in the index file
 Applicable
when records are sequentially ordered
on search-key (ordered files)
 Normally
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keeps only one key per data block
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Less space (can keep more of index in memory)
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Less maintenance overhead for insertions and
deletions.
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Sparse Indices
Ordered File
Sparse/Primary Index
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20
10
30
50
70
30
40
90
110
130
150
50
60
70
80
90
100
170
190
210
230
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Example
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Given the following data file EMPLOYEE(NAME, SSN, ADDRESS,
SAL)
Suppose that:
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record size R=150 bytes
block size B=512 bytes
r=30000 records
blocking factor Bfr= B/R=512/150=3 records/block
number of file blocks b=(r/Bfr)=(30000/3)=10000 blocks
For an index on the SSN field, assume the field size VSSN=9 bytes, assume
the record pointer size PR=7 bytes. Then:
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index entry size RI=(VSSN+ PR)=(9+7)=16 bytes
index blocking factor BfrI= B/RI= 512/16= 32 entries/block
number of index blocks b= (r/BfrI)=(30000/32)=938 blocks
binary search needs log2bI= log2938= 10 block accesses
This is compared to an average linear search cost of:
 (b/2)= 30000/2= 15000 block accesses
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If the file records are ordered, the binary search cost would be:
 log2b= log230000= 15 block accesses
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Types of Single-Level Indexes
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primary index:
 is specified on the ordering key field of an ordered file,
where every record has a unique value for that field.
 The index has the same ordering as the one of the file.
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clustering index:
 is specified on the ordering field of an ordered file.
 The index has the same ordering as the one of the file.
 An ordered file can have at most one primary index or one
clustering index, but not both.
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secondary index:
 is specified on any nonordering field of the file.
 The index has different ordering than the one of the file.
 A file can have several secondary indices in addition to its
primary/clustering index.
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Primary Indices
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Defined on an ordered data file
The data file is ordered on a key field
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Includes one index entry for each block in the data
file; the index entry has the key field value for the
first record in the block, which is called the block
anchor
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A similar scheme can use the last record in a block.
Finding a record is efficient – do a binary search
A primary index is a nondense (sparse) index, since
it includes an entry for each disk block of the data
file and the keys of its anchor record rather than for
every search value.
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Primary Index on the Ordering Key Field 
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Clustering Indices
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Defined on an ordered data file
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The data file is ordered on a non-key field unlike primary
index, which requires that the ordering field of the data file
have a distinct value for each record.
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Includes one index entry for each distinct value of the
clustering field (rather than for every record).
 Sparse index (nondense)
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The index entry points to the first data block that contains
records with that field value.
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A file can have at most one primary index or one clustering
index, but not both.
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Clustering Indices
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A clustering index on the
DEPNo ordering nonkey
field of an EMPLOYEE
file.
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Clustering Indices
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Clustering index with
a separate block cluster
for each group of
records that share the
same value for the
clustering field.
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Secondary Indices
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Secondary index:
 is specified on any nonordering field of the file.
 Non-ordering field can be a key (unique) or a non-key
(duplicates)
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The index has different ordering than the one of the
file.
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A file can have several secondary indices in addition to
its primary index.
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there is one index entry for each data record
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index record points either to the block in which the
record is stored, or to the record itself
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Hence, such an index is dense
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Secondary Indices
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A secondary index usually needs more storage space and
longer search time than does a primary index.
 It has larger number of entries.
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Sequential scan using primary index is efficient, but a
sequential scan using a secondary index is expensive
 each record access may fetch a new block from disk
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The index is an ordered file with two fields.
 The first field is of the same data type as some nonordering field of the data file that is an indexing field.
 The second field is either a block pointer or a record
pointer.
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Example of a Dense Secondary Index 
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A dense secondary index (with block
pointers) on a nonordering KEY field.
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Example of a Secondary Index
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A dense secondary index (with
record pointers) on a non-ordering
non-key field.
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Index Types and Indexing Fields
Data file ordered
by indexing field
Indexing field is key
Indexing field is nonkey
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Primary Index
Clustering Index
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Data file not
ordered by
indexing field
Secondary index
(Key)
Secondary index
(NonKey)
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Multi-Level Indexes
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Because a single-level index is an ordered file, we can
create a primary index to the index itself;
 In this case, the original index file is called the firstlevel index and the index to the index is called the
second-level index.
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We can repeat the process, creating a third, fourth, ..., top
level until all entries of the top level fit in one disk block
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A multi-level index can be created for any type of first-level
index (primary, secondary, clustering) as long as the firstlevel index consists of more than one disk block
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A Two-Level
Primary Index
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Multi-Level Indexes
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Such a multi-level index is a form of search tree
 However, insertion and deletion of new index
entries is a severe problem because every level of
the index is an ordered file.
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A Node in a search tree with pointers to subtrees below
it.
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Dynamic Multilevel Indexes Using B-Trees 
and B+-Trees
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Most multi-level indexes use B-tree or B+-tree data
structures because of the insertion and deletion problem
 This leaves space in each tree node (disk block) to
allow for new index entries
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These data structures are variations of search trees that
allow efficient insertion and deletion of new search values.
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In B-Tree and B+-Tree data structures, each node
corresponds to a disk block
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Each node is kept between half-full and completely full
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Dynamic Multilevel Indexes Using B-Trees 
and B+-Trees (cont.)
