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Database System ConceptsChapter 12: Indexing and Hashing

Departamento de Engenharia InformaticaInstituto Superior Tecnico

1st

Semester2008/2009

Slides (fortemente) baseados nos slides ociais do livroDatabase System Conceptsc Silberschatz, Korth and Sudarshan.

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Outline

1 Basic Concepts

2 B+ -Tree IndicesDenitionB+ -Tree NodesQueries on B+ -TreesUpdates on B+ -TreesB+ -Tree File Organization

3 Hash Indices

Static HashingDynamic Hashing

4 Indices on Multiple Keys

5 Indices and SQL

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Outline

1 Basic Concepts

2 B+ -Tree Indices

3 Hash Indices


5 Indices and SQL

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Indexing

Indexing mechanisms are used to speed up access todesired data.

For instance, an author catalog in a library

In a database, they serve the same purpose

An index le consists of records (called index entries) of the formsearch-key pointer

A search key is an attribute or set of attributes used tolook up records in a leIndex les are typically much smaller than the original leTwo basic kinds of indices:

Ordered indices: search keys are stored in sorted orderHash indices: search keys are distributed uniformly across

buckets using a hash function

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Index Evaluation Metrics

Indices are chosen according to:Access types supported efficiently

records with a specied value in the attributeor records with an attribute value falling in a speciedrange of values

Access timeInsertion time

Deletion timeSpace overhead

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Types of Indices

Primary index: in a sequentially ordered le, the indexwhose search key species the sequential order of the le

Also called clustering indexThe search key of a primary index is usually but notnecessarily the primary key.

Secondary index: an index whose search key species anorder different from the sequential order of the le

Also called non-clustering index

Index-sequential le: ordered sequential le with a primaryindex

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Dense Index Files

Dense index: index record appears for every search-keyvalue in the le

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Sparse Index Files

Sparse Index: contains index records for only somesearch-key values

Applicable when records are sequentially ordered onsearch-key

To locate a record with search-key value K we:Find index record with largest search-key value < K Search le sequentially starting at the record to which theindex record points

Less space and less maintenance overhead for insertions

and deletionsGenerally slower than dense index for locating recordsGood tradeoff: sparse index with an index entry for everyblock in le, corresponding to least search-key value in theblock

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Sparse Index Files (cont.)

Example of a sparse index le:

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Multilevel Index

If primary index does not t in memory, access becomesexpensiveTo reduce number of disk accesses to index records, treat

primary index kept on disk as a sequential le andconstruct a sparse index on itouter index - a sparse index of primary indexinner index - the primary index le

If even outer index is too large to t in main memory, yetanother level of index can be created, and so onIndices at all levels must be updated on insertion ordeletion from the le

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Multilevel Index (cont.)

Example of a multilevel index:

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Index Update: Deletion

If deleted record was the only record in the le with itsparticular search-key value, the search-key is deleted fromthe index also

Single-level index deletion:Dense indices: deletion of search-key is similar to lerecord deletionSparse indices:

if an entry for the search key exists in the index, it is

deleted by replacing the entry in the index with the nextsearch-key value in the le (in search-key order)If the next search-key value already has an index entry,the entry is deleted instead of being replaced

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Index Update: Insertion

Single-level index insertion:Perform a lookup using the search-key value appearing inthe record to be insertedDense indices - if the search-key value does not appear inthe index, insert itSparse indices - if index stores an entry for each block of the le, no change needs to be made to the index unless anew block is created

If a new block is created, the rst search-key value

appearing in the new block is inserted into the indexMultilevel insertion (as well as deletion) algorithms aresimple extensions of the single-level algorithms

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Secondary Indices

Frequently, one wants to nd all the records whose valuesin a certain eld (which is not the search-key of theprimary index) satisfy some condition

Example 1: In the account relation stored sequentially byaccount number, we may want to nd all accounts in aparticular branchExample 2: as above, but where we want to nd allaccounts with a specied balance or range of balances

We can have a secondary index with an index record foreach search-key value

Index record points to a bucket that contains pointers toall the actual records with that particular search-key value

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Secondary Indices (cont.)

