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7. Indexes
Section 7 # 1
Heap files allow record retrieval:by specifying the Record IDentifier, RID, orby scanning all records sequentially.
Sometimes, retrieval of records by specifying the values in one or more fields is needed (semantic search or value-based querying), e.g.,
Find all students in the CS dept; orFind students with gpa > 3;
Indexes are auxilliary files (separate from the data files they index) that enable answering these value-based queries efficiently.
An index contains a “search key attribute”, k, and “data entries”, k*, which lead us to the records containing the search key value.
the k*s can be actual records, or pointers to the records or pointers to pointers to the records (indirect pointers).
In these notes we will always assume the second alternative (pointer to the records).
Index Classification• Primary Indexes vs. Secondary Indexes:
– If the search key is, or at least contains, the clustered primary key, then the index is called a primary index, else it is called a secondary index.
• Clustered Indexes vs. Unclustered Indexes: – If closeness of key values implies closeness of the data records containing
those key values, the index is called a clustered index, else it is an unclustered index. (recall, since physical disk storage is not completely linear, "close" means
• 1st: close on one disk track• 2nd: on differnt tracks but on the same cylinder • 3rd: on the different cylinders but on close cylinders.
– A file can be clustered on at most 1 attribute (search key) but that attribute may be composite (made up of multiple attributes).
– The cost of retrieving data records through an index varies greatly based on whether index is clustered or not!
Section 7 # 2
Index Classification continued
• If there is at least one index entry per existing attribute value, then it is called dense, else sparse or non-dense.
Ashby
Cass
Smith
Sparse Indexon
Name
Anchor records of each page
• Every sparse index must be clustered! Sparse indexes are smaller.
• We show a sparse index on the Name attribute.
• The key values (name in this example) that do occur in a sparse index are called ANCHOR values (always the first key value that occurs on a page).
• In this example the anchor names are Ashby, Cass and Smith. Basu, Bristow, Daniels, Jones and Tracy are non-anchor names.
Ashby, 25, 3000
Smith, 44, 3000
22
25
30
40
44
44
50
Data File
Dense Indexon
Age
33
Bristow, 30, 2007
Basu, 33, 4003
Cass, 50, 5004
Tracy, 44, 5004
Daniels, 22, 6003
Jones, 40, 6003
Name, age, bonus
Section 7 # 3
key
Primary Index
Example: Assume the blocking factor (bfr) is 2, which means there is room for 2 records per page. STUDENT|S#|SNAME |LCODE ||17|BAID |NY2091||25|CLAY |NJ5101||32|THAISZ|NJ5102||38|GOOD |FL6321||57|BROWN |NY2092| |83|THOM |ND3450|
Primary Index on S# |S#|pg |17| 1 |32| 2 |57| 3
PRIMARY INDEX: I(k, k*) k ordered or clustered "key" field values from an ordered or clustered field of file
with the uniqueness property (individual value occurrences are "unique" i.e., each value can occur at most once.)
k* = pointer to page (sematic pointer - namely page number) containing the first record with key value, k.
