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

Tree-Structured Indexes

Introduction
As for any index, 3 alternatives for data entries k*: – Data record with key value k


§ Choice is orthogonal to the indexing technique used to locate data entries k*.
§ Tree-structured indexing techniques support both range searches and equality searches.
§ ISAM: static structure; B+ tree: dynamic, adjusts gracefully under inserts and deletes.

Range Searches “Find all students with gpa > 3.0’’
– If data is in sorted file, do binary search to find first such student, then scan to find others.
– Cost of binary search can be quite high. Simple idea: Create an `index’ file.
Index File
k1 k2
kN
Page 1
Page 2
Page 3
Page N
☛ Can do binary search on (smaller) index file!
Data File

ISAM (Indexed Sequential Access Method)
index entry
P0
K1
P1
K2
P2
Km
Pm
Index file may still be quite large. But we can apply the idea repeatedly!
Non-leaf Pages
Leaf Pages
Overflow page
Primary pages
☛ Leaf pages contain data entries.

Data Pages
Index Pages
Overflow pages
Comments on ISAM
§ File creation: Leaf (data) pages allocated sequentially, sorted by each key; then index pages allocated, then space for overflow pages.
§ Index entries: ; they `direct’ search for data entries, which are in leaf pages.
§ Search: Start at root; use key comparisons to go to leaf. Cost=logFN;
§ 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

Example ISAM Tree
Each node can hold 2 entries; no need for `next-
leaf-page’pointers. (Why?) Root
40
51
63
20
33
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*

Inserting 23*
Root
40
Index Pages
20
33
51
63
Primary Leaf Pages
Overflow Pages
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*
23*

Inserting 48*
Root
40
Index Pages
20
33
51
63
Primary Leaf Pages
Overflow Pages
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*
48*
23*

Inserting 41*
Root
40
Index Pages
20
33
51
63
Primary Leaf Pages
Overflow Pages
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*
48*
41*
23*

Inserting 42*
Root
40
Index Pages
20
33
51
63
Primary Leaf Pages
Overflow Pages
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*
48*
41*
23*
42*

Then Deleting 42*, 51*, 97*,55*
Root
40
Index Pages
20
33
51
63
Primary Leaf Pages
Overflow Pages
10*
15*
20*
27*
33*
37*
40*
46*
51*
55*
63*
97*
48*
41*
23*
42*

… After Deleting 42*, 51*, 97*, 55*
Root
40
51
63
20
33
10*
15*
20*
27*
33*
37*
40*
46*
63*
48*
41*
23*
☛ Note that 51* appears in index levels, but not in leaf!

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 (Direct search) Data Entries ("Sequence set") Example B+ Tree Search begins at root, and key comparisons direct it to a leaf. Search for 5*, 15*, all data entries >= 24* …
Root
13
17
24
30
2*
3*
5*
7*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
☛ Based on the search for 15*, we know it is not in the tree!

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: – Level1= 1page = 8Kbytes
– Level2= 133pages= 1Mbyte
– Level 3 = 17,689 pages = 133 MBytes

Inserting a Data Entry into a B+ Tree § Find correct leaf L.
§ Put data entry onto L.
– IfLhasenoughspace,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.
– Treegrowth:getswideroroneleveltallerattop.

Inserting 8*
Root
13
17
24
30
2*
3*
5*
7*
14*
16*
19*
20*
22*
24*
27*
29*
33*
34*
38*
39*
Node 5 is copied up.
17 is pushed up
5
17
2*
3*
5*
7*
8*
5
13
24
30

After Inserting 8*
Root
17
5
13
24
30
2*
3*
5*
7*
8*
14*
16*
19* 20*
22*
24*
27*
29*
33*
34*
38*
39*
vIn 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.

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!
§ IfLhasonlyd-1entries,
§ 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*
Root
17
5
13
24
30
2*
3*
5*
7*
8*
14*
16*
19* 20*
22*
24*
27*
29*
33*
34*
38*
39*

Deleting 20*
Root
17
5
13
24
30
2*
3*
5*
7*
8*
14*
16*
20*
22*
24*
27*
29*
33*
34*
38*
39*

After Deleting 20* …
Root
17
5
13
27
30
2*
3*
5*
7*
8*
14*
16*
22* 24*
27*
29*
33*
34*
38*
39*
Deleting 20* is done with re-distribution. Notice how middle key is copied up.

Deleting 24* …
Root
17
5
13
27
30
2*
3*
5*
7*
8*
14*
16*
22* 24*
27*
29*
33*
34*
38*
39*
30
22*
27*
29*
33*
34*
38*
39*

Root
After Deleting 24*
5
13
17
30
2*
3*
5*
7*
8*
14*
16*
22*
27*
29*
33*
34*
38*
39*

Example of Non-leaf Re-distribution
Root
22
30
5
13
17
20
2*
3*
5*
7*
8*
14*
16*
17* 18*
20*
21*
22* 27* 29*
33* 34* 38*
39*
Root
17
20
22
30
5
13
3*
5*
7*
8*
14*
16*
17* 18*
20*
21*
22*
27* 29*
33*
34*
38*
39*

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
Davey Jones
David Smith
……

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.
Root
Sorted pages of data entries; not yet in B+ tree
6*
9*
10*
11*
12*
13*
20*
22*
23*
31*
35*
36*
38*
41*
44*
3*
4*

Bulk Loading (Contd.)
Root
10
20
• 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!
Data entry pages not yet in B+ tree
6
12
23
35
3*
4*
6*
9*
10* 11*
12* 13*
20*22*
23* 31*
35*36*
38*41*
44*
20
Root
10
Data entry pages not yet in B+ tree
6
38
12
23
35
3*
4*
6*
9*
10*
11*
12* 13*
23* 31*
20* 22*
35* 36*
38* 41*
44*

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’).
– Indexpagescantypicallyholdmanymoreentriesthan leaf pages.
– Variablesizedrecordsandsearchkeysmeandiffernt 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.
– Onlyleafpagesmodified;overflowpagesneeded.
– Overflowchainscandegradeperformanceunlesssize of data set and data distribution stay constant.
B+ tree is a dynamic structure.
– Inserts/deletes leave tree height-balanced; log F N cost. – Highfanout(F)meansdepthrarelymorethan3or4.
– Almostalwaysbetterthanmaintainingasortedfile.

Summary (Contd.) – Typically,67%occupancyonaverage.
– 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.