B-Trees, Part 2 Hash-Based Indexes RG Chapter 10 Lecture 10.
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Transcript of B-Trees, Part 2 Hash-Based Indexes RG Chapter 10 Lecture 10.
B-Trees, Part 2Hash-Based
Indexes R&G Chapter 10Lecture 10
Administrivia• The new Homework 3 now available
– Due 1 week from Sunday– Homework 4 available the week after
• Midterm exams available here
Review• Last time discussed File Organization
– Unordered heap files– Sorted Files– Clustered Trees– Unclustered Trees– Unclustered Hash Tables
• Indexes– B-Trees – dynamic, good for changing data,
range queries– Hash tables – fastest for equality queries,
useless for range queries
Review (2)
• For any index, 3 alternatives for data entries k*:– Data record with key value k– <k, rid of data record with search key value k>– <k, list of rids of data records with search key
k>– Choice orthogonal to the indexing technique
Today: • Indexes
– Composite Keys, Index-Only Plans
• B-Trees – details of insertion and deletion
• Hash Indexes– How to implement with changing data sets
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.
Indexes with Composite Search Keys
• Composite Search Keys: Search on a combination of fields.– Equality query: Every field value is
equal to a constant value. E.g. wrt <sal,age> index:
• age=20 and sal =75– Range query: Some field value is
not a constant. E.g.:• age =20; or age=20 and sal > 10
• Data entries in index sorted by search key to support range queries.– Lexicographic order, or– Spatial order.
sue 13 75
bobcaljoe 12
10
208011
12
name age sal
<sal, age>
<age, sal> <age>
<sal>
12,2012,10
11,80
13,75
20,12
10,12
75,1380,11
11121213
10207580
Data recordssorted by name
Data entries in indexsorted by <sal,age>
Data entriessorted by <sal>
Examples of composite keyindexes using lexicographic order.
Composite Search Keys• To retrieve Emp records with age=30 AND
sal=4000, an index on <age,sal> would be better than an index on age or an index on sal.– Choice of index key orthogonal to clustering etc.
• If condition is: 20<age<30 AND 3000<sal<5000: – Clustered tree index on <age,sal> or <sal,age> is
best.• If condition is: age=30 AND 3000<sal<5000:
– Clustered <age,sal> index much better than <sal,age> index!
• Composite indexes are larger, updated more often.
Index-Only Plans• A number of
queries can be answered without retrieving any tuples from one or more of the relations involved if a suitable index is available.
SELECT D.mgrFROM Dept D, Emp EWHERE D.dno=E.dno
SELECT D.mgr, E.eidFROM Dept D, Emp EWHERE D.dno=E.dno
SELECT E.dno, COUNT(*)FROM Emp EGROUP BY E.dno
SELECT E.dno, MIN(E.sal)FROM Emp EGROUP BY E.dno
SELECT AVG(E.sal)FROM Emp EWHERE E.age=25 AND E.sal BETWEEN 3000 AND 5000
<E.dno><E.dno,E.eid>
Tree index!
<E.dno>
<E.dno,E.sal>Tree index!
<E. age,E.sal> or<E.sal, E.age>
Tree!
Index-Only Plans (Contd.)• Index-only plans
are possible if the key is <dno,age> or we have a tree index with key <age,dno>– Which is better?– What if we
consider the second query?
SELECT E.dno, COUNT (*)FROM Emp EWHERE E.age=30GROUP BY E.dno
SELECT E.dno, COUNT (*)FROM Emp EWHERE E.age>30GROUP BY E.dno
B-Trees: Insertion and Deletion• Insertion
– Find leaf where new record belongs– If leaf is full, redistribute*– If siblings too full, split, copy middle key up– If index too full, redistribute*– If index siblings full, split, push middle key up
• Deletion– Find leaf where record exists, remove it– If leaf is less than 50% empty, redistribute*– If siblings too empty, merge, remove key above– If index node above too empty, redistribute*– If index siblings too empty, merge, move above key
down
B-Trees: For Homework 2• Insertion
– Find leaf where new record belongs– If leaf is full, redistribute*– If siblings too full, split, copy middle key up– If index too full, redistribute*– If index siblings full, split, push middle key up
• Deletion– Find leaf where record exists, remove it– If leaf is less than 50% empty, redistribute*– If siblings too empty, merge, remove key above– If index node above too empty, redistribute*– If index siblings too empty, merge, move above key
downThis means that after deletion, nodes will often be < 50% full
B-Trees: For Homework 2 (cont)• Splits
– When splitting nodes, choose the middle key– If there are even number of keys, choose
middle
• Your code must Handle Duplicate Keys– We promise that there will never be more
than 1 page of duplicate values.– Thus, when splitting, if the middle key is
identical to the key to the left, you must find the closest splittable key to the middle.