Section 7 # 4
page 1
page 2
page 3
Clustering Index
ENROLL2 |S#|C#|GRADE|17|6 | 96 ||25|6 | 76 | |32|6 | 62 | |38|6 | 98 | |32|6 | 91 | |25|7 | 68 | |32|8 | 89 | |17|9 | 95 |
|C#|pg| Dense Clustering_Index on C# |6 | 1| |7 | 3| |8 | 4| |9 | 4|
|C#|pg| Non-dense Clustering_Index on C# |6 | 1| (indexing new anchor records only) |8 | 4|
There's no more search overhead with this 2nd type of non-dense clustering index, but
How can you know which page has C#=7? (search pages sequentially starting at page=1)
How can you know which page has C#=9? (search
pages sequentially starting at page=4)
is like a primary index except that the attribute need not be a key - but the file must be clustered on the attribute, k - and the pointer for any k is the 1st page with that k-value
Section 7 # 5
page 1
page 2
page 3
page 4
Secondary Index
S#|C#|GRADE ENROLL (unclustered C#) 32|8 | 89 |25|6 | 76 |32|6 | 62 |25|8 | 86 |38|6 | 98 |32|7 | 91 |17|5 | 96 |25|7 | 68 |17|8 | 95 |
C#|pg Secondary_Index, Option1 on C# 5 | 46 | 16 | 26 | 37 | 37 | 48 | 18 | 28 | 4
Option2: Use repeating groups of pointers (requires a variable length page attribute)
|C#|page |5 | 4 |6 | 1,2,3 |7 | 3,4 |8 | 1,2,4
Option3: Use 1 index entry for each value, with1 pointer to a linked list of record pointers. (1 level of indirection)|S#|C#|GRADE ENROLL (unclustered C#)|32|8 | 89 ||25|6 | 76 ||32|6 | 62 ||25|8 | 86 ||38|6 | 98 ||32|7 | 91 ||17|5 | 96 ||25|7 | 68 ||17|8 | 95 |
|C#| page |5 | -->|4| |6 | -->|1|->|2|->|3| |7 | -->|3|->|4| |8 | -->|1|->|2|->|4|
These indexes are the same as Clustering Indexes except, - the file need not be clustered on k, - k* points to the page or record containing k - every record must be indexed (all secondary indexes are dense). Why?
Option1: If there are multiple occurences of k, use multiple index entries for that k.
Section 7 # 6
page 1
page 4
page 2
page 3
page 5
Multi-level Index (made up of an index on an index)
STUDENT|S#|SNAME |LCODE | |17|BAID |NY2091| |25|CLAY |NJ5101||32|THAISZ|NJ5102| |38|GOOD |FL6321| |57|BROWN |NY2092| |83|THOM |ND3450| |91|PARK |MN7334| |94|SIVA |OR1123|
|S#|pg| (of index file) S#-index (nondense, primary) |17| 1| |32| 2| |57| 3| |91| 4|
2nd_LEVEL (a second level, nondense index) |S#|pg| |17| 1| |57| 2|
For any index, since it is a file clustered on the key, k, it can have a primary or clustering index on it. (constituting the second level of the multilevel index).
Section 7 # 7
page 1 of STUDENT
page 4
page 1 of First Level Index
page 2
ISAM
This Index file may still be quite large, but we can apply the idea repeatedly!
K*0
K1 K*
1K 2 K*
2K m
K*m
index entry
Non-leaf (inode
Leaf
Leaf pages contain data entries, <k,k*>. Non-Leaf or Inodes contain k values only
Pages
Overflow page
Primary pages
1 index every record by <k, k*>.
Section 7 # 8
Tree-structured or Multilevel indexing techniques: ISAM: (a variation of multilevel Secondary indexing) static structure;B+ tree: dynamic structure which adjusts gracefully under insert and delete.
Example ISAM TreeBlocking factor is 2 (each node has 2 k
entries)In any internal node or inode (non-leaf) add a
ptr for key_values < first k-value
Section 7 # 9
10,10* 15,15* 20,20* 27,27* 33,33* 37,37* 40,40* 46,46* 51,51* 55,55* 63,63* 97,97*
20 33 51 63
40
Root
Insert k=23
OverflowPages
Leaf
IndexPages
Pages
Primary
23,23*48,48*
42,42*
10,10* 15,15* 20,20* 27,27* 33,33* 37,37* 40,40* 46,46* 51,51* 55,55* 63,63* 97,97*
20 33 51 63
40
Need overflow page
Insert k=48
41,41*
Need overflow page
Insert k=41Insert k=42
Need overflow page
Section 7 # 10
Deleting 42
Note that 51 appears in index levels, but not in leaf!
Deleting 51
Deleting 97
23,23*48,48*
42,42*
10,10* 15,15* 20,20* 27,27* 33,33* 37,37* 40,40* 46,46* 51,51* 55,55* 63,63* 97,97*
20 33 51 63
40
41,41*
Section 7 # 11
B+ Tree: The Most Widely Used Index
• keeps tree height-balanced. • Minimum 50% occupancy (except for root). Each
node contains m entries, where d m 2d.– d is called the degree or order of the index.
