CS计算机代考程序代写 database chain concurrency Tree-Structured Indexes

§ Search: Start at root; use key comparisons to go to
leaf. Cost = log F N ;

§ F = # entries/index pg, N = # leaf pgs
§ Insert: Find leaf data entry belongs to, and put it

there.
§ Delete: Find and remove from leaf; if empty

overflow page, de-allocate.

☛ Static tree structure: inserts/deletes affect only leaf pages.

µ

Data Pages

Index Pages

Overflow pages

Example ISAM Tree
Each node can hold 2 entries; no need for `next-
leaf-page’ pointers. (Why?)

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40

Root

Inserting 23*

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40
Root

23*Overflow
Pages

Leaf

Index
Pages

Pages

Primary

Inserting 48*

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40
Root

23* 48*Overflow
Pages

Leaf

Index
Pages

Pages

Primary

Inserting 41*

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40
Root

23* 48*Overflow
Pages

Leaf

Index
Pages

Pages

Primary

41*

Inserting 42*

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40
Root

23* 48*Overflow
Pages

Leaf

Index
Pages

Pages

Primary

41*

42*

Then Deleting 42*, 51*, 97*,55*

10* 15* 20* 27* 33* 37* 40* 46* 51* 55* 63* 97*

20 33 51 63

40
Root

23* 48*Overflow
Pages

Leaf

Index
Pages

Pages

Primary

41*

42*

… After Deleting 42*, 51*, 97*, 55*

☛ Note that 51* appears in index levels, but not in leaf!

10* 15* 20* 27* 33* 37* 40* 46* 63*

20 33 51 63

40

Root

23* 48* 41*

B+ Tree: Most Widely Used Index
§ Insert/delete at log F N cost; keep tree height-

balanced. (F = fanout, N = # leaf pages)
§ Minimum 50% occupancy (except for root). Each

node contains d <= m <= 2d entries. § The parameter d is called the order of the tree. § Supports equality and range-searches efficiently. Index Entries Data Entries ("Sequence set") (Direct search) Example B+ Tree Search begins at root, and key comparisons direct it to a leaf. Search for 5*, 15*, all data entries >= 24* …

☛ Based on the search for 15*, we know it is not in the tree!

Root

17 24 30

2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

13

B+ Trees in Practice
Typical order: 100. Typical fill-factor: 67%.

– average fanout = 133

Typical capacities:
– Height 4: 1334 = 312,900,700 records
– Height 3: 1333 = 2,352,637 records

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

Inserting a Data Entry into a B+ Tree
§ 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, 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.

Inserting 8*
Root

17 24 30

2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

13

Node 5 is
copied up.

2* 3* 5* 7* 8*

5 17 is pushed up

5 24 30

17

13

After Inserting 8*

v In this example, we can avoid split by re-distributing
entries; however, this is usually not done in practice.

§ Redistributing I/O costs is not smaller than those
of splitting.

§ It has a chance that redistributing does not work;
thus costs for exploring redistribution are wasted.

2* 3*

Root
17

24 30

14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

135

7*5* 8*

Deleting a Data Entry from a B+ Tree

§ 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.

Deleting 19*

2* 3*

Root
17

24 30

14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*

135

7*5* 8*

Deleting 20*

2* 3*

Root
17

24 30

14* 16* 20* 22* 24* 27* 29* 33* 34* 38* 39*

135

7*5* 8*

After Deleting 20* …

Deleting 20* is done with re-distribution.
Notice how middle key is copied up.

2* 3*

Root

17

30

14* 16* 33* 34* 38* 39*

135

7*5* 8* 22* 24*

27

27* 29*

Deleting 24* …

2* 3*

Root

17

30

14* 16* 33* 34* 38* 39*

135

7*5* 8* 22* 24*

27

27* 29*

30

22* 27* 29* 33* 34* 38* 39*

After Deleting 24*

2* 3* 7* 14* 16* 22* 27* 29* 33* 34* 38* 39*5* 8*

Root
30135 17

Example of Non-leaf Re-distribution
Root

135 17 20

22

30

14* 16* 17* 18* 20* 33* 34* 38* 39*22* 27* 29*21*7*5* 8*3*2*

14* 16* 33* 34* 38* 39*22* 27* 29*17* 18* 20* 21*7*5* 8*3*

Root

135

17

3020 22

Prefix Key Compression
§ Important to increase fan-out. (Why?)
§ Key values in index entries only `direct traffic’; can often

compress them.
§ E.g., If we have adjacent index entries with search key values

Dannon Yogurt, David Smith and Devarakonda Murthy, we can
abbreviate David Smith to Dav. (The other keys can be
compressed too …)
§ Is this correct? Not quite! What if there is a data entry Davey Jones?

(Can only compress David Smith to Davi)
§ In general, while compressing, must leave each index entry greater

than every key value (in any subtree) to its left.
§ Insert/delete must be suitably modified.

Dannon Yogurt David Smith

Davey Jones ……

Bulk Loading of a B+ Tree
§ If we have a large collection of records, and we want

to create a B+ tree on some field, doing so by
repeatedly inserting records is very slow.

§ Bulk Loading can be done much more efficiently.
§ Initialization: Sort all data entries, insert pointer to

first (leaf) page in a new (root) page.

3* 4* 6* 9* 10* 11* 12* 13* 20* 22* 23* 31* 35* 36* 38* 41* 44*

Sorted pages of data entries; not yet in B+ tree
Root

Bulk Loading (Contd.)

• Index entries for leaf
pages always entered
into right-most index
page just above leaf
level. When this fills
up, it splits. (Split may
go up right-most path
to the root.)

• Much faster than
repeated inserts,
especially when one
considers locking!

3* 4* 6* 9* 10*11* 12*13* 20*22* 23* 31* 35*36* 38*41* 44*

Root

Data entry pages
not yet in B+ tree3523126

10 20

3* 4* 6* 9* 10* 11* 12*13* 20*22* 23* 31* 35*36* 38*41* 44*

6

Root

10

12 23

20

35

38

not yet in B+ tree
Data entry pages

Summary of Bulk Loading
Option 1: multiple inserts.

– Slow.
– Does not give sequential storage of leaves.

Option 2: Bulk Loading
– Has advantages for concurrency control.
– Fewer I/Os during build.
– Leaves will be stored sequentially (and linked, of

course).
– Can control “fill factor” on pages.

A Note on `Order’

Order (d) concept replaced by physical space
criterion in practice (`at least half-full’).

– Index pages can typically hold many more entries than
leaf pages.

– Variable sized records and search keys mean differnt
nodes will contain different numbers of entries.

– Even with fixed length fields, multiple records with the
same search key value (duplicates) can lead to
variable-sized data entries (if we use Alternative (3)).

Summary

Tree-structured indexes are ideal for range-searches,
also good for equality searches.
ISAM is a static structure.

– Only leaf pages modified; overflow pages needed.
– Overflow chains can degrade performance unless size

of data set and data distribution stay constant.
B+ tree is a dynamic structure.

– Inserts/deletes leave tree height-balanced; log F N cost.
– High fanout (F) means depth rarely more than 3 or 4.
– Almost always better than maintaining a sorted file.

Summary (Contd.)

– Typically, 67% occupancy on average.
– Usually preferable to ISAM, modulo locking

considerations; adjusts to growth gracefully.
– If data entries are data records, splits can change rids!

§ Key compression increases fanout, reduces height.
§ Bulk loading can be much faster than repeated

inserts for creating a B+ tree on a large data set.
§ Most widely used index in database management

systems because of its versatility. One of the most
optimized components of a DBMS.