Secondary index on balance eld of account

Secondary indices have to be dense

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Primary and Secondary Indices

Indices offer substantial benets when searching for records

When a le is modied, every index on the le must beupdated, Updating indices imposes overhead on databasemodicationSequential scan using primary index is efficient, but asequential scan using a secondary index is expensive

each record access may fetch a new block from disk

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B+ -TreeIndicesDenitionB+ -Tree NodesQueries onB+ -TreesUpdates onB+ -TreesB+ -Tree FileOrganization

Hash Indices


Indices andSQL

Outline

1 Basic Concepts

2 B+ -Tree IndicesDenition

B+

-Tree NodesQueries on B+ -TreesUpdates on B+ -TreesB+ -Tree File Organization

3 Hash Indices


5 Indices and SQL

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B+ -Tree Index Files

B+ -tree indices are an alternative to indexed-sequential lesDisadvantage of indexed-sequential les:

Performance degrades as le grows, since many overowblocks get createdPeriodic reorganization of entire le is required

Advantage of B+ -tree index les:Automatically reorganizes itself with small, local, changes,in the face of insertions and deletionsReorganization of entire le is not required to maintainperformance

Disadvantage of B+ -trees: extra insertion and deletionoverhead, space overheadAdvantages of B+ -trees outweigh disadvantages, and theyare used extensively

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B+ -Tree Index Files (cont.)

A B+ -tree is a rooted tree satisfying the following properties:All paths from root to leaf are of the same length

Each node that is not a root or a leaf has between n/ 2and n childrenA leaf node has between (n 1)/ 2 and n 1 valuesSpecial cases:

If the root is not a leaf, it has at least 2 childrenIf the root is a leaf (that is, there are no other nodes in thetree), it can have between 0 and n 1 values

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Example of a B+ -Tree

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

B+

-tree for account le (n = 5)

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Leaf Nodes in B+ -Trees

Properties of a leaf node:For i = 1 , 2, . . . , n 1, pointer P i either points to a lerecord with search-key value K i , or to a bucket of pointersto le records, each record having search-key value K i

Only needs bucket structure if search-key does not form aprimary key

If Li , L j are leaf nodes and i < j , Li s search-key values areless than L j s search-key valuesP n points to next leaf node in search-key order

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Non-Leaf Nodes in B+ -Trees

Non leaf nodes form a multi-level sparse index on the leaf nodesFor a non-leaf node with m pointers:

All the search-keys in the subtree to which P 1 points areless than K 1For 2 i m 1, all the search-keys in the subtree towhich P i points have values greater than or equal to K i 1and less than K i All the search-keys in the subtree to which P m points aregreater than K m 1

P 1 K 1 P 2 . . . P m 1 K m 1 P m

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Observations about B+ -trees

Since the inter-node connections are done by pointers,logically close blocks need not be physically closeThe non-leaf levels of the B+ -tree form a hierarchy of sparse indicesThe B+ -tree contains a relatively small number of levels(logarithmic in the size of the main le), thus searches canbe conducted efficiently

Insertions and deletions to the main le can be handledefficiently, as the index can be restructured in logarithmictime

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Queries on B+ -Trees

To nd 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 i ), then follow P i to

the child node3 Otherwise follow P m to the child node (k K m 1, wherethere are m pointers in the node)

2 If the node reached by following the pointer above is not aleaf node, repeat step 1 on the node

3 Else we have reached a leaf node1 If for some i , key K i = k follow pointer P i to the desired

record or bucket2 Else no record with search-key value k exists

+

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Queries on B+ -Trees (cont.)

In processing a query, a path is traversed in the tree fromthe root to some leaf nodeIf there are K search-key values in the le, the path is nolonger than logn / 2 (K )

A node is generally the same size as a disk block, typically4 kilobytes, and n is typically around 100 (40 bytes perindex entry)With 1 million search key values and n = 100, at mostlog50(1 000 000) = 4 nodes are accessed in a lookup

Contrast this with a balanced binary free with 1 millionsearch key valuesaround 20 nodes are accessed in alookupThe above difference is signicant since every node accessmay need a disk I/O, costing around 20 milliseconds!

