Struktur Data & Algoritme
(
Data Structures & Algorithms
)
Denny (
denny@cs.ui.ac.id)
Suryana Setiawan (
setiawan@cs.ui.ac.id)
Fakultas Ilmu Komputer Universitas Indonesia Semester Genap - 2004/2005
Version 2.0 - Internal Use Only
B Trees
SDA/BTREE/V2.0/2Outline
Motivation
B-Tree
B+Tree
SDA/BTREE/V2.0/3
Motivation
Perhatikan kasus berikut ini:
Kamu harus membuat program basisdata untuk menyimpan data
di yellow pages daerah Jakarta, misalnya ada 500.000 data.
Setiap entry terdapat nama, alamat, nomor telepon, dll. Asumsi
setiap entry disimpan dalam sebuah record yang besarnya 512 byte.
Total file size = 500,000 * 512 byte = 256 MB.
• terlalu besar untuk disimpan dalam memory (primary storage)
• perlu disimpan di disk (secondary storage)
Jika kita menggunakan disk untuk penyimpanan, kita harus menggunakan struktur blok pada disk untuk menyimpan basis data tsb.
Secondary storage dibagi menjadi blok-blok yang ukurannya
sama. Umumnya 512 byte, 2 KB, 4 KB, 8 KB.
Block adalah satuan unit transfer antar disk dengan memory.
Walaupun program hanya membaca 10 byte dari disk, 1 block akan dibaca dari disk dan disimpan ke memory.
Motivation
Misalnya 1 disk block 8.192 byte (8 KB)
Maka jumlah blok yang diperlukan = 256 MB / 8 KB
per block = 31,250 blocks.
Setiap blok menyimpan = 8,192 / 512 = 16 records.
A disk access is really expensive due to mechanical
limitation.
Disk access is approx. 10,000 times slower than main
memory. One disk access is worth about 200,000
instructions.
The number of disk accesses will dominate the
running time.
Goal: a multiway search tree that will minimize disk
accesses.
SDA/BTREE/V2.0/5
B-Tree
B-tree adalah (a,b)-tree di mana b = 2a - 1. a dan b
biasanya merupakan angka-angka yang cukup besar,
misalnya a = 100 dan b = 199.
B-tree banyak digunakan untuk external data
structure.
Setiap node berukuran sesuai dengan ukuran block
pada disk, misalnya 1 block = 8 KB.
Tujuannya: meminimalkan jumlah block transfer.
SDA/BTREE/V2.0/6
B Trees
The problem with Binary Trees is balance, the tree can easily deteriorate to a linked list. Consequently, the reduced search times are lost, this problem is overcome in B-Trees.
B stands for Balanced,
where all the leaves are the same distance from the root. B-Trees guarantee a predictable efficiency.
There are several varieties of B-Trees, most applications use the B+Tree.
SDA/BTREE/V2.0/7
B Tree
B Tree of degree m has the following properties:
All non-leaf nodes (except the root which is not bound
by a lower limit) have between ⎡m/2⎤ and m non-empty
children.
A non-leaf node that has n branches will contain n-1
keys.
All leaves are at the same level, that is the same depth
from the root.
1250 0625 1000 1277 1282 < >= 1291 1425 2000
B+Tree
B+Tree adalah variant dari B-tree:
semua key value disimpan dalam leaf.
disertakan suatu pointer tambahan untuk
menghubungkan setiap leaf node tersebut sebagai
suatu linear linked-list.
•Struktur ini memungkinkan akses sikuensial data
dalam B-tree tanpa harus turun-naik pada struktur hirarkisnya.
node internal digunakan sebagai ‘indeks’. Beberapa
SDA/BTREE/V2.0/9
B+Tree
1250 0625 1000 1425 2000 1250 1300 1425 1600 2000 0350 0625 1000 Leaf Nodes < >= SDA/BTREE/V2.0/10B+Tree Node Structure
P1 K1 P2 K2 P n-1 Kn-1 P n P1 K1 P2 K2 .. . . . P n-1 Kn-1 P n
. . . .
A high level node (internal node)
A leaf node (Every key value appears in a leaf node)
Pointer to subtree for keys>= K & < K Pointer to subtree for keys>= K 1 n-2 n-1 Pointer to subtree for keys>= K & < K1 2 Pointer to subtree for keys< K n-1 Pointer to record (block) with key K Pointer to record (block) with key K Pointer to leaf with smallest key greater thanK Pointer to
record (block)
SDA/BTREE/V2.0/11
Example of a B+Tree
1250 0625 1000 1425 2000 0350 0625 1300 1250 1300 1425 1600 2000 0350 0625 1000 1600 1425 2000 1000 1250 Leaf NodesActual Data Records
>= <
Queries on B+Trees
Find all records with a search-key value of k.
