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Multilevel Indexing and
B+ Trees
CENG 351 File Structures
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Problems with simple indexes
If index does not fit in memory:
1. Seeking the index is slow (binary search):
–
We don’t want more than 3 or 4 seeks for a search.
N
Log(N+1)
15 keys
4
1000
~10
100,000
~17
1,000,000
~20
2. Insertions and deletions take O(N) disk accesses.
CENG 351 File Structures
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Indexed Sequential Files
•
Provide a choice between two alternative
views of a file:
1. Indexed: the file can be seen as a set of
records that is indexed by key; or
2. Sequential: the file can be accessed
sequentially (physically contiguous
records), returning records in order by key.
CENG 351 File Structures
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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
CENG 351 File Structures
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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.
CENG 351 File Structures
5
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.
CENG 351 File Structures
6
Example: insertion
• Block size = 4
• Key : Last name
Block 1 ADAMS …
BIXBY …
CARSON … COLE …
• Insert “BAIRD …”:
Block 1
ADAMS … BAIRD …
Block 2 CARSON ..
BIXBY …
COLE …
CENG 351 File Structures
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Example: deletion
Block 1 ADAMS …
BAIRD …
BIXBY …
Block 2 BYNUM…
CARSON ..
CARTER ..
BOONE …
Block 3 DENVER… ELLIS …
Block 4 COLE…
DAVIS
• Delete “DAVIS”, “BYNUM”, “CARTER”,
CENG 351 File Structures
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Add an Index set
Key
Block
BERNE
CAGE
DUTTON
EVANS
FOLK
GADDIS
1
2
3
4
5
6
CENG 351 File Structures
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Tree indexes
• This simple scheme is nice if the index fits in
memory.
• If index doesn’t 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
…
CENG 351 File Structures
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Separators
Block
1
Range of Keys
Separator
ADAMS-BERNE
BOLEN
2
BOLEN-CAGE
CAMP
3
CAMP-DUTTON
EMBRY
4
EMBRY-EVANS
FABER
5
FABER-FOLK
FOLKS
6
FOLKS-GADDIS
CENG 351 File Structures
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root
EMBRY
Index set
BOLEN
FABER
CAMP-DUTTON
ADAMS-BERNE
1
CAMP
BOLEN-CAGE
3
FOLKS
EMBRY-EVANS
4
2
FOLKS-GADDIS
FABER-FOLK
6
5
CENG 351 File Structures
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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 B+tree is a B-tree in which data records are in
leaf nodes, and faster sequential access is possible.
CENG 351 File Structures
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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 it’s a leaf..
– All leaves are on the same level.
– An internal node with k keys has k+1 children
CENG 351 File Structures
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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 = n+1
 Requirements:
 K1 < K2 < … < K n
 For any search key value K in the subtree pointed by Pi,
If Pi = P0, we require K < K1
If Pi = Pn, Kn  K
If Pi = P1, …, Pn-1, Ki < K  Ki+1
CENG 351 File Structures
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B+ tree: leaf node structure
L K 1 r1 K 2
R
………………
K n rn
 Pointer L points to the left neighbor; R points to
the right neighbor
 K1 < K2 < … < Kn
 v  n  2v (v is the order of this B+ tree)
 We will use Ki* for the pair <Ki, ri> and omit L and
R for simplicity
CENG 351 File Structures
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Example: B+ tree with order of 1
• Each node must hold at least 1 entry, and at most
2 entries
Root
40
10*
15*
20
33
20*
27*
51
33*
37*
40*
46*
CENG 351 File Structures
51*
63
55*
63*
97*
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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 >= 24* ...
– The last one is a range search, we need to do the sequential scan, starting from
the first leaf containing a value >= 24.
