Index and Hashing manual and presentation foe students

7/27/2019 Index and Hashing manual and presentation foe students

1/96

Silberschatz, Korth and Sudarshan12. 1 Database System Concepts

Chapter 12: Indexing and HashingBasic Concepts

Ordered IndicesB+-Tree Index Files

B-Tree Index Files

Static Hashing

Dynamic HashingComparison of Ordered Indexing and Hashing

Index Definition in SQL

Multiple-Key Access


2/96


Data Dictionary Storage

Information about relationsnames of relationsnames and types of attributes of each relationnames and definitions of viewsintegrity constraints

User and accounting information, including passwordsStatistical and descriptive data

number of tuples in each relation

Physical file organization informationHow relation is stored (sequential/hash/) Physical location of relation

Information about indices (Chapter 12)

Data dictionary (also called system catalog ) stores metadata :

that is, data about data, such as


3/96


Data Dictionary Storage (Cont.)Catalog structure

Relational representation on diskspecialized data structures designed for efficient access, in memory

A possible catalog representation:

Relation_metadata = (relation_name, number_of_attributes,storage_organization, location )

Attribute_metadata = (attribute_name, relation_name, domain_type, position, length )

User_metadata = (user_name, encrypted_password, group )

Index_metadata = (index_name, relation_name, index_type,index_attributes )View_metadata = (view_name, definition )


4/96


Indexing: Basic ConceptsIndexing mechanisms are used to speed up access to desireddata.

E.g., author catalog in library

Search Key - attribute to set of attributes used to look uprecords in a file.

An index file consists of records (called index entries ) of the

form

Index files are typically much smaller than the original file

Two basic kinds of indices:Ordered indices: search keys are stored in sorted order

Hash indices: search keys are distributed uniformly acrossbuckets using a hash function .

search-key pointer


5/96


Basic concepts (Cont.)Index definition: Index is a file whose records (entries) have the

following structure:(search-key, pointer or set of pointers to data records ).

When the search key is unique the index has fixed size records,otherwise it may have variable size records.

What is the pointer?Block number or RID!


6/96


Index Evaluation MetricsAccess types supported efficiently. E.g.,

records with a specified value in the attributeor records with an attribute value falling in a specified range ofvalues.

Access time

Insertion timeDeletion time

Space overhead


7/96 Silberschatz, Korth and Sudarshan12. 7 Database System Concepts

Ordered Indices

In an ordered index , index entries are stored sorted on thesearch key value. E.g., author catalog in library.

Primary index : in a sequentially ordered file, the index whosesearch key specifies the sequential order of the file.

Also called clustering index

The search key of a primary index is usually but not necessarily theprimary key.

Secondary index : an index whose search key specifies an orderdifferent from the sequential order of the file. Also callednon-clustering index .

Index-sequential file : ordered sequential file with a primary index.

Indexing techniques evaluated on basis of:



Classes of Indexes

Unique/non-unique whether search key is unique/non-uniqueDense/Nondense every/not every search key (every record for Uniquekey) in the data file has a corresponding pointer in the index.

Clustered/Nonclustered order of index search key is equal/not equal tofile order.

Primary/Secondary search key equal/not equal to primary key. Also,the answer is a single/set of data file records.

Note difference in definition of Primary index (us vs. Silberschatz)

Note: for Unique key Dense index indexes every record, Non-densedoes not index every record

Note: Non-dense index must be clustered!



Dense Clustered Index FileDense index Index record appears for every search-key value

in the file. (note, for every search key vs. for every record)



Sparse Index FilesSparse Index : contains index records for only some search-key

values.Applicable when records are sequentially ordered on search-key

To locate a record with search-key value K we:Find index record with largest search-key value < K

Search file sequentially starting at the record to which the indexrecord points

Less space and less maintenance overhead for insertions anddeletions.

Generally slower than dense index for locating records.

