Chapter 18

Chapter 18

B-Trees

Lee, Hsiu-Hui

Ack: This presentation is based on the

lecture slides from Prof. Tsai, Shi-Chun as

• B-Tree 性質：

在 B-Tree 中若 node x 有 n[x] keys, 則 x 含有 n[x]+1 個 children.

M

D H

J K L B C F G

Q T X V W Y Z N P R S

• 典型之 B-Tree 運用情況：

資料量龐大無法全部存在 main memory

處理 B-Tree 之 algorithm 只將部分之資料 copy 至 main memory.

Memory Main Secondary Memory

(

disks )

(

RAM )

object some

pointer to

← a x

READ(x) -

DISK

of x fields the

modify and/or

access that

operations

Definition of B-trees

B-Tree T : is a rooted tree with the properties:

1

. Every node x has the following fields:

a. n[x]：# of keys currently stored in node x b.

c. leaf[x] is true if x is a leaf; false if x is an internal node.

2. Each internal node x contains n[x]+1 pointers C

[x] , C

[x] ,… , C

[x] to its children.

Leaf nodes have no children ( i.e. its C

[x] undefined).

3. If k

is any key stored in the subtree with root C

[x]

] [ ]

[ ]

[ x key x key

x key

≤

≤ L ≤

key1 key2

K₃ K₂

K₁

C₁[x] C₂[x] C₃[x x

] [ ]

[ ]

[ ≤ ≤ ≤ ≤ ≤ ≤

≤ key x k key x k key x k

k L

4. All leaves have the same depth, which is the tree’s height h.

5. t: minimum degree of the B-tree.

Every node (other than root) must have at least t-1 keys.

(have at least t children) Every node can contain at most 2t-1 keys.

(have at most 2t children)

A node is FULL, if it contains exactly 2t-1 keys

.

2-3-4 tree:

When t=2, every internal node has either 2, 3, or 4 children

Theorem 18.1

• The larger the value of t, the smaller the height of the B-tree.

If , then for any n -key B-tree T of height h and minimum degree ≥ 1

n

,

≥ 2 t

. log ²

+ 1

≤ _t ⁿ

h

Proof：

t-1 t-1

t-1 t-1 t-1 t-1

t t

t t t t

1

2

2t 2t²

− ∑ +

≥

=

h − i

t i

t n

1

2

1

) 1 ( 1

. 1 1 2

) 1 1 ( 2

1 ⎟⎟= −

⎠

⎞

⎜⎜

⎝

⎛

−

⋅ −

− +

=

^h t ^h

t t t

# of nodes

. log

. h

n t

t

h

≥

+ ≥

2 1

2

1

Basic operations on B-trees

• convention：

• Root of the B-tree is always in main memory.

• Any nodes that are passed as parameters must already have

had a DISK_READ operation performed on them.

• Operations：

• Searching a B-Tree.

• Creating an empty B-tree.

• Splitting a node in a B-tree.

• Inserting a key into a B-tree.

• Deleting a key from a B-tree.

B-Tree-Search(x,k)

Total CPU time

).

log (

)

( th O t n

O = _t

Ex. B-Tree-Search(x,R)

Creating an empty B-tree

) 1 ( O

Total CPU time

Splitting a node

Splitting a full node y ( have 2t-1 keys ) around its median key

into 2 nodes having (t-1) keys each.

full

B-Tree-Split-Child(x, i, y)

Insert a key in a B-Tree

• Splitting the root is the only way to increase the height of a B-tree.

• Example：Inserting keys into a B-Tree.

t=3

A C D E

(a) Initial tree

J K N O R S T U V Y Z G M P X

G M P T X

(c) Q inserted

A B C D E J K N O R S T U V Y Z G M P X

(b) B inserted

R S U V G M P T X

(d) L insert

A B C D E J K L N O Q R S U V Y Z P

G M T X

(e) F insert

P

C G M T X

• Deleting a key from a B-Tree：

1. K is in x and x is a leaf：

2. K is in x and x is an internal node：

a.

b.

( x has t keys ) ≥

K x

delete k from x.

K x

k’

≥ t keys

y

Recursively delete k’ and replace k by k’ in x.

K x

k’

≥ t keys

z

3. If K is not in internal node x：

a. If C_i[x] has only t-1 keys but has a sibling with t keys

b. If C_i[x] and all of C_i[x]’s siblings have t-1 keys, merge c_i with one sibling.

x

C_i[x]

k is in this subtree.

t-1

x

C_i[x] t

• Move a key from x down to C

[x].

• Move a key from C

[x]’s sibling to x.

• Move an appropriate child to C

[x] from its sibling

0 x

t-1

keys 0 2t-1

keys C_i[x] t-1

keys

• Example：Deleting a key from a B-Tree.

t=3

(a) Initial tree

(b) F delete：case 1

A B J K L N O Q R S U V Y Z

P

C G M T X

D E F

A B J K L N O Q R S U V Y Z

P

C G M T X

D E

(c) M delete：case 2a

P

(d) G deleted：case 2c

A B N O Q R S U V Y Z

P

C L T X

D E J H

(e) D deleted：case 3b

C L P T X

A B E J K N O Q R S U V Y Z

(e’) tree shrinks in height

C L P T X

A B E J K N O Q R S U V Y Z

(f) B delete：case 3a

E L P T X