Indexes and B-Trees
Lecture 9
R & G Chapters 8 & 9
“If I had eight hours to chop down a
tree, I'd spend six sharpening my ax.”
Abraham Lincoln
Administrivia
• Homework 1 Due Tonight, 10pm
• Homework 2 Available Today, Due 3 weeks
from today
• Midterm Exam 1 will be a week from Thursday
– It will be in class, at the usual time
– Next Tuesday’s class will be a review
• More SQL Exercises on the Class Website
Review
• Last two weeks:
– Formal Query Languages: Rel. Algebra & Calculus
– Actual Query Language: SQL
• This week: Indexes
– Tree Indexes (HW2)
– Hash Indexes
• Next week:
– Review
– Midterm 1
Review: Division A/B
sno
s1
s1
s1
s1
s2
s2
s3
s4
s4
pno
p1
p2
p3
p4
p1
p2
p2
p2
p4
A
pno
p2
pno
p2
p4
B2
pno
p1
p2
p4
sno
s1
s2
s3
s4
sno
s1
s4
A/B1
A/B2
sno
s1
B1
B3
A/B3
Review: Use NOT Exists for Division
Recall: X/Y means only give me X tuples that have a match in Y.
Find sailors who’ve reserved all boats.
X = set of sailors and Y = set of all boats with reservations.
SELECT S.sname
Find Sailors S such that ...
FROM Sailors S
WHERE NOT EXISTS
there is no boat B...
(SELECT B.bid
FROM Boats B
WHERE NOT EXISTS
(SELECT R.bid
without a reservation by
FROM Reserves R
Sailor S
WHERE R.bid=B.bid
AND R.sid=S.sid))
Division
SELECT S.sname
FROM Sailors S
WHERE NOT EXISTS
(SELECT B.bid
FROM Boats B
WHERE NOT EXISTS
(SELECT R.bid
FROM Reserves R
101
103
WHERE R.bid=B.bid
1
3
AND R.sid=S.sid))
2
Reserves
Sailors
sid
sid
bid
day
R
1
103
9/12
R
R
R
R
2
103
9/13
3
103
9/14
3
101
9/12
1
103
9/13
Boats
sname
rating
age
S 1
Frodo
7
22
S 2
Bilbo
2
39
S 3
Sam
8
27
bid
bname
color
B
101
Nina
red
B
103
Pinta
blue
Null Values
• Values are sometimes
– unknown (e.g., a rating has not been assigned) or
– inapplicable (e.g., no spouse’s name).
– SQL provides a special value null for such situations.
• The presence of null complicates many issues. E.g.:
– Special operators needed to check if value is/is not null.
– “rating>8” - true or false when rating is null? What about
AND, OR and NOT connectives?
– Need a 3-valued logic (true, false and unknown).
– Meaning of constructs must be defined carefully. (e.g.,
WHERE clause eliminates rows that don’t evaluate to true.)
– New operators (in particular, outer joins) possible/needed.
Null Values – 3 Valued Logic
(null > 0)
is null
(null + 1)
is null
(null = 0)
is null
null AND true
is null
AND
T
F
Null
OR
T
F
Null
T
T
F
Null
T
T
T
T
F
F
F
F
F
T
F
Null
NULL
Null
F
Null
NULL
T
Null
Null
Null Values in SQL
• “Where” clause must evaluate to true
• “IS NULL” operator, e.g. “where name is null”
• “IS NOT NULL” operator
• Outer Joins: Left, Right, Full
SELECT s.sid, s.name, r.bid
FROM Sailors s LEFT OUTER JOIN Reserves r
ON s.sid = r.sid
sid
22
31
95
sname rating age
Dustin
7
45.0
Lubber 8
55.5
Bob
3
63.5
s.sid
22
95
31
sid bid
day
22 101 10/10/96
95 103 11/12/96
s.name r.bid
Dustin
101
Bob
103
Lubber
SELECT r.sid, b.bid, b.name
FROM Reserves r RIGHT OUTER JOIN Boats b
ON r.bid = b.bid
sid bid
day
22 101 10/10/96
95 103 11/12/96
r.sid
bid
101
102
103
104
b.bid
22
95
101
102
103
104
bname
Interlake
Interlake
Clipper
Marine
b.name
Interlake
Interlake
Clipper
Marine
color
blue
red
green
red
Review – Buffer Management and Files
• Storage of Data
– Fields, either fixed or variable length...