• Supports equality and range-searches efficiently.
Index Entries
Data Entries("Sequence set")
(“Direct search set or index set”)
Section 7 # 12
Example B+ Tree (d=2)
Search begins at root, key comparisons direct it to a leaf (similar to ISAM except use comparisons to keys
and take left pointer iff < lowest key).
Root
17 24 30
2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
13
Search for 5
Search for 15
Search for all data entries 24
5*
15 is not in the file!
Leaves are doubly linked forfast sequential <, , , > search
Section 7 # 13
Example B+ Tree (contd.)
Root
17 24 30
2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
13
• Search for all data entries < 23• (note, this is the reason for the double linkage).
Section 7 # 14
Inserting a Data Entry into a B+ Tree• Find correct leaf L. • Put data entry in L.
– If L has enough space, done!– Else, must split L (into L and a new node L2)
• Redistribute entries, copy up (promote) middle key.• middle value which was promoted and is now the anchor key for L2).
• This can happen recursively (e.g., if there is no space for the promoted middle value in the inode to which it is promoted)
– To split inode, redistribute entries evenly, but push up (promote) middle key.
• So promote means Copy up at leaf; Move up at inode.
• Splits “grow” tree• only a root split increases height.
– Only tree growth possible: wider or 1 level taller at top.
Section 7 # 15
Inserting 8*
Observe how minimum occupancy is guaranteed in both leaf and index pg splits.
• Note difference between copy-up (leaf) and move-up (inode)
5 is promoted to parent node.(Note that 5 iscontinues to appear in the new leaf node, L2, as its anchor value.)
s copied up and
appears once in the index. Contrast
Entry to be inserted in parent node.(Note that 17 is moved up and only
this with a leaf split.)
2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
No room for 8, so split.
5* 7* 8*2* 3*
17 24 3013 2* 3* 5* 7*24 305 13
17
No room for 5, so split and move 17 up.5
Section 7 # 16
B+ Tree Before Inserting 8*
Note height_increase, balance and occupancy maintenance.
Root
17 24 30
2* 3* 5* 7* 14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
13
2* 3*
Root
17
24 30
14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
135
7*5* 8*
After Inserting 8*
Section 7 # 17
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 a sibling.
• Merge could propagate to root, and therefore decreasing height.
Section 7 # 18
Example Tree After Inserting 8*
• Deleting 19* is easy.• Deleting 20* is done with re-distribution of 24* (and
revision of anchor value (from 24 to 27) in inode.
2* 3*
Root17
24 30
14* 16* 19* 20* 22* 24* 27* 29* 33* 34* 38* 39*
135
7*5* 8*
Root
2* 3*
17
30
14* 16* 33* 34* 38* 39*
135
7*5* 8* 22* 24*
27
27* 29*
Then Deleting 19*, 20*
Section 7 # 19
... And Then Deleting 24*
• Must merge.
2* 3*
17
30
14* 16* 33* 34* 38* 39*
135
7*5* 8* 22* 24*
27
27* 29*
• Observe `toss’ of index entry, 27, now that inode is below min occupancy so merge it with its sibling
2* 3* 7* 14* 16* 22* 27* 29* 33* 34* 38* 39*5* 8*
Root30135 17
• and index entry, 17 can be `pulled down’ (sibling merge, followed by pull-down)
2* 3*
17
30
14* 16* 33* 34* 38* 39*
135
7*5* 8* 22*
27
27* 29*
Section 7 # 20
Multidimensional IndexMultidimensional data almost always requires multidimensional indexing for effective access. One dimensional indexes assume a single search column, attribute (search key) which can be a
composite column or key.