Good tradeoff: sparse index with an index entry for every block infile, corresponding to least search-key value in the block.

Basis of the ISAM index



Example of Sparse Clustered Index File



Multilevel IndexIf primary index does not fit in memory, access becomes

expensive.To reduce number of disk accesses to index records, treatprimary index kept on disk as a sequential file and construct asparse index on it.

outer index a sparse index of primary index

inner index the primary index file

If even outer index is too large to fit in main memory, yet anotherlevel of index can be created, and so on.

Indices at all levels must be updated on insertion or deletion from

the file.



Multilevel Index (Cont.)



A two level primary Non-dense clustered index resemblingIsam



Index Update: DeletionIf deleted record was the only record in the file with its particular

search-key value, the search-key is deleted from the index also.Single-level index deletion:

Dense indices deletion of search-key is similar to file recorddeletion.

Sparse indices if an entry for the search key exists in the index, itis deleted by replacing the entry in the index with the next search-key value in the file (in search-key order). If the next search-keyvalue already has an index entry, the entry is deleted instead ofbeing replaced.



Index Update: InsertionSingle-level index insertion:

Perform a lookup using the search-key value appearing in therecord to be inserted.

Dense indices if the search-key value does not appear in theindex, insert it.

Sparse indices if index stores an entry for each block of the file, no

change needs to be made to the index unless a new block iscreated. In this case, the first search-key value appearing in thenew block is inserted into the index.

Multilevel insertion (as well as deletion) algorithms are simpleextensions of the single-level algorithms



Secondary Indices

Frequently, one wants to find all the records whosevalues in a certain field (which is not the search-key ofthe primary index satisfy some condition.

Example 1: In the account database stored sequentiallyby account number, we may want to find all accounts in aparticular branch

Example 2: as above, but where we want to find allaccounts with a specified balance or range of balances

We can have a secondary index with an index recordfor each search-key value; index record points to abucket that contains pointers to all the actual recordswith that particular search-key value.



Primary and Secondary IndicesSecondary indices have to be dense.

Indices offer substantial benefits when searching for records.When a file is modified, every index on the file must be updated,Updating indices imposes overhead on database modification.

Sequential scan using primary index is efficient, but a sequentialscan using a secondary index is expensive

each record access may fetch a new block from disk since index isusually un-clustered

D U l d S d I d



Dense Unclustered Secondary Indexon balance field of account- Access List



B+-Tree Index Files

All paths from root to leaf are of the same length

Each node that is not a root or a leaf has between [ n /2] andn children.

A leaf node has between [( n 1)/2] and n 1 values

Special cases:If the root is not a leaf, it has at least 2 children.

If the root is a leaf (that is, there are no other nodes in thetree), it can have between 0 and ( n 1) values.

A B+-tree is a rooted tree satisfying the following properties:



B+-Tree Node StructureTypical node

Ki are the search-key values

P i are pointers to children (for non-leaf nodes) or pointers to records

or buckets of records (for leaf nodes).The search-keys in a node are ordered

K 1 < K 2 < K 3 < . . . < K n 1



Leaf Nodes in B +-Trees

For i = 1, 2, . . ., n 1, pointer P i either points to a file record withsearch-key value K i , or to a bucket of pointers to file records,each record having search-key value K i . Only need bucketstructure if search-key does not form a primary key.

If Li , L j are leaf nodes and i < j, L i s search -key values are less

than L j s search -key valuesP n points to next leaf node in search-key order

Properties of a leaf node:



Non-Leaf Nodes in B +-TreesNon leaf nodes form a multi-level sparse index on the leaf nodes.

For a non-leaf node with m pointers:All the search-keys in the subtree to which P 1 points are less thanK 1

For 2 i n 1, all the search-keys in the subtree to which P i pointshave values greater than or equal to K i 1 and less than K i

P n points to the subtree with keys greater than K n 1 and less than thefirst key in next leaf


24/96



25/96


-Key)

B+-tree for account file (n = 3)

Note, index points to FIRST key!


