Chapter 12: Indexing and Hashing

Indexing
❑ Basic Concepts
❑ Ordered Indices
❑ B+-Tree Index Files
❑ B-Tree Index Files

Database System Concepts, 6th Ed.

©Silberschatz, Korth and Sudarshan
See www.db-book.com for conditions on re-use
 Basic Concepts
● Indexing mechanisms used to speed up access to desired data.
● E.g., author catalog in library
● Search Key - attribute to set of attributes used to look up records in a file.
● An index file consists of records (called index entries) of the form

● Index files are typically much smallerpointer

search-key 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 across “buckets”
using a “hash function”.

Database System Concepts - 6th Edition 11.2 ©Silberschatz, Korth and Sudarshan
 Index Evaluation Metrics
● Access types supported efficiently. E.g.,
● records with a specified value in the attribute
● or records with an attribute value falling in a specified range of values.
● Access time
● Insertion time
● Deletion time
● Space overhead

Database System Concepts - 6th Edition 11.3 ©Silberschatz, Korth and Sudarshan
 Ordered Indices

● In an ordered index, index entries are stored sorted on the search key value.
E.g., author catalog in library.
● Primary index: in a sequentially ordered file, the index whose search key
specifies the sequential order of the file.
● Also called clustering index
● The search key of a primary index is usually but not necessarily the
primary key.
● Secondary index: an index whose search key specifies an order different from
the sequential order of the file. Also called
non-clustering index.
● Index-sequential file: ordered sequential file with a primary index.

Database System Concepts - 6th Edition 11.4 ©Silberschatz, Korth and Sudarshan
 Dense Index Files
● Dense index — Index record appears for every search-key value in
the file.
● E.g. index on ID attribute of instructor relation

Database System Concepts - 6th Edition 11.5 ©Silberschatz, Korth and Sudarshan
 Dense Index Files (Cont.)
● Dense index on dept_name, with instructor file sorted on dept_name

Database System Concepts - 6th Edition 11.6 ©Silberschatz, Korth and Sudarshan
 Sparse Index Files
● Sparse 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 index record
points

Database System Concepts - 6th Edition 11.7 ©Silberschatz, Korth and Sudarshan
 Sparse Index Files (Cont.)
● Compared to dense indices:
● Less space and less maintenance overhead for insertions and deletions.
● Generally slower than dense index for locating records.
● Good tradeoff: sparse index with an index entry for every block in file, corresponding to least search-key
value in the block.

Database System Concepts - 6th Edition 11.8 ©Silberschatz, Korth and Sudarshan
 Secondary Indices Example

Secondary index on salary field of instructor

● Index record points to a bucket that contains pointers to all the actual
records with that particular search-key value.
● Secondary indices have to be dense

Database System Concepts - 6th Edition 11.9 ©Silberschatz, Korth and Sudarshan
 Primary and Secondary Indices
● Indices offer substantial benefits when searching for records.
● BUT: Updating indices imposes overhead on database modification --when a file is modified,
every index on the file must be updated,
● Sequential scan using primary index is efficient, but a sequential scan using a secondary index
is expensive
● Each record access may fetch a new block from disk
● Block fetch requires about 5 to 10 milliseconds, versus about 100 nanoseconds for memory
access

Database System Concepts - 6th Edition 11.10 ©Silberschatz, Korth and Sudarshan
 Multilevel Index
● If primary index does not fit in memory, access becomes expensive.
● Solution: treat primary index kept on disk as a sequential
file and construct a sparse 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 another
level of index can be created, and so on.
● Indices at all levels must be updated on insertion or deletion from the
file.

Database System Concepts - 6th Edition 11.11 ©Silberschatz, Korth and Sudarshan
 Index Update: Deletion

● If 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 entry deletion:
● Dense indices – deletion of search-key is similar to file record
deletion.
● Sparse indices –
4 if an entry for the search key exists in the index, it is deleted by
replacing the entry in the index with the next search-key value in
the file (in search-key order).
4 If the next search-key value already has an index entry, the entry
is deleted instead of being replaced.
Database System Concepts - 6th Edition 11.12 ©Silberschatz, Korth and Sudarshan
 Index Update: Insertion
● Single-level index insertion:
● Perform a lookup using the search-key value appearing in the
record to be inserted.
● Dense indices – if the search-key value does not appear in the
index, 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
is created.
4 If a new block is created, the first search-key value appearing
in the new block is inserted into the index.
● Multilevel insertion and deletion: algorithms are simple extensions
of the single-level algorithms

Database System Concepts - 6th Edition 11.13 ©Silberschatz, Korth and Sudarshan
 Secondary Indices
● Frequently, one wants to find all the records whose values in a
certain field (which is not the search-key of the primary index)
satisfy some condition.
● Example 1: In the instructor relation stored sequentially by ID,
we may want to find all instructors in a particular department
● Example 2: as above, but where we want to find all instructors
with a specified salary or with salary in a specified range of
values
● We can have a secondary index with an index record for each
search-key value

Database System Concepts - 6th Edition 11.14 ©Silberschatz, Korth and Sudarshan
 B+-Tree Index Files

B+-tree indices are an alternative to indexed-sequential files.