1.Start with the root node
1. Examine the node for the smallest search-key value
> k.
2. If such a value exists, assume it is Kj. Then follow Pi
to the child node
3. Otherwise k ≥ Km–1, where there are m pointers in
the node. Then follow Pm to the child node.
2.
If the node reached by following the pointer above is
not a leaf node, repeat the above procedure on the
node, and follow the corresponding pointer.
3.
Eventually reach a leaf node. If for some i, key K
i= k
follow pointer P
ito the desired record (or bucket).
SDA/BTREE/V2.0/13
Queries on B+Trees: Range Query
Find all records with a search-key value > k and < l
(range query).
Find all records with a search-key value of k.
while the next search-key value < l, follow the
corresponding pointer to the records.
•when the current search-key is the last search-key in
a node, follow the last pointer Pn to the next leaf
node.
SDA/BTREE/V2.0/14
Insertion on B+Trees
Find the leaf node in which the search-key value
would appear
If the search-key value is already there in the leaf
node (non-unique seach-key),
record is added to data file, and
if necessary search-key and the corresponding pointer
is inserted into the leaf node
SDA/BTREE/V2.0/15
Insertion on B+Trees
If the search-key value is not there, then add the record to the data file:
If there is room in the leaf node, insert (key-value, pointer) pair in
the leaf node (should be sorted)
Otherwise, split the node (along with the new (key-value, pointer)
entry) as shown in the next slides.
Splitting a node:
Take the new (search-key value, pointer) pairs (including the one
being inserted) in sorted order. Place the first ⎡n/2⎤ in the original node, and the rest in a new node.
When splitting a leaf, promote the middle/median key in the
parent of the node being split, but retain the copy in the leaf.
When splitting an internal node, promote the middle/median key
in the parent of the node being split, but DO NOT retain the copy in the leaf.
If the parent is full, split it and propagate the split further up.
67
Building a B+Tree
67, 123, 89, 18
, 34, 87, 99, 104, 36, 55, 78, 9
< data records leaf node root node 18 89 123The split at
leaf nodes
67 89 123
promote but retain a copy < root node 18 67 89 89 123 >= split why promote 89, not 67?
SDA/BTREE/V2.0/17
67, 123, 89, 18
,
34, 87
, 99, 104, 36, 55, 78, 9
89 123 < root node 18 34 67 89 >= 67 87 < root node 18 34 67 89 >= 89 123 split SDA/BTREE/V2.0/1867, 123, 89, 18
,
34, 87,
99, 104
, 36, 55, 78, 9
67 87 < root node 18 34 67 89 >= 89 99 123 67 87 < root node 18 34 67 89 104 89 99 104 123 splitSDA/BTREE/V2.0/19
67, 123, 89, 18
,
34, 87, 99, 104
,
36, 55,
78, 9
67 87 < 18 34 36 67 89 104 89 99 104 123 67 87 < 18 34 36 67 89 99 104 123 36 55 104 89 67 36 89 104The split at
non-leaf nodes
promote and do not
retain a copy
double node split
split
split
The splitting of nodes proceeds upwards till a node that is not full is found. In the worst case the root node may be split increasing the height of the tree by 1.
Observations about B+Trees
The B+Tree contains a relatively small number of
levels (logarithmic in the number of data), thus
searches can be conducted efficiently.
In processing a query, a path is traversed in the tree
from the root to some leaf node.
If there are K search-key values in the file, the path is
no longer than ⎡log
⎡m/2⎤(K)⎤, where b is index blocking
factor
SDA/BTREE/V2.0/21
Deletion on B+Trees
Remove (search-key value, pointer) from the leaf
node
If the node has too few entries due to the removal
(minimum requirement:
⎡m/2⎤ children), and the
entries in the node and a sibling
fit
into a single node,
then
Merge the two nodes into a single node
Delete the pair (K
i–1, P
i), where P
iis the pointer to the
deleted node, from its parent, recursively using the
above procedure.
SDA/BTREE/V2.0/22
Deletion on B+Trees
Otherwise, if the node has too few entries due to the
removal, and the entries in the node and a sibling
does 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.
The node deletions may cascade upwards till a node
which has
⎡n/2 ⎤ or more pointers is found.
SDA/BTREE/V2.0/23
Summary
B Tree is mostly used as an external data structure
for databases.
B Tree of degree m has the following properties:
All non-leaf nodes (except the root which is not bound
by a lower limit) have between ⎡m/2⎤ and m children.
A non-leaf node that has n branches will contain n-1
keys.
All leaves are at the same level, that is the same depth
from the root.
B+Tree is a variant from B Tree where all key values
are stored in leaves
Further Reading
applet simulasi B+Tree
http://www.cs.msstate.edu/~cs2314/global/BTreeAnimation/visual ization.html
SDA/BTREE/V2.0/25