26/96


Example of B +-tree

Leaf nodes must have between 2 and 4 values( (n 1)/2 and n 1, with n = 5).

Non-leaf nodes other than root must have between 3and 5 children ( (n /2 and n with n =5).

Root must have at least 2 children.

B+-tree for account file (n= 5)


27/96


Observations about B +-treesSince the inter- node connections are done by pointers, logically

close blocks need not be physically close. The non-leaf levels of the B +-tree form a hierarchy of sparseindices.

The B +-tree contains a relatively small number of levels(logarithmic in the size of the main file), thus searches can beconducted efficiently.Insertions and deletions to the main file can be handledefficiently, as the index can be restructured in logarithmic time(as we shall see).

Differences between B+ and B TreeDifferences between B-tree Index and B-tree file


28/96


Queries on B +-TreesFind all records with a search-key value of k.1. Start with the root node

1. Examine the node for the smallest search-key value > k.

2. If such a value exists, assume it is K j . Then follow P i to thechild node

3. Otherwise k K m 1, where there are m pointers in the

node. Then follow P m to the child node.2. If the node reached by following the pointer above is not a leaf

node, repeat the above procedure on the node, and follow thecorresponding pointer.

3. Eventually reach a leaf node. If for some i , key K i = k followpointer P

i to the desired record or bucket. Else no record with

search-key value k exists.


29/96


Queries on B +-Trees (Cont.)In processing a query, a path is traversed in the tree fromthe root to some leaf node.

If there are K search-key values in the file, the path is nolonger than log n /2 (K ) .

A node is generally the same size as a disk block,typically 4 kilobytes, and n is typically around 100 (40

bytes per index entry).With 1 million search key values and n = 100, at mostlog 50(1,000,000) = 4 nodes are accessed in a lookup.

Contrast this with a balanced binary free with 1 millionsearch key values around 20 nodes are accessed in alookup

above difference is significant since every node accessmay need a disk I/O, costing around 20 milliseconds!


30/96


Updates on B +-Trees: InsertionFind the leaf node in which the search-key value would appear

If the search-key value is already there in the leaf node, record isadded to file and if necessary a pointer is inserted into thebucket.

If the search-key value is not there, then add the record to themain file and create a bucket if necessary. Then:

If there is room in the leaf node, insert (key-value, pointer) pair in theleaf node

Otherwise, split the node (along with the new (key-value, pointer)entry) as discussed in the next slide.


31/96


Updates on B +-Trees: Insertion (Cont.)Splitting a node:

take the n(search-key value, pointer) pairs (including the one beinginserted) in sorted order. Place the first n /2 in the original node,and the rest in a new node.

let the new node be p, and let k be the least key value in p. Insert(k,p ) in the parent of the node being split. If the parent is full, split itand propagate the split further up.

The splitting of nodes proceeds upwards till a node that is not fullis found. In the worst case the root node may be split increasingthe height of the tree by 1.

Result of splitting node containing Brighton and Downtown oninserting Clearview

U d B + T I i (C )


32/96


Updates on B +-Trees: Insertion (Cont.)

B+-Tree before and after insertion of Clearview

-i AS k )


33/96


points to LAST key)


34/96


Updates on B +-Trees: DeletionFind the record to be deleted, and remove it from themain file and from the bucket (if present)Remove (search-key value, pointer) from the leaf nodeif there is no bucket or if the bucket has become emptyIf the node has too few entries due to the removal, andthe entries in the node and a sibling fit into a singlenode, then

Insert all the search-key values in the two nodes into asingle node (the one on the left), and delete the othernode.Delete the pair ( K i 1, P i ), where P i is the pointer to thedeleted node, from its parent, recursively using the

above procedure.


35/96


Updates on B +-Trees: DeletionOtherwise, if the node has too few entries due to the removal,

and the entries in the node and a sibling do not fit into a singlenode, then Redistribute the pointers between the node and a sibling such that

both have more than the minimum number of entries.