Data structures, that support queries into multidimensional data specifically, fall in two categories:
1. Hash-table-like (e.g., Grid files and Partitioned Hash Functions)
2. Tree-like, eg,multi-key indexes, kd-trees, quad-trees (for sets of points); R-trees (for sets of regions as well as sets of points) ), Predicate-trees (P-trees) for vertical compressed, representations of data
Section 7 # 21
Hash-like Structures for Multidimensional e.g., Data Grid Files
Partition the POINTS in the space into a grid. In each dimension "grid lines" partition space into stripes. Points that fall directly on a grid line belong to the stripe above or to the right of it. Example: 12 customer(age,salary) 2-dimensional data records.(age,salary): (24,60) (46,60) (50,80) (50,100) (50,120) (70,100) (84,140) (30,260) (26,400) (44,360) (50,280) (60,260)
If vertical grid lines are drawn at age=40, age=65, horizontal at SAL=90K, SAL=224KThe points are hashed by ranges: 40 56400K380K360K340K320K300K280K260K240K220K200K180K160K140K120K100K 80K 60K 40K 20K 0K 0 10 20 30 40 50 60 70 80 90 100 AGE
* * *** *
*
*
**
* *
Grid hash functionage sal
pointsrange range 0-39 0-89K (24,60) 40-55 0-89K (46,60) (50,80) 40-55 90-223K (50,100)
(50,120) 56-99 90-223K (70,100) (84,140) 0-39 224-400K (30,260) (26,400) 40-55 224-400K (44,360) (50,280) 56-99 224-400K (60,260)
Inserting into Grid files: If there is room, insert, else (two methods) 1. add overflow block and chain it to the primary block, or 2. reorganize the structure by adding or moving grid lines (similar to dynamic hashing)
A problem with Grid files is that the number of buckets grows exponentially with dimension and the grid may become sparse. Section 7 # 22
is a sequence of hash functions, h=(h1,...hn) such that hi produces the ith segment of bits in the hash key, that is, h(a) is the concatenation of bit subsequences, h1(a)h2(a)..hn(a).
Example: The data file is CUSTOMER(AGE,SAL) consisting again of (24,60) (46,60) (50,80) (50,100) (50,120) (70,100) (84,140) (30,260) (26,400) (44,360) (50,280) (60,260)
Use 2 hash functions and 3 bits, the 1st bit is for age with hash function, mod2(tens_digit of age) and the last 2 bits are for salary with hash function, mod4(hundreds_digit of sal)
The lookup table is: Partitioined hash functionkey points
0 0 0 (24,060) (46,060) (26,400) 0 0 1 (84,140) 0 1 0 (60,260)0 1 1 (44,360) 1 0 0 (50,080)1 0 1 (50,100) (50,120) (70,100) 1 1 0 (30,260) (50,280) 1 1 1
Hash-like Structures for Multidimensional e.g., Partitioned hash Files
Section 7 # 23
Assume several attributes representing "dimensions" of the data points (data cube tuples) - uses a multi-level index, e.g., suppose there are 2 attributes: Provides a second level of Indexes on 2nd attribute to all tuples with same 1st attribute value
Tree-like Structures for Multidimensional e.g., Multi-key Index
/|--> / |-->
Index on .--> < |--> 1st attr / \ |..