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An insertion into a node that is not full is quite
efficient
 If a node is full the insertion causes a split into
two nodes
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Splitting may propagate to other tree levels
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A deletion is quite efficient if a node does not
become less than half full
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If a deletion causes a node to become less than half
full, it must be merged with neighboring nodes
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Difference between B-tree and B+-tree
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In a B-tree, pointers to data records exist at all
levels of the tree
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In a B+-tree, all pointers to data records exists at the
leaf-level nodes
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A B+-tree can have less levels (or higher capacity
of search values) than the corresponding B-tree
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B-tree Structures
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B+-Tree Index
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A B+-tree, of order f (fan-out --- maximum node capacity),
is a rooted tree satisfying the following:
 All paths from root to leaf are of the same length
(balanced tree)
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Each non-leaf node (except the root) has between f/2
and up to f tree pointers (f-1 key values).
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A leaf node has between f/2 and f-1 data pointers
(plus a pointer for sibling node).
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If the root is not a leaf, it has at least 2 children.
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If the root is a leaf (that is, there are no other nodes in
the tree), it can have between 0 and f-1 values.
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The Nodes of a B+-tree
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B+-Tree Non-leaf Node Structure
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Ki are the search-key values, K1  K2  K3  …  Kf-1
 all keys in the subtree to which P1 points are  K1.
 all keys in the subtree to which Pf points are  Kf-1.
 for 2  i  f-1, all keys in the subtree to which Pi points
have values  Ki-1 and  Ki.
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Pi are pointers to children nodes (tree nodes).
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B+-Tree Leaf Node Structure
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for i = 1, 2, …, f-1, pointer Pri is a data pointer, that either
points to a
 file record with search-key value Ki, or
 block of record pointers that point to records having
search-key value Ki. (if search-key is not a key)
Pnext points to next leaf node in search-key order.
Within each leaf node, K1  K2  K3  …  Kf-1
If Li, Lj are leaf nodes and i  j, then
 Li’s search-key values  Lj’s search-key values
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57
81
95
To record
with key 57
To record
with key 81
To record
with key 95
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Sample Leaf Node
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From non-leaf node
to next leaf
in sequence
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95
81
57
to keys
 57
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Sample Non-Leaf Node
to keys
57  k  81
to keys
81  k  95
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to keys
 95
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110
130
179
11
35
Root
180
200
150
156
179
120
130
100
101
110
30
35
3
5
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Example of a B+-Tree
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f=4
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Number of pointers/keys for B+-Tree 
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Non-leaf
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Leaf
30
35
min. node
120
150
180
Full node
3
5
11
f=4
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Observations about B+-Trees
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In a B+-tree, data pointers are stored only at the
leaf nodes of the tree
 hence, the structure of leaf nodes differs from
the structure of internal nodes.
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The leaf nodes have an entry for every value of the
search field, along with a data pointer to the
record.
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Some search field values from the leaf nodes are
repeated in the internal nodes.
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B+-Trees: Search
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Let a be a search key value and T the pointer to the
root of the tree that has f pointer.
Search(a, T)
If T is non-leaf node:
 for the first i that satisfy a  Ki, 1  i  f-1
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call Search(a, Pi),
 else call Search(a, Pf).
Else
//T is a leaf node
 if no value in T equals a, report not found.
 else if Ki in T equals a, follow pointer Pri to
read the record/block.
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B+-Trees: Search
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In processing a query, a path is traversed in the tree
from the root to some leaf node.
If there are n search-key values in the file,
 the path is no longer than log f/2(n) (worst
case).
With 1 million search key values and f = 100, at
most log50(1000000) = 4 nodes are accessed in a
lookup.
Contrast this with a balanced binary tree with 1
million search key values -- around 20 nodes are
accessed in a lookup.
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B+-Trees: Insertion
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Find the leaf node in which the search-key value
would appear
If the search-key value is found in the leaf node,
 add the record to main file and if necessary
 add to the block a pointer to the record
If the search-key value is not there,
 add the record to the main file and then:
 If there is room in the leaf node, insert (keyvalue, pointer) pair in the leaf node
 Otherwise, split the node along with the new
(key-value, pointer) entry
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B+-Trees: Insertion
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Splitting a node:
 take the f (search-key value, pointer) pairs
(including the one being inserted) in sorted order.
 place the first (f+1)/2 in the original node x, and
the rest in a new node y.
 let k be the largest key value in x.
 insert (k, y) in the parent node in their correct
sequence.
 If the parent is full
 the entries in the parent node up to Pj, where j =
(f+1)/2 are kept, while the jth search value is
moved to the parent, no replicated.
 A new internal node will hold the entries from
Pj+1 to the end of the entries in the node.
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B+-Trees: Insertion
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The splitting of nodes proceeds upwards till a node
that is not full is found.
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In the worst case the root node may be split
increasing the height of the tree by 1.
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B+-Trees: Deletion
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Find the record to be deleted, and remove it from
the main file and from the bucket (if present).
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Remove (search-key value, pointer) from the leaf
node.
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If the node has too few entries due to the removal,
and the entries in the node and a sibling fit into a
single node, then
 Insert all the search-key values in the two nodes
into a single node (the one on the left), and
delete the other node.
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B+-Trees: Deletion
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 Delete
the pair (Ki-1, Pi), where Pi is the pointer to
the deleted node, from its parent, recursively
using the above procedure.
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Otherwise, if the node has too few entries due to the
removal, and the entries in the node and a sibling
DO NOT fit into a single node, then
 Redistribute the pointers between the node and a
sibling such that both have more than the
minimum number of entries.
 Update the corresponding search-key value in the
parent of the node.
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B+-Trees: Deletion
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The node deletions may cascade upwards till a node
which has f/2 or more pointers is found.
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If the root node has only one pointer after deletion, it
is deleted and the sole child becomes the root.
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Example of a
Deletion in a
B+-tree
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Extra Reading
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Read Examples 1 to 7.
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