Update the corresponding search-key value in the parent of the

node.The node deletions may cascade upwards till a node which hasn/2 or more pointers is found. If the root node has only one

pointer after deletion, it is deleted and the sole child becomes theroot.

E l f B + T D l ti


36/96


Examples of B +-Tree Deletion

The removal of the leaf node containing Downtown did notresult in its parent having too little pointers. So the cascadeddeletions stopped with the deleted leaf nodes parent.

Before and after deleting Downtown


37/96


Examples of B +-Tree Deletion (Cont.)

Node with Perryridge becomes underfull (actually empty, in this special case)and merged with its sibling.

As a result Perryridge nodes parent became underfull, and was merged with itssibling (and an entry was deleted from their parent)Root node then had only one child, and was deleted and its child became the new

root node

Deletion of Perryridge from result of previous example


38/96


Example of B +-tree Deletion (Cont.)

Parent of leaf containing Perryridge became underfull, and borrowed apointer from its left sibling

Search- key value in the parents parent changes as a result

Before and after deletion of Perryridge from earlier example


39/96


40/96


41/96


B+-Tree File Organization

Index file degradation problem is solved by using B +-Treeindices. Data file degradation problem is solved by using B +-Tree File Organization.

The leaf nodes in a B +-tree file organization store records,instead of pointers.

Since records are larger than pointers, the maximum number ofrecords that can be stored in a leaf node is less than the numberof pointers in a nonleaf node.

Leaf nodes are still required to be half full.

Insertion and deletion are handled in the same way as insertion

and deletion of entries in a B +-tree index.Simplest to implement sequential file organization no need for

overflows!


42/96


B+-Tree File Organization (Cont.)

Good space utilization important since records use more space thanpointers.To improve space utilization, involve more sibling nodes in redistributionduring splits and merges

Involving 2 siblings in redistribution (to avoid split / merge where possible)results in each node having at least entries

Example of B +-tree File Organization

3/2n


43/96


B-Tree Index FilesSimilar to B+-tree, but B-tree allows search-key values toappear only once; eliminates redundant storage of searchkeys.

Search keys in nonleaf nodes appear nowhere else in the B-tree; an additional pointer field for each search key in anonleaf node must be included.

Generalized B-tree leaf node

Nonleaf node pointers Bi are the bucket or file recordpointers.

B Tree Index File Example


44/96


B-Tree Index File Example

B-tree (above) and B+-tree (below) on same data


45/96


B-Tree Index Files (Cont.)Advantages of B-Tree indices:

May use less tree nodes than a corresponding B +-Tree.

Sometimes possible to find search-key value before reaching leaf node.

Disadvantages of B-Tree indices : Only small fraction of all search-key values are found early

Non-leaf nodes are larger, so fan-out is reduced (because of the extra pointer!).Thus B-Trees typically have greater depth than corresponding B +-Tree

Insertion and deletion more complicated than in B +-Trees

Implementation is harder than B +-Trees.

Reading in sorted order is hard

Typically, advantages of B-Trees do not out weigh disadvantages .


46/96


Assume a file with 10**5 records, a key is 8 bytes, a pointer is 4bytes, a block (page) is 1K. Compute time to access a singlerecord via index or a range of 1000 recordsNumber of entries in a node: 1024/12 = 85 lets round to 80.meaning 2d = 80, d =40 (we dont distinguish between number of pointers and number of keys (-1 ) with large numbers)

With 3 levels we can store at least 2*40**2 keys and at most wecan store upto 80**3, so we need h=3. In average we need about47 keys per node (lets round to 50)

Assuming root is in memory we need 2+1 page accesses to get toa record

To access 1000records we need 2 to get to the first leave +20leaves. Then depending whether the file is clustered on this indexor not:

Clustered: 22 + number of blocks to store 1000 records (need tocompute)

Unclustered: 1022

B+-tree Example computation

B+ trees Implementations issues


47/96


B -trees Implementations issues

Fixed vs. variable size keys.