Root
13
2*
3*
5*
7*
14* 15*
17
24
19* 20* 22*
30
24* 27* 29*
CENG 351 File Structures
33* 34* 38* 39*
18
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 H+1 pages to reach a leaf node
• Let F be the (average) number of pointers in a
node (for internal node, called fanout )
Height
– Level 1 = 1 page = F0 page
0
– Level 2 = F pages = F1 pages
1
– Level 3 = F * F pages = F2 pages
– Level H+1 = …….. = 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( FD 1)
CENG 351 File Structures
level
1
2
19
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) < 4
The cost is 5 pages read
CENG 351 File Structures
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How to Insert a Data Entry into a
B+ Tree?
• Let’s look at several examples first.
CENG 351 File Structures
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Inserting 16*, 8* into Example B+ tree
Root
2*
3*
5*
7*
8*
13
17
24
30
15* 16*
14*
You overflow
13
2*
3*
5*
7*
17
24
30
8*
One new child (leaf node)
generated; must add one more
pointer to its parent, thus one more
CENG 351 File Structures
key value as well.
22
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
continues to appear in the leaf.)
5
2*
5
3*
13
5*
17
24
7*
30
CENG 351 File Structures
8*
You overflow!
23
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.
5
13
17
24
30
We split this node, redistribute entries evenly,
and push up middle key.

17
5
13
24
CENG 351 File Structures
Entry to be inserted in parent node.
(Note that 17 is pushed up and only
appears once in the index. Contrast
this with a leaf split.)
30
24
Example B+ Tree After Inserting 8*
Root
17
5
2*
3*
24
13
5*
7* 8*
14* 15*
19* 20* 22*
30
24* 27* 29*
33* 34* 38* 39*
Notice that root was split, leading to increase in height.
CENG 351 File Structures
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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.
CENG 351 File Structures
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Deleting a Data Entry from a B+
Tree
• Examine examples first …
CENG 351 File Structures
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Delete 19* and 20*
Root
17
5
2*
3*
24
13
5*
7* 8*
14* 16*
19* 20* 22*
30
24* 27* 29*
33* 34* 38* 39*
22*
22* 24*
27* 29*
Have we still forgot something?
CENG 351 File Structures
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Deleting 19* and 20* (cont.)
Root
17
5
2*
3*
•
•
•
•
27
13
5*
7* 8*
14* 16*
22* 24*
30
27* 29*
33* 34* 38* 39*
Notice how 27 is copied up.
But can we move it up?
Now we want to delete 24
Underflow again!CENG
But351can
we
redistribute
this
time?
File Structures
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Deleting 24*
• Observe the two leaf
nodes are merged, and
27 is discarded from
their parent, but …
• Observe `pull down’ of
index entry (below).
New root
2*
3*
5*
5
7*
8*
13
14* 16*
30
22*
17
27*
29*
33*
34*
38*
39*
30
22* 27* 29*
CENG 351 File Structures
33* 34* 38* 39*
30
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.
CENG 351 File Structures
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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
5
2* 3*
5* 7* 8*
13
14* 16*
17
30
20
17* 18*
20* 21*
CENG 351 File Structures
22* 27* 29*
33* 34* 38* 39*
32
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;
we’ve re-distributed 17 as well for illustration.
Root
17
5
2* 3*
5* 7* 8*
13
14* 16*
20
17* 18*
20* 21*
CENG 351 File Structures
22
30
22* 27* 29*
33* 34* 38* 39*
33
Primary vs Secondary Index
• Note: We were assuming the data items
were in sorted order
– This is called primary index
• Secondary index:
– Built on an attribute that the file is not sorted
on.
• Can have many different indexes on the
same file.
CENG 351 File Structures
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A Secondary B+-Tree index
Root
17
5
2
3
5
7
8
13
14 16
22* 3* 14*
8*
20
17 18
2* 16* 5* 39*
20 21
22
30
22 27 29
33 34 38 39
20* 7* 38* 29*
Actual data
buckets
More…
• Hash-based Indexes
– Static Hashing
– Extendible Hashing
– Linear Hashing
• Grid-files
• R-Trees
• etc…
CENG 351 File Structures
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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 B+tree 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)
CENG 351 File Structures
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Summary
• Tree-structured indexes are ideal for rangesearches, 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.
CENG 351 File Structures
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