/|/ \|--> / | / | /|--> / | / |--> --> < |----> < |--> \ | \ |.. \ |\ \|--> \ | \ \| \ /|--> \ \ / |--> \ `>< |--> \ \ |.. \ \|--> \ `-> . . . indexes on
2nd attr
Take the (age, salary) points again (24,60) (24,260) (24,400) (50,80) (50,100) (50,120) (50,280) (60,100) (60,260) (84,140)
. - - - - - - - - - - -> (24,060)
/ .- - - - - - -> (24,060)
/ / .- - - -> (24,400)
___/_________/______/ .--> |_60_|_260_|_400_____| / .- - - - - - -- - - - - -> (50,080)age / ____/________________24----' .-> |_80_|_100|_120_|_280_|- - - --> (50,280)50-------' \ `- - - - - - - --> (50,120)60. _______ `- - - - - - - - - - -> (50,100)84 `-->|100|260|- - - - - - - - - - - - - --> (60,260) \ `- - - - - - - - - - - - - - - - --> (60,100) \ _____ `- >|_140_|- - - - - - - - - - - - - - --> (84,140)
Section 7 # 24
Interior nodes have (Attribute, Value, LowPointer, HighPontr) - Value is a value which splits data points - The example below will show (a, V, down, up) with pointers going down for LowPointer and up for HighPointer. (goes up on greater or equal actually). - Attributes used for different levels are different and ROTATE among the dimensions (round robin). - The leaves are blocks of records (assume data blocks hold 2 records, i.e., the blocking factor, bfr, is 2). - to search: decide along the tree until you reach a leaf (going up on greater or equal) - to insert: decide along the tree until you reach the proper leaf if there is room there, insert; else split the block and divide its contents according to the appropriate attribute (next one in the rotation). Example: (insert into kd-tree in this order using age first then salart, sal): age,sal (50,80) (84,140) (30,260) (44,360) (50,120) (70,100) (24,60) (26,400) (50,280) (46,60) (60,260) (50,100)insert the first 2 pairs (no tree yet, since just 1 leaf block): 50, 80 84, 140
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index
age sal 30, 260 (leaf is full so split it and divide the contents by sal=150)
30,260 sal /
{ ,150} < \ 50,80 84,140 Section 7 # 25
30,260 sal /
{ ,150} < \ 50,80 84,140
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index continued
age sal 44,360 (leaf is not full so insert)
30,260 44,360
sal / { ,150} <
\ 50,80 84,140
age sal 50,120 (leaf is full so split, divide contents by age=55) 30,260
44,360 sal /
{ ,150} < \ 84,140 \age / { 55, } <
\ 50,80
50,120 Section 7 # 26
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index continued age sal
50,120 30,260 44,360
sal / { ,150} <
\ 84,140 \age / { 55, } <
\ 50,80
50,120
age sal 70,100 (leaf is not full so insert) 30,260
44,360 sal /
{ ,150} < 84,140\ 70,100 \age / { 55, } <
\ 50,80
50,120
age sal 24,060 (leaf is full so split, divide by sal=75) 30,260
44,360 sal /
{ ,150} < 84,140\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060
Section 7 # 27
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index continued 30,260
44,360 sal /
{ ,150} < 84,140\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060
age sal 26,400 (leaf full split, div by age=28)
30,260 44,360 age / { 28, }< / \ / 26,400 /
sal / { ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060
age sal 50,280 (leaf full split, div by sal=300)
44,360 sal /
(300, }< age / \ { 28, }< 30,260 / \ 50,280 / 26,400 /
sal / { ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060
Section 7 # 28
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index continued 44,360
sal / (300, }< age / \ { 28, }< 30,260 / \ 50,280 / 26,400 /
sal / { ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060
age sal 46,060 (leaf not full so insert)
44,360 sal /
(300, }< age / \ { 28, }< 30,260 / \ 50,280 / 26,400 /
sal / { ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060 46,060
Section 7 # 29
Tree-like Structures for Multidimensional e.g., k dimensional (kd tree) Index continued 44,360
sal / (300, }< age / \ { 28, }< 30,260 / \ 50,280 / 26,400 /
sal / { ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060 46,060
age sal 60,260 (leaf full split by age=40)
44,360 sal /
(300, }< age / \ 30,260 { 28, }< \ age / / \ { 40, }< / 26,400 \ / 50,280
sal / 60,260{ ,150} < 84,140
\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060 46,060
Section 7 # 30
Tree-like Structures for Multidime.g., k dim (kd tree) Index continued 44,360
sal / (300, }< age / \
30,260 { 28, }< \ age / / \ { 40, }< / 26,400 \ /
50,280sal / 60,260
{ ,150} < 84,140\ 70,100 \age / { 55, } <
\ 50,080 \ 50,120
\ sal / { ,75)<
\ 24,060 46,060
age sal 50,100 (full split age=50 full again split sal=90)
44,360 sal /
(300, }< age / \ 30,260 { 28, }< \ age / / \ { 40, }< / 26,400 \ / 50,280
sal / 60,260{ ,150} < 84,140 50,120
\ 70,100 50,100
\age / sal / { 55, } < { , 90 }<
\ age / \ \ { 50, }< 50,080
\ sal / \ { ,75)<
\ 24,060 46,060 Section 7 # 31
Tree-like Structures for Multidimensional datasets e.g., Quad tree indexes - Interior nodes (Inodes) correspond to rectangulars in 2-D (more generally, they can be constructed to represent
hypercubes higher dimensional space)
- If the number of points in the rectangle fits in a block, it's a leaf, else the rectangle is treated as interior node with children corresponding to its 4 quadrants.
