FALL 2004CENG 351 File Structures1 Multilevel Indexing and B+ Trees Reference: Folk et al. Chapters:...
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Transcript of FALL 2004CENG 351 File Structures1 Multilevel Indexing and B+ Trees Reference: Folk et al. Chapters:...
FALL 2004 CENG 351 File Structures 1
Multilevel Indexing and B+ Trees
Reference: Folk et al. Chapters: 9.1, 9.2, 10.1, 10.2,10.3,10.10
FALL 2004 CENG 351 File Structures 2
Problems with simple indexesIf 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.
2. Insertions and deletions take O(N) disk accesses.
N Log(N+1)
15 keys 4
1000 ~10
100,000 ~17
1,000,000 ~20
FALL 2004 CENG 351 File Structures 3
Multilevel Indexing
• Build an index of an index file:– Build a simple index for the file sorting keys using
external sorting.
– Build an index for this index.
– Build another index for the previous index, and so on.
• Note: the index of an index stores the largest key in the record it is pointing to.
• Insertion and deletion are still problematic.
FALL 2004 CENG 351 File Structures 4
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.
FALL 2004 CENG 351 File Structures 5
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
FALL 2004 CENG 351 File Structures 6
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.
FALL 2004 CENG 351 File Structures 7
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.
FALL 2004 CENG 351 File Structures 8
Example: insertion• Block size = 4
• Key : Last name
ADAMS … BIXBY … CARSON … COLE …Block 1
•Insert “BAIRD …”:
ADAMS … BAIRD … BIXBY …
CARSON .. COLE …
Block 1
Block 2
FALL 2004 CENG 351 File Structures 9
Example: deletionADAMS … BAIRD … BIXBY … BOONE …Block 1
BYNUM… CARSON .. CARTER ..
DENVER… ELLIS …
Block 2
Block 3
• Delete “DAVIS”, “BYNUM”, “CARTER”,
COLE… DAVISBlock 4
FALL 2004 CENG 351 File Structures 10
Add an Index set
Key Block
BERNE 1CAGE 2DUTTON 3EVANS 4FOLK 5GADDIS 6
FALL 2004 CENG 351 File Structures 11
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– …
FALL 2004 CENG 351 File Structures 12
Separators Block Range of Keys Separator
1 ADAMS-BERNEBOLEN
2 BOLEN-CAGECAMP
3 CAMP-DUTTONEMBRY
4 EMBRY-EVANSFABER
5 FABER-FOLKFOLKS
6 FOLKS-GADDIS
FALL 2004 CENG 351 File Structures 13
ADAMS-BERNE ADAMS-BERNE
ADAMS-BERNEADAMS-BERNE
ADAMS-BERNE ADAMS-BERNE
BOLEN CAMP
EMBRY
FABER FOLKS
1
2
3
5
4 6
Index set
root
FALL 2004 CENG 351 File Structures 14
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.
FALL 2004 CENG 351 File Structures 15
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
FALL 2004 CENG 351 File Structures 16
B+ tree: Internal/root node structure
P0 K1 P1 K2 ……………… Pn-1 Kn Pn
Requirements:
K1 < K2 < … < Kn
For any search key value K in the subtree pointed by Pi,If Pi = P0, we require K < K1If Pi = Pn, Kn KIf Pi = P1, …, Pn-1, Ki K < Ki+1
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
FALL 2004 CENG 351 File Structures 17
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
B+ tree: leaf node structure
L K1 r1 K2 ……………… Kn rn R
FALL 2004 CENG 351 File Structures 18
Example: B+ tree with order of 1
• Each node must hold at least 1 entry, and at most 2 entries
10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*
20 33 51 63
40
Root
FALL 2004 CENG 351 File Structures 19
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
17 24 30
2* 3* 5* 7* 14* 15* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
13
FALL 2004 CENG 351 File Structures 20
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 ) – Level 1 = 1 page = F0 page– Level 2 = F pages = F1 pages– 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( )
1
2
levelHeight 0
2
1F
D
FALL 2004 CENG 351 File Structures 21
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
FALL 2004 CENG 351 File Structures 22
How to Insert a Data Entry into a B+ Tree?
• Let’s look at several examples first.
FALL 2004 CENG 351 File Structures 23
Inserting 16*, 8* into Example B+ treeRoot
17 24 3013
2* 3* 5* 7* 8*
2* 5* 7*3*
17 24 3013
8*
You overflow
One new child (leaf node) generated; must add one more pointer to its parent, thus one more key value as well.
14* 15* 16*
FALL 2004 CENG 351 File Structures 24
Inserting 8* (cont.)• Copy up the
middle value (leaf split)
2* 3* 5* 7* 8*
5
Entry to be inserted in parent node.(Note that 5 iscontinues to appear in the leaf.)
s copied up and
13 17 24 30
You overflow! 5 13 17 24 30
FALL 2004 CENG 351 File Structures 25
(Note that 17 is pushed up and onlyappears once in the index. Contrast
Entry to be inserted in parent node.
this with a leaf split.)
5 24 30
17
13
Insertion into B+ tree (cont.)
5 13 17 24 30
• 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.
FALL 2004 CENG 351 File Structures 26
Example B+ Tree After Inserting 8*
Notice that root was split, leading to increase in height.
2* 3*
Root
17
24 30
14* 15* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
135
7*5* 8*
FALL 2004 CENG 351 File Structures 27
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.
FALL 2004 CENG 351 File Structures 28
Deleting a Data Entry from a B+ Tree
• Examine examples first …
FALL 2004 CENG 351 File Structures 29
Delete 19* and 20*
2* 3*
Root
17
24 30
14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
135
7*5* 8*
22*
27* 29* 22* 24*
You underflow
Have we still forgot something?
FALL 2004 CENG 351 File Structures 30
Deleting 19* and 20* (cont.)
• 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?
2* 3*
Root
17
30
14* 16* 33* 34* 38* 39*
135
7*5* 8* 22* 24*
27
27* 29*
FALL 2004 CENG 351 File Structures 31
Deleting 24*• Observe the two leaf
nodes are merged, and 27 is discarded from their parent, but …
• Observe `pull down’ of index entry (below).
30
22* 27* 29* 33* 34* 38* 39*
2* 3* 7* 14* 16* 22* 27* 29* 33* 34* 38* 39*5* 8*
30135 17
You underfl
ow
Merge w
ith sib
ling!
New root
FALL 2004 CENG 351 File Structures 32
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.
FALL 2004 CENG 351 File Structures 33
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
135 17 20
22
30
14* 16* 17* 18* 20* 33* 34* 38* 39*22* 27* 29*21*7*5* 8*3*2*
FALL 2004 CENG 351 File Structures 34
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.
14* 16* 33* 34* 38* 39*22* 27* 29*17* 18* 20* 21*7*5* 8*2* 3*
Root
135
17
3020 22
FALL 2004 CENG 351 File Structures 35
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)
FALL 2004 CENG 351 File Structures 36
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.