???? ???? ??? ???? ? ??????? B

1
??? ???

• ???? ???? ??? ???? ? ??????? B

2
????? ???

• ????? ? ???? ??? B ???? ?? ????? ???? ?? ???? ??
  ?? ????? ??? ???.
• ????? ?? ???? ????????? ????? ?? ???? ??? ??
  ????? ??? ????? ???? ??????? ???? ?????? ??? ????
  AVL ???? ???? ??? (paged).
• ????? ???? ??? ??? ?????? ? ??? ???? ? ???????
  ???? ??? ?????.
• ??? ????? ??? ???? ?? ???? ???? ??? B ?????? ??
  ????? ????? ????? ?? ?? ?? ????? ??? ?????.
• ????? ??? ???? ???? ??? B
• ????? ???? BTreeNode? ????? ??? ??? ???? ??? B ??
  ?????.
• ????? ???? BTree? ????? ???? ???? ??? B ? ????
  ????? ?? ??.
• ????? ????? ???? ???? ?????? ???? ??? B
• ????? ??????? ???? ?? ? ???? ??? B ?????.
• ????? ???????? ??? ???? ???? B? ??? ?? ???? ??
  ???? ????? ???? B ? ???? B ?? ???????? ??? ?????
  ?? ??? ????? ??? ???.

3

• ???????? ?? ??????? ?????? ?????????? ??
• ???????? ?? ??????? ??? ?????????? ??? ????? ??
  ???? ? ?????, ???? ?? ? ???? ??? ???? ??

4
????

• ???? ???? ??? ????? ???? ?? ????? ????? ??? ???
  ?? ?????? ?? ????? ????? ??? ???

5
???? ??? ?? ?? ???? ????? ?? ???

• ????? ?? ??? ???? ???? ???? ???? ?????? ??????
  ????.
• ??? ? ??? ???? ?? ???? ????? ???? ????? ???.

6
?????? ???? ???

• ??????? ?? ???? ??? ?????? ???? ??

?? ???? ?????? ???? ?? ??? ?? ??? ?? ???? ????
????? ??? ?????? ?? ?? ???? ???? ???? ?? ?? ??? .
?? ??? ????? ?? ?? ?? ??????? ?? ???? ??? ?????
?? ???? ?? ?? ???? ?? ??? ???? . ? ???? ??? ????
?? ?? ???? ?? ??? ????? ????? ???? , ?? ?????
??? ??????? ?? ???? ????? ?? ????? ???? .
7
??? ??? ????? ?? ?? ???? ?????? ???? ?? ?? ????
?? ???.
8
?????? ???? ???

• ??????? ?? ???? ?????? ???

9
?? ???? ??????? AVL

• ?? ????? ???? ?????? ????? ???? ?? ?????? ?? ??
  ??? ???? ??? ???? ?? ????? ?????? ?? ?? ?????
  ????? ?? ????.
• ???? ????? ????? ??? ?? ???? AVL ????? ??? ?? ???
  ??? ?????? ???? ??? ???? ???.

10
??????? ???? B

• ?????? ????? ?? ????? ? ???????? ???? ???? ???
  ????? " ???? B ????? ?? "?? ????? ???? ??????
  ????? ???????? ? ???? ??? ???????? ????? ?? ?????
  ??? ? 1960 ???? ?? ???. ???? ???? ???? ???? ???
  ???? ????? ? ??????? ???? ?? ?? ???? ???? ???? ??
  ????? ?????? ???? ?? ????? ???? ?? ????? ???. ??
  ???? ????? ???? ? ?. ?? ????? ????? ?? ???? ????
  ?????? ??? ?? ?????. ?? ??? 1972 ?? ?? ????? ??
  ?? ??? " ?????? ??? ? ??????? ???? ??? ???? ??? ?
  ???? " ????? ????? ?? ???? B ?? ?? ???? ?????
  ???.

11
???? B

• ?? ??? ???? B ?? ????? ???? ??? .?? ???? ?? ???
  ????? ?? ????? ?????? ?? ?????? ???? ???? ?????
  ?? ????? ???? ??? ????.

12
???? ????

• ?? ??? ?? ??? ???? ?? ????? ????? ???
• ????? ????-???? ?? ???? ????? ??????? ?? ???.
• ????? ?????? ???? ?? ????? ???? ?? ?????? ?? ???.
• ?? ??? ?? ????? ?? ????? ????? ???? ?? ???
  ??????? ??? .
• ?????? ??? ????? ????

13
???? ?? ?? ?? B ?? ????? m

• ?? ???? ?? ???? m ????? ????.
• ?? ???? ??? ???? ?? ? ????? ????? m/2?????
  ????.
• ???? ?? ??? ?? ????? ????.
• ???? ????? ?? ?? ??? ???? ?????.
• ??? ?????, ?? ???? ???? ? ???? ??? ?? ???? ??? ??
  ??? ?? ???? ?? ????? ?? ???.

