This document discusses different data structures used for indexing in databases, including B-trees, B+ trees, hashing techniques, and linear hashing. B-trees allow searches, insertions and deletions in logarithmic time and are well-suited for storage systems that read and write large blocks of data. B+ trees are balanced trees where every path from the root to a leaf is the same length, and are used to index database records. Hashing techniques like external hashing map database records to disk pages using a hash function for efficient retrieval, while dynamic hashing allows hash tables to grow and shrink as records are added or removed. Linear hashing is a dynamic hashing method that expands and contracts hash

1. TO:
K.Padma Priya M.Sc .,M.Phil.,
Assistant professor,
Department of Maths(CA),
SBK College,
Arupukottai.
B TREE, B+ TREE ,HASHING

2. B -TREE
 In “computer science “,B tree is a self balancing “tree data
structure” that maintains soreted data and allows searches,
sequential access , insertions,and deletions in “logarithmic
time”.
 The B tree generalizes the “binary search tree”, allowing
for nodes with more than two children.
 Unlike other “self balancing binary search tree”, the B
tree is well suited for storage systems that read and write
relatively large blocks of data, such as discs.

3. DEFINITION:
 According to knuth’s definition,a B tree of order m is a tree
which satisfies the following properties:
1. Every node has at most m children.
2. Every non-leaf node has at least [m/2] child
nodes.
3. The root has at least two children if it is not a
leaf node.
4. A non-leaf node with k children contains k-1
keys.
All leaves appear in the same level and carry no
information.

4. Each internal node’s keys act as separation values
which divide its subtrees.
For example , if an internal node has 3 child nodes
then it must have 2 keys: a1 and a2 .
All values in the leftmost subtree will be less than
a1, all values in the middle subtree will be between a1
and a2 , and all values in the rightmost subtree will be
greater than a2.

6. B+ TREE
 Index lookups and the sequential scans take more
times as more records are there in the files.
 AB+ tree index takes the form of a balanced tree in
which every path from of data is the B+ tree index leaf
is the same length.
 Each non-leaf node in the tree has between ‘n/2’
and ‘n’ children . Where ‘n’ is fixed for a particular tree.
 The range of values in each leaf do not ovelap.

7.  B+ tree pointers either to a file with the associated
search key values or to a bucket of pointer.
 The leaf nodes in the search key order thus
following efficient sequential processing of the file.
 We use last pointer in each leaf node to chain
together the leaf node in the search key order.

9. HASHING
 File using hashing techniques are called ‘hash files’.
 In most case the files is also the key field of the file
in case it is called “hash key”.
 The idea behind hashing is to provide a function
called “hash function” or “randomizing function”.

10. INTERNAL HASHING:
 Hashing in implemented as a hash table using array
of records .
 In internal hashing the hash table is in memory
,where each slot holds only one entry.
 This type of hashing is covered in a separate lesson.
 This lesson covers the applications of hashing
techniques for indexing records on disk , where slots
are called buckets and refer to pages on disk .

11. EXTERNAL HASHING:
 Each bucket may hold multiple data entries.
 It is used to create hashed files, in which records
are positioned based on a hash function on same
fields .
 When searching for a record with specific fields or
search key in a data base , we can use hashing to find
the records containing that key on disk.
 This is done with a Hash function ,which takes
the key and computer an integer.
 This integer can be used to map to the record on
disk.

12.  Direct file- the integer map directly to the record .
The operation system must provide support for this
type of file .
 Heap file – the integer map to the id of the page
containing the record , where the data page is searched
sequentially .
 Lookup table – translate relative page address to
physical page address.

13. DYNAMIC HASHING
 The dynamic hashing method is used to overcome
the problem of static hashing like bucket overflow.
 In this method, data bucket grow or shrink as the
records increases. This method is also know as
extendable hashing method.
 This method makes hashing dynamic, it allows
insertion or deletion without resulting in poor
performance.

14. ADVANTAGES:
 In this method , the performance does not
decrease as the data grow in the system. It simply
increases the size of memory to accommodate the
data.
 In this method, memory is well utilized as it grow
and shrinks with the data . There will not be any
unused memory lying.
 This method is good for dynamic database
where data grows and shrinks frequently.

15. DISADVANTAGES: In this method , if the data size increases then the
bucket size is also increased.
 These addresses of data will be maintained in
the bucket address table.
 This is because the data address will keep
changing as buckets grow and shrink.
 If there is a huge increase in data , maintaining
the bucket address table becomes tedious.

16.  In this case , the bucket overflow situation will also
occur. But it might take little time to reach this
situation than static hashing.

17. LINEAR HASHING:
 Linear hashing is a dynamic data structure which
implements a “ hash table” and grows or shrinks
one bucket at a time .
 It was invented by “witold litwin” in 1980.
 It has been analyzed by baeza-yates and soza –
pollman .
 It is the first in a number of schemes know as
dynamic hashing such as larson’s linear hashing with
partial expansions , linear hashing with priority
splitting , linear hashing with partial expansions and
priority splitting , or recursive linear hashing.

18.  The file structure of a dynamic hashing data
structure adapts itself to changes in the size of the
file ,so expensive periodic file recorganization is
avoided.
 A linear hashing file expands by splitting a pre-
determined bucket into two and contracts by
merging two predetermined bucket into one.
 In L.H* , each bucket resides at a different server.
 Key based operations in LH and LH* take
maximum constant time independent of the number
of bucket and hence of records .