B-Tree Demo
Hashing• Static and dynamic hashing techniques
exist; trade-offs based on data change over time
• Static Hashing– Good if data never changes
• Extendable Hashing – Uses directory to handle changing data
• Linear Hashing– Avoids directory, usually faster
Static Hashing• # primary pages fixed, allocated
sequentially, never de-allocated; overflow pages if needed.
• h(k) mod N = bucket to which data entry with key k belongs. (N = # of buckets)
h(key) mod N
hkey
Primary bucket pages Overflow pages
20
N-1
Static Hashing (Contd.)• Buckets contain data entries.
• Hash fn works on search key field of record r. Must distribute values over range 0 ... N-1.– h(key) = (a * key + b) usually works well.– a and b are constants; lots known about how to
tune h.
• Long overflow chains can develop and degrade performance. – Extendible and Linear Hashing: Dynamic techniques
to fix this problem.
Extendible Hashing• Situation: Bucket (primary page) becomes full.
• Why not re-organize file by doubling # of buckets?– Reading and writing all pages is expensive!
• Idea: Use directory of pointers to buckets, – double # of buckets by doubling the directory, – splitting just the bucket that overflowed!– Directory much smaller than file, doubling much
cheaper. – No overflow pages!– Trick lies in how hash function is adjusted!
Extendible Hashing Details• Need directory with pointer to each bucket
• Need hash function to incrementally double range– can just use increasingly more LSBs of h(key)
• Must keep track of global “depth”– how many times directory doubled so far
• Must keep track of local “depth”– how many time each bucket has been split
Example• Directory is array of size 4.• To find bucket for r, take
last `global depth’ # bits of h(r); we denote r by h(r).– If h(r) = 5 = binary
101, it is in bucket pointed to by 01.
Insert: If bucket is full, split it (allocate new page, re-distribute). If necessary, double the directory. (As we will see, splitting a bucket does not always require doubling; we can tell by comparing global depth with local depth for the split bucket.)
13*00
01
10
11
2
2
2
2
2
LOCAL DEPTH
GLOBAL DEPTH
DIRECTORY
Bucket A
Bucket B
Bucket C
Bucket D
DATA PAGES
10*
1* 21*
4* 12* 32* 16*
15* 7* 19*
5*
Insert h(r)=20 (Causes Doubling)
00011011
2 2
2
2
LOCAL DEPTH 2
DIRECTORY
GLOBAL DEPTHBucket A
Bucket B
Bucket C
Bucket D
1* 5* 21*13*
32*16*
10*
15* 7* 19*
4* 12*
19*
2
2
2
000001010
011100101
110111
3
3
3DIRECTORY
Bucket A
Bucket B
Bucket C
Bucket D
Bucket A2(`split image'of Bucket A)
32*
1* 5* 21*13*
16*
10*
15* 7*
4* 20*12*
LOCAL DEPTH
GLOBAL DEPTH
Now Insert h(r)=9 (Causes Split Only)
19*
3
2
2
000001010
011100101
110111
3
3
3
DIRECTORY
Bucket A
Bucket B
Bucket C
Bucket D
Bucket A2
32*
1* 9*
16*
10*
15* 7*
4* 20*12*
LOCAL DEPTH
GLOBAL DEPTH
3
Bucket B25* 21* 13*
19*
2
2
2
000001010
011100101
110111
3
3
3DIRECTORY
Bucket A
Bucket B
Bucket C
Bucket D
Bucket A2(`split image'of Bucket A)
32*
1* 5* 21*13*
16*
10*
15* 7*
4* 20*12*
LOCAL DEPTH
GLOBAL DEPTH
Points to Note• 20 = binary 10100. Last 2 bits (00) tell us r belongs
in A or A2. Last 3 bits needed to tell which.– Global depth of directory: Max # of bits needed to tell
which bucket an entry belongs to.– Local depth of a bucket: # of bits used to determine if
splitting bucket will also double directory
• When does bucket split cause directory doubling?– If, before insert, local depth of bucket = global depth.