● Disadvantage of indexed-sequential files

● performance degrades as file grows, since many overflow blocks
get created.
● Periodic reorganization of entire file is required.
● Advantage of B+-tree index files:
● automatically reorganizes itself with small, local, changes, in the
face of insertions and deletions.
● Reorganization of entire file is not required to maintain
performance.
● (Minor) disadvantage of B+-trees:
● extra insertion and deletion overhead, space overhead.
● Advantages of B+-trees outweigh disadvantages
● B+-trees are used extensively

Database System Concepts - 6th Edition 11.15 ©Silberschatz, Korth and Sudarshan
 Example of B+-Tree

Database System Concepts - 6th Edition 11.16 ©Silberschatz, Korth and Sudarshan
 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/2⎤ and n
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 the
tree), it can have between 0 and (n–1) values.

Database System Concepts - 6th Edition 11.17 ©Silberschatz, Korth and Sudarshan
 B+-Tree Node Structure
● Typical node

● Ki are the search-key values

● Pi 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
K1 < K2 < K3 < . . . < Kn–1
(Initially assume no duplicate keys, address duplicates later)

Database System Concepts - 6th Edition 11.18 ©Silberschatz, Korth and Sudarshan
 Leaf Nodes in B+-Trees

Properties of a leaf node:

● For i = 1, 2, . . ., n–1, pointer Pi points to a file record with search-key
value Ki,
● If Li, Lj are leaf nodes and i < j, Li’s search-key values are less than or
equal to Lj’s search-key values
● Pn points to next leaf node in search-key order

Database System Concepts - 6th Edition 11.19 ©Silberschatz, Korth and Sudarshan
 Non-Leaf Nodes in B+-Trees
● Non 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 P1 points are less than K1
● For 2 ≤ i ≤ n – 1, all the search-keys in the subtree to which Pi points have values greater
than or equal to Ki–1 and less than Ki
● All the search-keys in the subtree to which Pn points have values greater than or equal to
Kn–1

Database System Concepts - 6th Edition 11.20 ©Silberschatz, Korth and Sudarshan
 Example of B+-tree

B+-tree for instructor file (n = 6)

● Leaf nodes must have between 3 and 5 values

(⎡(n–1)/2⎤ and n –1, with n = 6).
● Non-leaf nodes other than root must have between 3 and 6
children (⎡(n/2⎤ and n with n =6).
● Root must have at least 2 children.

Database System Concepts - 6th Edition 11.21 ©Silberschatz, Korth and Sudarshan
 Updates on B+-Trees: Insertion
1. Find the leaf node in which the search-key value would appear
2. If the search-key value is already present in the leaf node
1. Add record to the file
2. If necessary add a pointer to the bucket.
3. If the search-key value is not present, then
1. add the record to the main file (and create a bucket if necessary)
2. If there is room in the leaf node, insert (key-value, pointer) pair in the leaf node
3. Otherwise, split the node (along with the new (key-value, pointer) entry) as discussed in
the next slide.

Database System Concepts - 6th Edition 11.22 ©Silberschatz, Korth and Sudarshan
 Updates on B+-Trees: Insertion (Cont.)
● Splitting a leaf node:
● take the n (search-key value, pointer) pairs (including the one being inserted) 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 it and propagate the split further up.
● Splitting of nodes proceeds upwards till a node that is not full is found.
● In the worst case the root node may be split increasing the height of the tree by 1.

Result of splitting node containing Brandt, Califieri and Crick on inserting Adams
Next step: insert entry with (Califieri,pointer-to-new-node) into parent

Database System Concepts - 6th Edition 11.23 ©Silberschatz, Korth and Sudarshan
 B+-Tree Insertion

B+-Tree before and after insertion of “Adams”

Database System Concepts - 6th Edition 11.24 ©Silberschatz, Korth and Sudarshan
 B+-Tree Insertion

B+-Tree before and after insertion of “Lamport”

Database System Concepts - 6th Edition 11.25 ©Silberschatz, Korth and Sudarshan
 Insertion in B+-Trees (Cont.)
● Splitting a non-leaf node: when inserting (k,p) into an already full internal node N
● Copy N to an in-memory area M with space for n+1 pointers and n keys
● Insert (k,p) into M
● Copy P1,K1, …, K ⎡n/2⎤-1,P ⎡n/2⎤ from M back into node N
● Copy P⎡n/2⎤+1,K ⎡n/2⎤+1,…,Kn,Pn+1 from M into newly allocated node N’
● Insert (K ⎡n/2⎤,N’) into parent N
● Read pseudocode in book!