+

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Insertion on a B+ -Tree

Find the leaf node in which the search-key value wouldappearIf the search-key value is already there in the leaf node,

record is added to le and if necessary a pointer is insertedinto the bucket.If the search-key value is not there, then add the record tothe main le and create a bucket if necessary. Then:

If there is room in the leaf node, insert (key-value, pointer)pair in the leaf nodeOtherwise, split the node (along with the new (key-value,pointer) entry)

S li i N d

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Splitting a Node

To split a node:take the n (search-key,pointer) pairs (including the onebeing inserted) in sorted orderplace the rst n/ 2 in the original node, and the rest in anew nodelet the new node be p , and let k be the smallest key valuein p Insert (k ,p ) in the parent of the node being splitIf the parent is full, split it and propagate the split furtherup

The splitting of nodes proceeds upwards till a node that isnot full is foundIn the worst case the root node may be split increasing theheight of the tree by 1

I i E l

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Insertion Example

B+

-Tree before and after insertion of Clearview

S litti N d

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Splitting a Node

Result of splitting node containing Brighton and Downtown oninserting Clearview

(example )

D l ti B+ T

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Deletion on a B+ -Tree

Find the record to be deleted, and remove it from themain le and from the bucket (if present)Remove (search-key value,pointer) from the leaf node if

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 t into a single node,then

Insert all the search-key values in the two nodes into a

single node and delete the other node.Delete the pair (K i 1,P i ), where P i is the pointer to thedeleted node, from its parent, recursively using the aboveprocedure.

Deletion on a B+ Tree(cont )

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Deletion on a B -Tree(cont.)

If the parent node has too few entries due to the removal,but the entries in the parent and a sibling do not t into asingle node, then

Redistribute the pointers between the node and a sibling

such that both have more than the minimum number of entriesUpdate the corresponding search-key value in the parent of the node

The node deletions may cascade upwards till a node whichhas n/ 2 or more pointers is foundIf the root node has only one pointer after deletion, it isdeleted and the sole child becomes the root

Deletion Example

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Deletion Example

B+

-Tree before and after deletion of Downtown

Deletion Example (cont )

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Deletion Example (cont.)

B+ -Tree before and after deletion of Perryridge

Deletion Example (cont )

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Deletion Example (cont.)

B+

-Tree before and after deletion of Perryridge

B+ -Tree File Organization

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B -Tree File Organization

Index le degradation problem is solved by using B+

-treeindices. Data le degradation problem is solved by usingB+ -tree File OrganizationThe leaf nodes in a B+ -tree le organization store records,instead of pointersInsertion and deletion are handled in the same way asinsertion and deletion of entries in a B+ -tree index

Outline

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1 Basic Concepts

2 B+ -Tree Indices

3 Hash IndicesStatic HashingDynamic Hashing


5 Indices and SQL

Static Hashing

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Static Hashing

A bucket is a unit of storage containing one or morerecords (a bucket is typically a disk block)In a hash le organization we obtain the bucket of a recorddirectly from its search-key value using a hash functionHash 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 and as deletionRecords with different search-key values may be mappedto the same bucket; thus entire bucket has to be searchedsequentially to locate a record

Hash File Organization

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Hash File Organization

Hash le organization of account le, using branch name as key

There are 10 bucketsThe binary representation of

the i

th character is assumedto be the integer i The hash function returnsthe sum of the binaryrepresentations of thecharacters modulo 10E.g. h(Perryridge) = 5,h(Round Hill) = 3,h(Brighton) = 3

Hash Functions

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Hash Functions

Worst hash function maps all search-key values to thesame bucketAn ideal hash function is uniform, i.e., each bucket isassigned the same number of search-key values from the

set of all possible valuesIdeal hash function is random, so each bucket will havethe same number of records assigned to it irrespective of the actual distribution of search-key values in the leTypical hash functions perform computation on theinternal binary representation of the search-key.

For example, for a string search-key, the binaryrepresentations of all the characters in the string could beadded and the sum modulo the number of buckets couldbe returned

Handling of Bucket Overows

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a d g o uc et Ove ows

Bucket overow can occur because of Insufficient bucketsSkew in distribution of records. This can occur due to two

reasons:multiple records have same search-key valuechosen hash function produces non-uniform distribution of key values

Although the probability of bucket overow can be

reduced, it cannot be eliminated; it is handled by usingoverow buckets

Handling of Bucket Overows (cont.)