– Stored in Records...
– Stored in Pages...
– Stored in Files
• If data won’t fit in RAM, store on Disk
– Need Buffer Pool to hold pages in RAM
– Different strategies decide what to keep in pool
Today: File Organization
• How to keep pages of records on disk
• but must support operations:
– scan all records
– search for a record id “RID”
– search for record(s) with certain values
– insert new records
– delete old records
Alternative File Organizations
Many alternatives exist, tradeoffs for each:
– Heap files:
• Suitable when typical access is file scan of all records.
– Sorted Files:
• Best for retrieval in search key order
• Also good for search based on search key
–
Indexes: Organize records via trees or hashing.
•
•
Like sorted files, speed up searches for search key fields
Updates are much faster than in sorted files.
Indexes
• Often want to get records byvalues in one or more fields,
e.g.,
– Find all students in the “CS” department
– Find all students with a gpa > 3
• An index on a file is a:
– Disk-based data structure
– Speeds up selections on the search key fields for the index.
– Any subset of the fields of a relation can be index search key
– Search key is not the same as key
• (e.g. doesn’t have to be unique ID).
• An index
– Contains a collection of key/data entry pairs
– Supports efficient retrieval of all records with a given search key
value k.
Index Classification
• What selections does it support?
• What does index actually store?
– 3 alternatives:
• Data record with key value k
• <k, rid of data record>
• <k, list of rids of data records>
•
•
•
•
Clustered vs. Unclustered Indexes
Single Key vs. Composite Indexes
Tree-based, hash-based, other
Can have multiple (different) indexes per file.
– E.g. file sorted by age, with a hash index on salary
and a B+tree index on name.
First Question to Ask About an Index
• What kinds of selections does it support?
– Selections of form field <op> constant
– Equality selections (op is =)
– Range selections (op is one of <, >, <=, >=, BETWEEN)
– More exotic selections:
• 2-dimensional ranges (“east of Berkeley and west of Truckee
and North of Fresno and South of Eureka”)
– Or n-dimensional
• 2-dimensional distances (“within 2 miles of Soda Hall”)
– Or n-dimensional
• Ranking queries (“10 restaurants closest to VLSB”)
• Regular expression matches, genome string matches, etc.
• One common n-dimensional index: R-tree
What data is held by the index?
• Alternative 1:
Actual data record (with key value k)
– Index structure is file organization for data records
(like Heap files or sorted files).
– At most one index on a table can use Alternative 1.
– Saves pointer lookups
– Can be expensive to maintain with insertions and
deletions.
What data is held by the index? (Contd.)
Alternative 2
<k, rid>
and Alternative 3
<k, list of rids>
– Easier to maintain than Alt 1.
– At most one index can use Alternative 1; any others
must use Alternatives 2 or 3.
– Alternative 3 more compact than Alternative 2, but
leads to variable sized data entries even if search keys
are of fixed length.
– Even worse, for large rid lists the data entry might
have to span multiple pages!
Clustered and Unclustered
• Clustered vs. unclustered:
– If order of data records is the same as, or `close to’,
order of index data entries, then called clustered index.
• A file can be clustered on at most one search key.
• Cost to retrieve data records with index varies
greatly based on whether index clustered or not!
• Alternative 1 implies clustered, but not vice-versa.
Clustered vs. Unclustered Index
• Suppose that Alternative (2) is used for data entries,
and that the data records are stored in a Heap file.
– To build clustered index, first sort the Heap file (with some
free space on each block for future inserts).
– Overflow blocks may be needed for inserts. (Thus, order of
data recs is `close to’, but not identical to, the sort order.)
CLUSTERED
Index entries
direct search for
data entries
Data entries
UNCLUSTERED
Data entries
(Index File)
(Data file)
Data Records
Data Records
Unclustered vs. Clustered Indexes
• What are the tradeoffs????
• Clustered Pros
– Efficient for range searches
– May be able to do some types of compression
– Possible locality benefits (related data?)
• Clustered Cons
– Expensive to maintain (on the fly or sloppy with
reorganization)
Hash-Based Indexes
• Good for equality selections.