Internal search (sequential, binary, jump).

Key compression.

Secondary index implementation (Access list vs. compositekeys).

Bulk loading.


48/96


Both direct and sequential access, including Range queries.

Always Balanced2-3 page accesses even for very large files.

At least 50% storage utilization and in average 70%

Dynamic, no need for Reorganization.

B+-tree Advantages


49/96


Memory based Hashing

1) Hash function maps key space to a smaller address space address = H(Key)

2) Collision resolution required to handle two keys hashed intothe same address

M b d H hi C lli i


50/96


Memory based Hashing CollisionResolution techniques

1) Linear overflow or open addressing find first empty slot andinsert.

2) Independent or chained overflow create overflow chains inoverflow areas.


51/96



52/96


53/96


Linear overflowP = n/(m-n) (1 for m=1.5n)

Clearly when n approaches m performance of linear overflow ismuch worse!

For loading factor 0.9 (n = 0.9m) p = 5 ! (avg. of 5 overflowrecords per hash!)


54/96


Disk based Static Hashing

A bucket is a unit of storage containing one or more records (a

bucket is typically a disk block).In a hash file organization we obtain the bucket of a recorddirectly from its search-key value using a hash function.

Hash function h is a function from the set of all search-keyvalues K to the set of all bucket addresses B.

Hash function is used to locate records for access, insertion aswell as deletion.

Records with different search-key values may be mapped tothe same bucket; thus entire bucket has to be searched

sequentially to locate a record.

k b d h h


55/96


Disk based Hashing - Hash FunctionsWorst hash function maps all search-key values to the samebucket; this makes access time proportional to the number of

search-key values in the file.An ideal hash function is uniform , i.e., each bucket is assignedthe same number of search-key values from the set of all possible values.

Ideal hash function is random , so each bucket will have thesame number of records assigned to it irrespective of the actual distribution of search-key values in the file.

Typical hash functions perform computation on the internalbinary representation of the search-key.

For example, for a string search-key, the binary representations ofall the characters in the string could be added and the sum modulothe number of buckets could be returned. .


56/96

l f h l ( )


57/96


Example of Hash File Organization (Cont.)

There are 10 buckets,

The binary representation of the I-th character is assumed to

be the integer i. The hash function returns the sum of the binaryrepresentations of the characters modulo 10

E.g. h(Perryridge) = 5 h(Round Hill) = 3 h(Brighton) = 3

Hash file organization of account file, using branch-name as key(See figure in next slide.)


58/96

H dli f B k O fl


59/96


Handling of Bucket OverflowsBucket overflow can occur because of

Insufficient buckets

Skew in distribution of records. This can occur due to tworeasons:

multiple records have same search-key value

chosen hash function produces non-uniform distribution of keyvalues

Although the probability of bucket overflow can be reduced, itcannot be eliminated; it is handled by using overflow buckets .

Handling of Bucket Overflows (Cont.)


60/96


a d g o Buc et Ove ows (Co t.)

Overflow chaining the overflow buckets of a given bucket arechained together in a linked list.

Above scheme is called closed hashing . An alternative, called open hashing , which does not use overflowbuckets, is not suitable for database applications.


61/96


P f f di k b d H hi


62/96


Performance of disk-based Hashing

Searching within a bucket

does not cost I/O!

H h I di


63/96


Hash IndicesHashing can be used not only for file organization, but also for index-structure creation.

A hash index organizes the search keys, with their associated recordpointers, into a hash file structure.

Strictly speaking, hash indices are always secondary indicesif the file itself is organized using hashing, a separate primary hash index onit using the same search-key is unnecessary.

However, we use the term hash index to refer to both secondary indexstructures and hash organized files.