• Insert causes local depth to become > global depth; • directory is doubled by copying it over and `fixing’ pointer to
split image page. • (Use of least significant bits enables efficient doubling via
copying of directory!)
Directory DoublingWhy use least significant bits in directory?
Allows for doubling via copying!
13*00
01
10
11
2
2
2
2
2
10*
1* 21*
4* 12* 32* 16*
15* 7* 19*
5* 13*
000
001
010
011
32
2
2
2
10*
1* 21*
4* 12* 32* 16*
15* 7* 19*
5*
100
101
110
111
Deletion• Delete: If removal of data entry makes bucket
empty, can be merged with `split image’. If each directory element points to same bucket as its split image, can halve directory.
13*00
01
10
11
2
2
2
2
2
10*
1* 21*
4* 12* 32* 16*
15*
5* 13*00
01
10
11
2
1
2
2
1* 21*
4* 12* 32* 16*
15*
5*Delete 10
Deletion (cont)• Delete: If removal of data entry makes bucket
empty, can be merged with `split image’. If each directory element points to same bucket as its split image, can halve directory.
13*
00
01
10
11
2 1
1
1* 21*
4* 12* 32* 16*
5*
Delete 15
13*00
01
10
11
2
1
2
2
1* 21*
4* 12* 32* 16*
15*
5*
13*
0
1
1 1
1
1* 21*
4* 12* 32* 16*
5*
Comments on Extendible Hashing
• If directory fits in memory, equality search answered with one disk access; else two.– 100MB file, 100 bytes/rec, 4K pages contains
1,000,000 records (as data entries) and 25,000 directory elements; chances are high that directory will fit in memory.
– Directory grows in spurts, and, if the distribution of hash values is skewed, directory can grow large.
• Biggest problem:– Multiple entries with same hash value cause problems!– If bucket already full of same hash value, will keep
doubling forever! So must use overflow buckets if dups.
Linear Hashing• This is another dynamic hashing scheme, an
alternative to Extendible Hashing.
• LH handles the problem of long overflow chains without using a directory, and handles duplicates.
• Idea: Use a family of hash functions h0, h1, h2, ...– hi(key) = h(key) mod(2iN); N = initial # buckets– h is some hash function (range is not 0 to N-1)– If N = 2d0, for some d0, hi consists of applying h and
looking at the last di bits, where di = d0 + i.– hi+1 doubles the range of hi (similar to directory doubling)
• Let’s start with N = 4 Buckets– Start at “round” 0, “next” 0, have 2round buckets– Each time any bucket fills, split “next” bucket
– If (O ≤ hround(key) < Next), use hround+1(key) instead
Linear Hashing Example
13*
10*
1* 21*
4* 12* 32* 16*
15*
5*
Start
13*
10*
1* 21*
4* 12*
32* 16*
15*
5*
Add 9Nex
t9*Nex
t
13*
10*
1* 21*
4* 12*
32* 16*
15*
5*
Add 20
9*Next
20*
Nround Nround
Nround
• Overflow chains do exist, but eventually get split• Instead of doubling, new buckets added one-at-a-time
Linear Hashing Example (cont)
13*
10*
1* 21*
4* 12*
32* 16*
15*
5*
Add 6
9*Next
20*
6*
13*
10*
1*
21*
4* 12*
32* 16*
15*
5*
Add 17
9*
Next
20*
6*
Nround Nround
Linear Hashing (Contd.)• Directory avoided in LH by using overflow pages,
and choosing bucket to split round-robin.– Splitting proceeds in `rounds’. Round ends when all
NR initial (for round R) buckets are split. Buckets 0 to Next-1 have been split; Next to NR yet to be split.