14
????? ?? ???? B
Note references to actual record only occur in
the leaf nodes. The interior nodes are only
higher level indexes (this is why there are
duplications in the tree)
15
????? ?? ???? B

• Problem 1 Look for L
• Problem 2 Look for S

16
??? ????? ?? ???? B

• ?? ???? ?????? ??? ?? ????.
• ?? ?? ?? ??? ??? ?? ???? ?? ???? ?? ?? ???? ???
  ?? ?????.

17
????? ??? ????

• ????? ?? ??? ??? ?? ??????? ?? ??? FINDLEAF ???
  ?? ?????.
• ??? ????? ????? ? ????? ???? ?? ???? ?? ?? ????.
• ????? ?? ???? ???? , ?? ????? ?? ???? ???? ?????
  ??? ???.

18
Example
a) Insert of C, S, D, T into the initial node.
b) Insertion of A
c) Insertions of M and P ?
19
c) Insertion of I
d) Insertions of B, W, N and G
e) Insertion of U?
20
Example
e) Insertion of U?
e) Insertion of R?
21
Example
e) Insertion of R?
22
Example
23
Example
24
????? ???? ???? ???? ?? Split
After inserting C, S, D, T
Inserting A
25
????? ???? ????? ?? ???? ??? ????? ??
Inserting R
26
Worst-Case Search Depth I

• Given 1,000,000 keys and a B-Tree of order 512,
  what is the maximum number of disk accesses
  necessary to locate a key in the tree? In other
  words, how deep will the tree be?
• Each key appears in the leaf gt What is the
  maximum height of a tree with 1,000,000 leaves?
• The maximum height will be reached if all pages
  (or nodes) in the tree has the minimum allowed
  number of descendents
• For a B-Tree of order m, the minimum number of
  descendents from the root page is 2. It is ?m/2?
  for all the other pages.

27
Worst-Case Search Depth II
Level the minimum number of descendants
1 2
2 2 ?m/2?
3 2 ?m/2? ?m/2?
4 2 ?m/2?3
.
d 2 ?m/2?d-1
28
Worst-Case Search Depth III

• For any level d of a B-Tree, the minimum number
  of descendants extending from that level is
  .
• 2 ?m/2? d-1
• For a tree with N keys in its leaves, we have
• N? 2 ?m/2? d-1
• ? d ? 1 log?m/2? (N/2)
• For m 512 and N 1,000,000, we thus get
  
• d ? 3.37

29
Deletion from a B-Tree Rules for Deleting a key
k from a node n-

• If n has more than the number of keys and the k
  is not the largest in n, simply delete k from n.
• If n has more than the minimum number of keys and
  the k is the largest in n, delete k and modify
  the higher level indexes to reflect the new
  largest key in n.

30
Deletion from a B-Tree Rules for Deleting a key
k from a node n

• If n has exactly the minimum number of keys and
  one of the siblings of n has few enough keys,
  merge n with its sibling and delete a key from
  the parent node.
• If n has exactly the minimum number of keys and
  one of the siblings of n has extra keys,
  redistribute by moving some keys from a sibling
  to n, and modify the higher level indexes to
  reflect the new largest keys in the affected
  nodes.

31
Deletion from a B-Tree Example

• Problem 1 Delete C
• Problem 2 Delete P
• Problem 3 Delete H

32
Redistribution during Insertion

• Redistribution during insertion is a way to
  avoid, or at least postpone, the creation of new
  pages.
• Redistribution allows us to place some of the
  overflowing keys into another page instead of
  splitting an overflowing page.
• B Trees formalize this idea

33
???? ??????? B

• ?? ???? ?? ???? m ????? ????.
• ?? ???? ??? ???? ?????(2m-1)/3 ????? ????.
• ???? ????? ?? ????? ????.
• ???? ????? ?? ?? ??? ???? ?????.

34
??? LRU

• ??? ??? ????? ?????? ??? ?? ?????? ?? ?????
  ?????? ?? ?? ???? ??? ?? ??????? ?????? ?????? ??
  ?? ???? ???? ?????? ????? ????.

35
??????? ???? ???? B

• ???? B ????? ???????? ??? ? ??? ???? ?????? ??
  ?????? ?????? ?? log K I ?? ???? ?? ?? ????? ??
  ???? I ?????? ? ???? ?? ???? ?? ??? ? K ?? ???
  ????? ?????? ?? ?????? ??? ?? ?????? ? ???? ??
  ???? ?? ???? ?? ???? ?? ??????? ? ???????? ?????
  ?? ???? ????? ????.

36
PAGE FAULT

• ?????? ??????? ?? ???? ???? ?????? ???? ?? ?? ??
  ???? ???? ?????.

37
?? ??? ???? ??? ???? ???? ????

• ?????? ?? ???? ?? ?? ???? ??????? ????? ???.
• ?? ???? ???? ?? ???? ???? ??? ??? ???? ?????
  ??????? ?? ??? ???.