- to insert into the quad treee index: search to find the proper leaf; if there's room, insert; else split node into 4 quadrants, divide contents appropriately.
Example: Build the Quad-tree index as it would develop, assuming (age,sal) arrive in this order: age,sal (24,60) (46,60) (50,80) (50,100) (50,120) (70,100) (84,140) (30,260) (26,400) (44,360) (50,280) (60,260)
Insert (24,60) (46,60)
400K380K360K340K320K300K280K260K240K220K200K180K160K140K120K100K 80K 60K 40K 20K 0K 0 10 20 30 40 50 60 70 80 90 100 AGE
* *
The only leaf node is:age sal 24,06046,060
Section 7 # 32
Tree-like Structures for Multidim datasets e.g., Quad tree indexes
400K380K360K340K320K300K280K260K240K220K200K180K160K140K120K100K 80K 60K 40K 20K 0K 0 10 20 30 40 50 60 70 80 90 100 AGE
* * *
24,06046,060
insertage sal 50, 080 (leaf full split (e.g., at age=50 and sal=200) divide contents by quadrant
.-NW /
/---NE age,sal /{50,200} <
\ \---SW 24,060 \ 46,060 \ `SE 50,080
Section 7 # 33
Tree-like Structures for Multidim datasets e.g., Quad tree indexes
400K380K360K340K320K300K280K260K240K220K200K180K160K140K120K100K 80K 60K 40K 20K 0K 0 10 20 30 40 50 60 70 80 90 100 AGE
* * *
insert 50, 100 (not full insert
.-NW /
/---NE age,sal /{50,200} <
\ \---SW 24,060 \ 46,060 \ `SE 50,080 50,100
.-NW /
/---NE age,sal /{50,200} <
\ \---SW 24,060 \ 46,060 \ `SE 50,080
insert 50, 120 full split SE at 75,100
.-NW /
/---NE age,sal /{50,200} <
\ \---SW 24,060 \ 46,060 \ `SE(75,100)<
.-NW 50,100 / 50,120 /---NE / < \ \---SW 50,080 \ \ `SE
**
ETC.
Section 7 # 34
Tree-like Structures for Multidim datasets: Region tree (Rtree) indexes - inodes of an R-tree correspond to interior regions, (which can be overlapping) (usually regions are
rectangles, tho, not necessarily)
- R-tree regions have subregions that represent the contents of their children
- And the subregions need not cover the region they subdivide (but all data must be within a subregion) Example, Consider the spatial image:
Example: Consider the spatial image:
100________________________________________________________ | | | | | | | | | .---------. | | | | | | | school | | | |_________| | | | | | |---------------------------. | | road1 | .-------. | |---------------------------| |house2 | | | |r | |_______| | | .------.________ |o_|_____________________________| | |house1|________ |a_|________pipeline_____________| | |______| |d | | | |2 | | | | | | | | | | 0 `-------------------------------------------------------' 0 100
Assume a leaf can hold 6 regions (bfr=6)
and that the 6 regions or objects above are together on 1 leaf block, whose region is shown as the outer red rectangle
Thus the R-tree has a root and 1 leaf:
( (0,0), (100,90) ) (corners of outer
red region)
road1 road2 house1 school house2 pipeline
(a full leaf with 6 objects)
Section 7 # 35
Rtree indexes cont.