• Index is a collection of buckets. Bucket = primary
page plus zero or more overflow pages.
• Hashing function h:
– h(r) = bucket in which record r belongs.
– h looks at the search key fields of r.
• If Alternative (1) is used, the buckets contain the
data records; otherwise, they contain <key, rid>
or <key, rid-list> pairs.
B+ Tree Indexes
Non-leaf
Pages
Leaf
Pages
Leaf pages contain data entries, and are chained (prev & next)
 Non-leaf pages contain index entries and direct searches:

index entry
P0
K 1
P1
K 2
P 2
K m Pm
Comparing File Organizations
•
•
•
•
•
Heap files (random order; insert at eof)
Sorted files, sorted on <age, sal>
Clustered B+ tree file, Alternative (1), search
key <age, sal>
Heap file with unclustered B + tree index on
search key <age, sal>
Heap file with unclustered hash index on
search key <age, sal>
Operations to Compare
•
•
•
•
•
•
Scan: Fetch all records from disk
Fetch all records in sorted order
Equality search
Range selection
Insert a record
Delete a record
Cost Model for Analysis
I/O cost 150,000 times more than hash function
– We ignore CPU costs, for simplicity
B: The number of data pages
R: Number of records per page
F: Fanout of B-tree
Average-case analysis; based on several simplistic
assumptions.
 Good enough to show the overall trends!
Assumptions in Our Analysis
• Heap Files:
– Equality selection on key; exactly one match.
• Sorted Files:
– Files compacted after deletions.
• Indexes:
– Alt (2), (3): data entry size = 10% size of record
– Hash: No overflow buckets.
•
–
80% page occupancy => File size = 1.25 data size
Tree: 67% occupancy (this is typical).
•
Implies file size = 1.5 data size
I/O Cost of
Operations
Heap File
Scan all
records
Get all in
sort order
B: Number of data pages (packed)
R: Number of records per page
S: Time required for equality search
B
4B
Equality
Search
0.5 B
Range
Search
B
Insert
2
Delete
0.5B + 1
I/O Cost of
Operations
B: Number of data pages (packed)
R: Number of records per page
S: Time required for equality search
Sorted File
Scan all
records
B
Get all in
sort order
B
Equality
Search
log2 B
Range
Search
S + # matching
pages
Insert
S+B
Delete
S+B
I/O Cost of
Operations
B:
R:
F:
S:
Number of data pages (packed)
Number of records per page
Fanout of B-Tree
Time required for equality search
Clustered Tree
Scan all
records
1.5 B
Get all in
sort order
1.5 B
Equality
Search
logF (1.5 B)
Range
Search
S + #matching
pages
Insert
S+1
Delete
0.5B + 1
I/O Cost of
Operations
B:
R:
F:
S:
Number of data pages (packed)
Number of records per page
Fanout of B-Tree
Time required for equality search
Unclustered Tree
Scan all
records
B
(ignore index)
Get all in
sort order
4B
(ignore index)
Equality
Search
logF (.15 B) + 1
Range
Search
S + #matching
records
Insert
S+2
Delete
S+2
I/O Cost of
Operations
B: Number of data pages (packed)
R: Number of records per page
S: Time required for equality search
Hash Index
Scan all
records
B
(ignore index)
Get all in
sort order
4B
(ignore index)
Equality
Search
2
Range
Search
B
(ignore index)
Insert
4
Delete
S+2
B: The number of data pages
R: Number of records per page
F: Fanout of B-Tree
S: Time required for equality search
* Don’t Use Index
I/O Cost of
Operations
Heap File
Sorted File
Clustered Tree
Unclustered Tree
Hash Index
Scan all
records
B
B
1.5 B
B*
B*
Get all in
sort
order
4B
B
1.5 B
4B*
4B*
Equality
Search
0.5 B
log2 B
logF (1.5 B)
logF (.15 B)
+1
2
Range
Search
B
S+
S+
#matching #matching
pages
pages
S+
#matching
records
B*
Insert
2
S+B
S+1
S+2
4
Delete
0.5B + 1
S+B
0.5B + 1
S+2
S+2
Index Selection Guidelines
• Attributes in WHERE clause are candidates for index keys.