I dont agree ! You may have Hash index as primary and the file is clusteredon another key or a Heap file

Note the difference between a Hash Index and a Hash file whether thebuckets contain Pointers to data records or the records themselves! (sameas difference between A B+-tree index and a B+-tree file! )

E l f H h I d


64/96


Example of Hash Index

Deficiencies of Static Hashing


65/96


Deficiencies of Static HashingIn static hashing, function h maps search-key values to a fixedset of B of bucket addresses.

Databases grow with time. If initial number of buckets is too small,performance will degrade due to too much overflows.

If file size at some point in the future is anticipated and number ofbuckets allocated accordingly, significant amount of space will bewasted initially.

If database shrinks, again space will be wasted. One option is periodic re-organization of the file with a new hash

function, but it is very expensive.

These problems can be avoided by using techniques that allowthe number of buckets to be modified dynamically.


66/96

Dynamic Hashing


67/96


Dynamic HashingGood for database that grows and shrinks in sizeAllows the hash function to be modified dynamicallyExtendable hashing one form of dynamic hashing

Hash function generates values over a large range typically b-bitintegers, with b = 32.At any time use only a prefix of the hash function to index into atable of bucket addresses.

Let the length of the prefix be i bits, 0 i 32.

Bucket address table size = 2 i. Initially i = 0Value of i grows and shrinks as the size of the database grows andshrinks.Multiple entries in the bucket address table may point to a bucket.

Thus, actual number of buckets is < 2 i The number of buckets also changes dynamically due tocoalescing and splitting of buckets.

General Extendable Hash Structure


68/96


General Extendable Hash Structure

I is called: Global depth; I j is called: Local depth

In this structure, i 2 = i 3 = i , whereas i 1 = i 1 (seenext slide for details)


69/96

Updates in Extendable Hash Structure


70/96


Updates in Extendable Hash Structure

If i > i j (more than one pointer to bucket j )

allocate a new bucket z , and set i j and i z to the old i j -+ 1. make the second half of the bucket address table entries pointing

to j to point to z

remove and reinsert each record in bucket j.

recompute new bucket for K j and insert record in the bucket (furthersplitting is required if the bucket is still full)

If i = i j (only one pointer to bucket j ) increment i and double the size of the bucket address table.

replace each entry in the table by two entries that point to the same

bucket. recompute new bucket address table entry for K j

Now i > i j so use the first case above.

To split a bucket j when inserting record with search-key value K j :

Updates in Extendable Hash Structure(C )


71/96


(Cont.)

When inserting a value, if the bucket is full after several splits(that is, i reaches some limit b) create an overflow bucket insteadof splitting bucket entry table further.To delete a key value,

locate it in its bucket and remove it.The bucket itself can be removed if it becomes empty (withappropriate updates to the bucket address table).Coalescing of buckets can be done (can coalesce only with abuddy bucket having same value of i j and same i j 1 prefix, if it ispresent)Decreasing bucket address table size is also possible

Note: decreasing bucket address table size is an expensiveoperation and should be done only if number of buckets becomesmuch smaller than the size of the table

Example

2LOCAL DEPTHBucket A


72/96


Example

Directory is array of size 4.

To find bucket for r , take last ` global depth # bits of h (r ); we denote r byh (r ).

If h (r ) = 5 = binary 101, it is inbucket 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 abucket does not always require doubling; we can tell bycomparing global depth with local depth for the split bucket.)

13*00

01

10

11

2 2

2

2

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 in previous slide


73/96


(Causes Doubling)

20*

00

01

10

11

2 2

2

2

LOCAL DEPTH 2

2

DIRECTORY

GLOBAL DEPTHBucket A

Bucket B

Bucket C

Bucket D

Bucket A2(`split image'of Bucket A)

1* 5* 21*13*

32*16*

10*

15* 7* 19*

4* 12*

19*

2

2

2

000

001

010

011

100

101

110

111

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

Use of Extendable Hash Structure:E l


74/96


Example

Initial Hash structure, bucket size = 2


75/96

Example (Cont )


76/96


Example (Cont.)Hash structure after insertion of Mianus record

Example (Cont )


77/96


Example (Cont.)