100________________________________________________________ | | | | | | | | | .---------. | | | | | | | school | | | |_________| | | | | |
|---------------------------. | | road1 | .-------. | |---------------------------| |house2 | | | |r | |_______| | | .------.________ |o_|_____________________________| | |house1|________ |a_|________pipeline_____________| | |______| |d | | | |2 | | | | | | | | | | 0 `-------------------------------------------------------' 0 100
(0,0), (100,90)
road1 | road2 | house1
Now suppose a local cellular phone company adds a POP as shown.
POP
Split the full leaf putting 4 objects in 1 new leaf and 3 in the other
(minimize overlap and split ~evenly)
school | house2 | pipeline (0,0), (60,50) (20,20), (100,80)
POP
Section 7 # 36
Rtree indexes cont.
100________________________________________________________ | | | | | | | | | .---------. | | | | | | | school | | | |_________| | | | | |
|---------------------------. | | road1 | .-------. | |---------------------------| |house2 | | | |r | |_______| | | .------.________ |o_|_____________________________| | |house1|________ |a_|________pipeline_____________| | |______| |d | | | |2 | | | | | | | | | | 0 `-------------------------------------------------------' 0 100
(0,0), (100,90)
road1 | road2 | house1
POP
school | house2 | pipeline
(0,0), (60,50) (20,20), (100,80)
POP
Now suppose we insert house3
house3
Since house3 is not in either region (and both have room) we must decide to expand one of them.
If we pick the green, expanding it to (0,20), (100,80) we add 1600 units2
If we pick the purple, expanding it to ((0,0), (80,50) we add 1000 units2
so to minimize we pick the purple.
(80,50)
house2 house3
Note that house2 is in both regions.
Section 7 # 37
Similar to B-tree, except - only the common parts of key values are embedded in inodes - a single bit is used to make the navigation direction decision at each level (0 for up
and 1 for down). (zero-based bit positions are used) Example: (in this example, the tree structure is being built left-to-right) Starting with an empty structure,
Binary Radix Tree Index (AKA a trie) an additional index structure (e.g., used in IBM AS/400 systems)
INSERT JAY | LA | 25 | STAR (assigned RRN=1 to it)
INSERT JON | LA | 45 | HOOD (assigned RRN=2 1st letters are teh same (J) so the common pat is embedded in the root 2nd letters: A and O, bit 3 (zero-based count) is 1st difference (and makes the decision)
0123 4567 bit positionsDBCDIC for A=1100 0001EBCDIC for O=1101 0110
1 | JAY | LA | 25 | STAR
CUSTOMER FILERRN nam loc age job
2 | JON | LA | 45 | HOODJAY 1
nam_trie_INDEX
namepart RRN
b3 J<
ON 2
Section 7 # 38
Binary Radix Tree Index (AKA a trie) cont.
INSERT JAN | RO | 93 | DOC (assigned RRN=3
0123 4567 bit positionsDBCDIC for N=1101 0101EBCDIC for Y=1110 0000
1 | JAY | LA | 25 | STAR
CUSTOMER FILERRN nam loc age job
2 | JON | LA | 45 | HOOD
nam_trie_INDEX
JAY 1b3 J<
ON 2
N 3b2 A<
Y 1
b3
J<
ON 2
3 | JAN | RO | 93 | DOC
Section 7 # 39
Binary Radix Tree Index (AKA a trie) cont.
INSERT SUE | RO | 16 | PROG (assigned RRN=4
0123 4567 bit positionsDBCDIC for J= 1101 0001EBCDIC for Y= 1110 0010
1 | JAY | LA | 25 | STAR
CUSTOMER FILERRN nam loc age job
2 | JON | LA | 45 | HOOD
nam_trie_INDEX
N 3b2 A<
Y 1
b3
J<
ON 2
b2
<
SUE 2
3 | JAN | RO | 93 | DOC4 | SUE | RO | 16 | PROG
Section 7 # 40
Thank you.
Section 7