– Exact match condition suggests hash index.
– Range query suggests tree index.
• Clustering is especially useful for range queries; can also help on
equality queries if there are many duplicates.
• Multi-attribute search keys should be considered when a WHERE
clause contains several conditions.
– Order of attributes is important for range queries.
– Such indexes sometimes enable index-only strategies
•
For index-only strategies, clustering is not important!
• Choose indexes that benefit as many queries as possible.
• Since only one index can be clustered per table, choose it based
on important queries that would benefit the most from
clustering.
B+ Tree: The Most Widely Used Index
• Supports equality and range-searches efficiently.
• Insert/delete at log F N cost; keep tree heightbalanced. (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.
Index Entries
(Direct search)
Data Entries
("Sequence set")
Example B+ Tree
• Search begins at root, and key comparisons
direct it to a leaf (as in ISAM).
• Search for 5*, 15*, all data entries >= 24* ...
Root
13
2*
3*
5*
7*
14* 16*
17
24
19* 20* 22*
30
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:
– 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.
Example B+ Tree - Inserting 8*
Root
13
2*
3*
5*
7*
14* 16*
17
24
19* 20* 22*
30
24* 27* 29*
33* 34* 38* 39*
Example B+ Tree - Inserting 8*
Root
17
5
2* 3*
24
13
5* 7* 8*
14* 16*
19* 20* 22*
30
24* 27* 29*
33* 34* 38* 39*
 Notice that root was split, leading to increase in height.
 In this example, we can avoid split by re-distributing
entries; however, this is usually not done in practice.
Inserting 8* into Example B+ Tree
• Observe how
minimum
occupancy is
guaranteed in
both leaf and
index pg splits.
• Note difference
between copyup and push-up;
be sure you
understand the
reasons for this.
…
Entry to be inserted in parent node.
(Note that 5 is
s copied up and
continues to appear in the leaf.)
5
2*
3*
5*
17
5
13
24
7*
8*
…
Entry to be inserted in parent node.
(Note that 17 is pushed up and only
appears once in the index. Contrast
this with a leaf split.)
30
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.
B-Tree Demo
Example Tree (including 8*)
Delete 19* and 20* ...
Root
17
5
2* 3*
24
13
5* 7* 8*
14* 16*
• Deleting 19* is easy.
19* 20* 22*
30
24* 27* 29*
33* 34* 38* 39*
Example Tree (including 8*)
Delete 19* and 20* ...
Root
17
5
2* 3*
27
13
5* 7* 8*
14* 16*
22* 24*
30
27* 29*
33* 34* 38* 39*
• Deleting 19* is easy.
• Deleting 20* is done with re-distribution.
Notice how middle key is copied up.
... And Then Deleting 24*
• Must merge.
• Observe `toss’ of
index entry (on right),
and `pull down’ of
index entry (below).
30
22*
27*
29*
33*
34*
38*
39*
Root
5
2*
3*
5*
7*
8*
13
14* 16*
17
30
22* 27* 29*
33* 34* 38* 39*
Summary
• Alternative file organizations, tradeoffs for each
• If selection queries are frequent, sorting the file
or building an index is important.
– Hash-based indexes only good for equality search.
– Sorted files and tree-based indexes best for range
search; also good for equality search. (Files rarely
kept sorted in practice; B+ tree index is better.)
• Index is a collection of data entries plus a way to
quickly find entries with given key values.
Summary (Contd.)
• Data entries can be actual data records, <key,
rid> pairs, or <key, rid-list> pairs.
– Choice orthogonal to indexing technique used to
locate data entries with a given key value.
• Can have several indexes on a given file of data
records, each with a different search key.
• Indexes can be
– clustered, unclustered
– B-tree, hash table, etc.
Summary (Contd.)
• Understanding the nature of the workload for the
application, and the performance goals, is essential
to developing a good design.
– What are the important queries and updates? What
attributes/relations are involved?
• Indexes must be chosen to speed up important
queries (and perhaps some updates!).
– Index maintenance overhead on updates to key fields.
– Choose indexes that can help many queries, if possible.
– Build indexes to support index-only strategies.
– Clustering is an important decision; only one index on a
given relation can be clustered!
– Order of fields in composite index key can be important.