Hash structure after insertion of three Perryridge records

Using OVERFLOW BUCKET!

Example (Cont )


78/96


Example (Cont.)Hash structure after insertion of Redwood and Round Hillrecords

Performance of Extendable Hashing


79/96


Performance of Extendable Hashing

Assume a file with 10**7 records, record size is 200 bytes, key

size is 12 bytes, pointer size is 8 bytes.Assume Data block is 4K and blocks are in average half full

Assume Index blocks are also 4K and are in average half full

Therefore an Index block contain about 100 entries and a data

block contains about 10 records.With an Hash-index organization, the number of entries for thedirectory is: 10**7 / 100 = 10**5, therefore size of address is 17bits and size of directory 2**17 *4 = 512K. Access time is 2 (1 forIndex +1 for data file) ( or 2**17*8 = 1M )

With Hash file organization, the number of entries for thedirectory is 10**7/10 = 10**6, therefore size of address is 20 bitsand size of directory is 2**20*4 = 4M. Access time is only 1!

Extendable Hashing vs Other Schemes


80/96


Extendable Hashing vs. Other SchemesBenefits of extendable hashing:

Hash performance does not degrade with growth of file Minimal space overhead

Disadvantages of extendable hashing Extra level of indirection to find desired record

Bucket address table may itself become very big (larger thanmemory)

Need a tree structure to locate desired record in the structure!

Changing size of bucket address table is an expensive operation

Linear hashing is an alternative mechanism which avoids these

disadvantages at the possible cost of more bucket overflows


81/96


82/96


Linear Hashing (Contd.)

Directory avoided in LH by using overflow pages, and choosingbucket to split round-robin.

Splitting proceeds in `rounds . Round ends when all NR initial (for roundR) buckets are split. Buckets 0 to Next-1 have been split; Next to NR yetto be split.

Current round number is Level . Search: To find bucket for data entry r, find h Level (r ):

If h Level (r ) in range ` Next to NR , r belongs here.

Else, r could belong to bucket h Level (r ) or bucket h Level (r ) + NR;

must apply h Level +1(r ) to find out.


83/96


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 use

h Level+1

`split image' bucket.to decide if entry is in

created (through splitting`split image' buckets:



84/96



Insert : Find bucket by applying hLevel

/ hLevel+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, usually but not necessarilywhen an overflow occurs. Since buckets are split round- robin, long overflow chains dont develop! Doubling of directory in Extendible Hashing is similar; switching of hashfunctions is implicit in how the # of bits examined is increased.

Example of Linear Hashing


85/96


p g

On split, h Level+1 is used to re-distribute entries.

0hh

1

( This inf o is for i llu stration only!)

Level=0, N=4

00

01

10

11

000

001

010

011

(Th e actual contents of th e li near hashed file)

Next=0

PRIMARYPAGES

Data entry r with h(r)=5

Primary bucket page

44* 36*32*

25*9* 5*

14* 18* 10* 30*

31* 35* 11*7*

0hh

1

Level=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


86/96


p

0hh1

22*

00

01

10

11

000

001

010

011

00100

Next=3

01

10

101

110

Level=0PRIMARYPAGES

OVERFLOWPAGES

32*

9*

5*

14*

25*

66* 10*18* 34*

35*31* 7* 11* 43*

44*36*

37* 29*

30*

0hh 1

37*

00

01

10

11

000

001

010

011

00100

10

101

110

Next=0

Level=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* page = 2K

Comparison of Ordered Indexing and Hashing


87/96


Cost of periodic re-organization

Relative frequency of insertions and deletionsIs it desirable to optimize average access time at the expense ofworst-case access time?