Califieri

Adams Brandt Califieri Crick Adams Brandt Crick

Database System Concepts - 6th Edition 11.26 ©Silberschatz, Korth and Sudarshan
 Examples of B+-Tree Deletion

Before and after deleting “Srinivasan”

● Deleting “Srinivasan” causes merging of under-full leaves

Database System Concepts - 6th Edition 11.27 ©Silberschatz, Korth and Sudarshan
 Examples of B+-Tree Deletion (Cont.)

Deletion of “Singh” and “Wu” from result of previous example

● Leaf containing Singh and Wu became underfull, and borrowed a value Kim
from its left sibling
● Search-key value in the parent changes as a result

Database System Concepts - 6th Edition 11.28 ©Silberschatz, Korth and Sudarshan
 Example of B+-tree Deletion (Cont.)

Before and after deletion of “Gold” from earlier example

● Node with Gold and Katz became underfull, and was merged with its sibling
● Parent node becomes underfull, and is merged with its sibling
● Value separating two nodes (at the parent) is pulled down when merging
● Root node then has only one child, and is deleted
Database System Concepts - 6th Edition 11.29 ©Silberschatz, Korth and Sudarshan
 Updates on B+-Trees: Deletion
● Find the record to be deleted, and remove it from the main file 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 empty
● If the node has too few entries due to the removal, and the entries in the node
and a sibling fit into a single node, then merge siblings:
● Insert all the search-key values in the two nodes into a single node (the
one on the left), and delete the other node.
● Delete the pair (Ki–1, Pi), where Pi is the pointer to the deleted node, from
its parent, recursively using the above procedure.

Database System Concepts - 6th Edition 11.30 ©Silberschatz, Korth and Sudarshan
 Updates on B+-Trees: Deletion
● Otherwise, if the node has too few entries due to the removal, but the entries in the node and a sibling do
not fit into a single node, then redistribute pointers:
● 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 has ⎡n/2⎤ or more pointers is found.
● If the root node has only one pointer after deletion, it is deleted and the sole child becomes the root.

Database System Concepts - 6th Edition 11.31 ©Silberschatz, Korth and Sudarshan
 B+-Tree File Organization
● Index file degradation problem is solved by using B+-Tree indices.
● 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.
● Leaf nodes are still required to be half full
● Since records are larger than pointers, the maximum number of records
that can be stored in a leaf node is less than the number of pointers in a
nonleaf node.
● Insertion and deletion are handled in the same way as insertion and deletion of
entries in a B+-tree index.

Database System Concepts - 6th Edition 11.32 ©Silberschatz, Korth and Sudarshan
 B-Tree Index Files
● Similar to B+-tree, but B-tree allows search-key values to appear
only once; eliminates redundant storage of search keys.
● Search keys in nonleaf nodes appear nowhere else in the B-tree; an
additional pointer field for each search key in a nonleaf node must
be included.
● Generalized B-tree leaf node

● Nonleaf node – pointers Bi are the bucket or file record pointers.

Database System Concepts - 6th Edition 11.33 ©Silberschatz, Korth and Sudarshan
 B-Tree Index File Example

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

Database System Concepts - 6th Edition 11.34 ©Silberschatz, Korth and Sudarshan
 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. 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.
● Typically, advantages of B-Trees do not out weigh disadvantages.

Database System Concepts - 6th Edition 11.35 ©Silberschatz, Korth and Sudarshan
 Multiple-Key Access
● Use multiple indices for certain types of queries.
● Example:
select ID
from instructor
where dept_name = “Finance” and salary = 80000
● Possible strategies for processing query using indices on single
attributes:
1. Use index on dept_name to find instructors with department
name Finance; test salary = 80000
2. Use index on salary to find instructors with a salary of $80000;
test dept_name = “Finance”.
3. Use dept_name index to find pointers to all records pertaining to
the “Finance” department. Similarly use index on salary. Take
intersection of both sets of pointers obtained.

Database System Concepts - 6th Edition 11.36 ©Silberschatz, Korth and Sudarshan
 Indices on Multiple Keys
● Composite search keys are search keys containing more than one attribute
● E.g. (dept_name, salary)
● Lexicographic ordering: (a1, a2) < (b1, b2) if either
● a1 < b1, or
● a1=b1 and a2 < b2

Database System Concepts - 6th Edition 11.37 ©Silberschatz, Korth and Sudarshan
 Indices on Multiple Attributes

Suppose we have an index on combined search-key

(dept_name, salary).

● With the where clause

where dept_name = “Finance” and salary = 80000
the index on (dept_name, salary) can be used to fetch only records that
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 handle
where dept_name = “Finance” and salary < 80000
● But cannot efficiently handle
where dept_name < “Finance” and balance = 80000
● May fetch many records that satisfy the first but not the second
condition

Database System Concepts - 6th Edition 11.38 ©Silberschatz, Korth and Sudarshan
 Other Features
● Covering indices
● Add extra attributes to index so (some) queries can avoid fetching the actual records
4 Particularly useful for secondary indices
– Why?
● Can store extra attributes only at leaf

Database System Concepts - 6th Edition 11.39 ©Silberschatz, Korth and Sudarshan