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g ( )

Overow chaining - the overow buckets of a given bucket arechained together in a linked list

Above scheme is called closed hashingAn alternative, called open hashing (linear probing), which doesnot use overow buckets, is not suitable for database

applications

Hash Indices

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Hashing can be used not only for le organization, butalso for index-structure creationA hash index organizes the search keys, with their

associated record pointers, into a hash le structureStrictly speaking, hash indices are always secondary indices

if the le itself is organized using hashing, a separateprimary hash index on it using the same search-key is

unnecessaryhowever, we use the term hash index to refer to bothsecondary index structures and hash organized les

Example of Hash Index

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p

Deciencies of Static Hashing

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g

In static hashing, function h maps search-key values to axed set of B of bucket addressesDatabases grow with time. If initial number of buckets istoo small, performance will degrade due to too muchoverowsIf le size at some point in the future is anticipated andnumber of buckets allocated accordingly, signicantamount of space will be wasted initiallyIf database shrinks, again space will be wastedOne option is periodic re-organization of the le with anew hash function, but it is very expensiveThese problems can be avoided by using techniques thatallow the number of buckets to be modied dynamically

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Extendable Hash Structure

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Use of Extendable Hash Structure

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Each bucket j stores a value i j ; all the entries that point tothe same bucket have the same values on the rst i j bitsTo locate the bucket containing search-key K j :

1 Compute h(K j ) = X 2 Use the rst i high order bits of X as a displacement into

bucket address table, and follow the pointer to appropriatebucket

To insert a record with search-key value K j follow same procedure as look-up and locate the bucket,say j If there is room in the bucket j insert record in the bucketElse the bucket must be split and insertion re-attempted

Overow buckets used instead in some cases

Updates in Extendable Hash Structure

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To split a bucket j when inserting record with search-key valueK j :If i = i j (only one pointer to bucket j )

increment i and double the size of the bucket address tablereplace each entry in the table by two entries that point to

the same bucketrecompute new bucket address table entry for K j Now i > i j so use the following case

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 entriespointing to j to point to z remove and reinsert each record in bucket j recompute new bucket for K j and insert record in thebucket (further splitting is required if the bucket is still full)

Updates in Extendable Hash Structure (cont.)

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When inserting a value, if the bucket is full after severalsplits (that is, i reaches some limit b ) create an overowbucket instead of splitting bucket entry table furtherTo delete a key value,

locate it in its bucket and remove itthe 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 witha buddy bucket having same value of i j and same i j 1prex, if it is present)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 bucketsbecomes much smaller than the size of the table

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An Example (cont.)

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Hash structure after insertion of one Brighton and twoDowntown records

An Example (cont.)

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Hash structure after insertion of one Mianus record

An Example (cont.)

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Hash structure after insertion of three Perryridge records

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Extendable Hashing vs. Other Schemes

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Benets of extendable hashing:Hash performance does not degrade with growth of leMinimal space overhead

Disadvantages of extendable hashingExtra level of indirection to nd desired recordBucket address table may itself become very big (largerthan memory)Changing size of bucket address table is an expensiveoperation

Outline

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1 Basic Concepts

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Multiple-Key Access

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Use multiple indices for certain types of queriesExampleselect account number from account where branch name = Perryridge and balance = 1000

Possible strategies for processing query using indices onsingle attributes:

1 Use index on branch name to ndbranch name = Perryridge; test for balance = $1000

2 Use index on balance to nd balance = $1000; test forbranch name = Perryridge

3 Use branch name index to nd pointers to all recordspertaining to the Perryridge branch; similarly, use index onbalance ; take intersection of both sets of pointers obtained

Indices on Multiple Keys

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Composite search keys are search keys containing morethan one attribute

E.g. (branch name , balance )Lexicographic ordering: (a1, a2) < (b 1, b 2) if either

a1 < b 1, ora1 = b 1 and a2 < b 2

Examples

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With the where clause

where branch name = Perryridge and balance = 1000

the index can be used to fetch only records that satisfyboth conditions

Using separate indices in less efficient we may fetchmany records that satisfy only one of the conditions.Can also efficiently handle

where branch name = Perryridge and balance < 1000

But cannot efficiently handle

where branch name < Perryridge and balance = 1000

May fetch many records that satisfy the rst but not the

second condition

Outline

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Index Denition in SQL

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Create an index

create index < index-name> on < relation-name> (< attribute-list > )

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

Use create unique index to indirectly specify and enforcethe condition that the search key is a candidate key

Not really required if SQL unique integrity constraint issupported

To drop an index

drop index < index-name>

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Indices andSQL

End of Chapter 12

Documents

Ch12 Indexing (1)