– Current round number also called Level.
– Search: To find bucket for data entry r, find hround(r):• If hround(r) in range `Next to NR’ , r belongs here.• Else, r could be hround(r) or hround(r) + NR;
– must apply hround+1(r) to find out.
Overview of LH File • In the middle of a round.
Levelh
Buckets that existed at thebeginning of this round:
this is the range of
NextBucket to be split
of other buckets) in this round
Levelh search key value )(
search key value )(
Buckets split in this round:If is in this range, must useh Level+1
`split image' bucket.to decide if entry is in
created (through splitting`split image' buckets:
Linear Hashing (Contd.)• Insert: Find bucket by applying hround / hround+1:
– If bucket to insert into is full:• Add overflow page and insert data entry.• (Maybe) Split Next bucket and increment Next.
• Can choose any criterion to `trigger’ split.
• Since buckets are split round-robin, long overflow chains don’t develop!
• Doubling of directory in Extendible Hashing is similar; switching of hash functions is implicit in how the # of bits examined is increased.
Another Example of Linear Hashing
• On split, hLevel+1 is used to re-distribute entries.
0hh
1
(This infois for illustrationonly!)
Round=0, N=4
00
01
10
11
000
001
010
011
(The actual contentsof the linear hashedfile)
Next=0PRIMARY
PAGES
Data entry rwith h(r)=5
Primary bucket page
44* 36*32*
25*9* 5*
14* 18*10*30*
31*35* 11*7*
0hh
1
Round=0
00
01
10
11
000
001
010
011
Next=1
PRIMARYPAGES
44* 36*
32*
25*9* 5*
14* 18*10*30*
31*35* 11*7*
OVERFLOWPAGES
43*
00100
Example: End of a Round
0hh1
22*
00
01
10
11
000
001
010
011
00100
Next=3
01
10
101
110
Round=0PRIMARYPAGES
OVERFLOWPAGES
32*
9*
5*
14*
25*
66* 10*18* 34*
35*31* 7* 11* 43*
44*36*
37*29*
30*
0hh1
37*
00
01
10
11
000
001
010
011
00100
10
101
110
Next=0
Round=1
111
11
PRIMARYPAGES
OVERFLOWPAGES
11
32*
9* 25*
66* 18* 10* 34*
35* 11*
44* 36*
5* 29*
43*
14* 30* 22*
31*7*
50*
LH Described as a Variant of EH• The two schemes are actually quite similar:
– Begin with an EH index where directory has N elements.– Use overflow pages, split buckets round-robin.– First split is at bucket 0. (Imagine directory being
doubled at this point.) But elements <1,N+1>, <2,N+2>, ... are the same. So, need only create directory element N, which differs from 0, now.
• When bucket 1 splits, create directory element N+1, etc.• So, directory can double gradually. Also, primary
bucket pages are created in order. If they are allocated in sequence too (so that finding i’th is easy), we actually don’t need a directory! Voila, LH.
Summary• Hash-based indexes: best for equality searches,
cannot support range searches.
• Static Hashing can lead to long overflow chains.
• Extendible Hashing avoids overflow pages by splitting a full bucket when a new data entry is to be added to it. (Duplicates may require overflow pages.)– Directory to keep track of buckets, doubles periodically.– Can get large with skewed data; additional I/O if this
does not fit in main memory.
Summary (Contd.)• Linear Hashing avoids directory by splitting
buckets round-robin, and using overflow pages. – Overflow pages not likely to be long.– Duplicates handled easily.– Space utilization could be lower than Extendible
Hashing, since splits not concentrated on `dense’ data areas.• Can tune criterion for triggering splits to trade-off
slightly longer chains for better space utilization.
• For hash-based indexes, a skewed data distribution is one in which the hash values of data entries are not uniformly distributed!