Expected type of queries:Hashing is generally better at retrieving records having a specifiedvalue of the key.If range queries are common, ordered indices are to be preferred

Index Definition in SQL


88/96


QCreate an index

create index on )

E.g.: create index b-index on branch(branch-name)

Use create unique index to indirectly specify and enforce thecondition that the search key is a candidate key is a candidate

key.Not really required if SQL unique integrity constraint is supported

To drop an indexdrop index

Multiple-Key Access


89/96


p yUse multiple indices for certain types of queries.Example:

select account-number

from account where branch-name = Perryridge and balance - 1000

Possible strategies for processing query using indices on singleattributes:

1. Use index on branch-name to find accounts with balances of $1000; testbranch- name = Perryridge .

2. Use index on balance to find accounts with balances of $1000; test branch-name = Perryridge .

3. Use branch-name index to find pointers to all records pertaining to thePerryridge branch. Similarly use index on balance . Take intersection ofboth sets of pointers obtained.

Indices on Multiple Attributes


90/96


p

With the where clausewhere branch-name = Perryridge and balance = 1000the index on the combined search-key will fetch only recordsthat satisfy both conditions.Using separate indices in less efficient we may fetch many

records (or pointers) that satisfy only one of the conditions.Can also efficiently handlewhere branch-name - Perryridge and balance < 1000

But cannot efficiently handlewhere branch-name < Perryridge and balance = 1000

May fetch many records that satisfy the first but not thesecond condition.

Suppose we have an index on combined search-key(branch-name, balance ).

Indices on Multiple Attributes


91/96


p

Example: find all objects within the following boundaries:

0


92/96


y

A classic problem in computer science!

Data requested in sorted ordere.g., find students in increasing gpa order

Sorting is first step in bulk loading B+ tree index.

Sorting useful for eliminating duplicate copies in a collection of records(Why?)

Sort-merge join algorithm involves sorting.

Problem: sort 1Gb of data with 1Mb of RAM.why not virtual memory?

2-Way Sort: Requires 3 Buffers


93/96


y q

Pass 1: Read a page, sort it, write it.only one buffer page is used

Pass 2, 3 , , etc.: three buffer pages used.

Main memory buffers

INPUT 1

INPUT 2OUTPUT

DiskDisk

Two-Way External Merge Sort


94/96


y g

Each pass we read + write eachpage in file.

N pages in the file => the number ofpasses

So toal cost is:

Idea: Div ide and conq uer : sortsubfiles and merge

log 2 1 N

2 12 N N log

Input file

1-page runs

2-page runs

4-page runs

8-page runs

PASS 0

PASS 1

PASS 2

PASS 3

9

3,4 6,2 9,4 8,7 5,6 3,1 2

3,4 5,62,6 4,9 7,8 1,3 2

2,34,6

4,78,9

1,35,6 2

2,34,46,78,9

1,23,56

1,22,33,44,56,67,8

General External Merge Sort


95/96


To sort a file with N pages using B buffer pages:Pass 0: use B buffer pages. Produce sorted runs of B pages each.

Pass 2 , , etc.: merge B-1 runs. N B/

B Main memory buffers

INPUT 1

INPUT B-1

OUTPUT

DiskDisk

INPUT 2

. . . . . .. . .

More than 3 buffer pages. How can we utilize them?

Cost of External Merge Sort


96/96

Number of passes:

Cost = 2N * (# of passes)E.g., with 5 buffer pages, to sort 108 page file:

Pass 0: = 22 sorted runs of 5 pages each (last run is only3 pages)

Pass 1: = 6 sorted runs of 20 pages each (last run is only

8 pages)Pass 2: 2 sorted runs, 80 pages and 28 pages

Pass 3: Sorted file of 108 pages

For example: 0 passes (main memory) = 1N (writing output notconsidered)

1 pass = 3N

1 1log / B N B

108 5/

22 4/

Index and Hashing manual and presentation foe students

Documents