13. Indexing MTrees - Data Structures using C++ by Varsha Patil

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 Indexing techniques
 B-trees which prove invaluable for problems of external
information retrieval
 A class of trees called tries, which share some properties
of table lookup
 Important uses of trees in many search techniques

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 A file is a collection of records, each record having one or
more fields
 The fields used to distinguish among the records are
known as keys
 File organization describes the way where the records are
stored in a file
 File organization is concerned with representing data
records on an external storage media

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 The file organization breaks down into two more aspects:
 Directory—for collection of indices
 File organization—for the physical organization of
records

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 File organization is the way records are organized on a
physical storage
 One of such organizations is sequential (ordered and
unordered)
 In this general framework, processing a query or updating
a request would proceed in two steps:
 The indices would be interrogated to determine the
parts of the physical file to be searched
 These parts of the physical file will be searched

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 An index, whether it is a book or a data file index (in
computer memory), is based on the basic concepts such as
keys and reference fields
 The index to a book provides a way to find a topic quickly

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 An index, whether it is a book or a data file index (in
computer memory), is based on the basic concepts such as
keys and reference fields
 The index to a book provides a way to find a topic quickly

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 This is the simplest type of index organization. It is useful
only for the primary key index of a sequentially ordered
file
 In a sequentially ordered file, the physical sequence of
records is ordered by the key, called the primary key
 The cylinder-surface index consists of a cylinder index
and several surface indexes

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For each cylinder, there is a surface index. If the disk has S
usable surfaces, then each surface index has s entries.
The total number of surface index entries is C.S
Emp. No. Emp. Name Cylinder Surface
1
2
3
4
5
6
7
8
Abolee
Anand
Amit
Amol
Rohit
Santosh
Saurabh
Shila
1
1
1
1
2
2
2
2
1
1
2
2
1
1
2
2

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Let there be two surfaces and two records stored per track.
The file is organized sequentially on the field ‘Emp. name’
The cylinder index is shown in following table
Emp. No. Highest Key Value
1
2
Amol
Shila

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 This method of maintaining a file and index is referred to
as ISAM (indexed sequential access method)
 It is the simplest file organization for single key files but
not useful for multiple key files

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 The operations related to hashed indexes are the same as
those for hash tables

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 A multiway search tree is a tree of order m, where each
node has utmost m children
 Fig. shows way search tree:
d e p v
w x y zrh j k lb c
qia f g m n o s t u

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 A B-tree is a balanced M-way tree. A node of the tree
contains many records or keys of records and pointers to
children
 To reduce disk access, the following points are applicable:
 Height is kept minimum
 All leaves are kept at the same level
 All other than leaves must have at least minimum
number of children

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 Definition:
 A B-tree of order m is an m-way tree with the following
properties:
 The number of keys in each internal node is one less than
the number of its non-empty children, and these keys
partition the keys in the children in the fashion of the
search tree
 All leaves are on the same level
 All internal nodes except the root have utmost m non-empty
children and at least [m/2] non-empty children
 The root is either a leaf node, or it has from two to m
children
 A leaf node contains no more than m − 1 keys

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Ptr1 Key1 Ptr2 Key2 Ptri Keyi …….. Keyn-1 Ptrn
X XX
X<Key1 Keyi-1<X<Keyi X>Keyn-1

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 Search a node
 Insertion of a key into a B-tree
 Deletion from a B-tree

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 B+ trees are internal data structures
 That is, the nodes contain whatever information is
associated with the key as well as the key values

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 The structure of a B+ tree can be understood from the
following points:
 A B+ tree is in the form of a balanced tree where every
path from the root of the tree to a leaf of the tree is of
the same length
 Each non-leaf node (internal node) in the tree has
between [n/2] and n children, where n is fixed
 The pointer (Ptr) can point to either a file record or a
bucket of pointers which each point to a file record
 Searching time is less in B+ trees but has some
problem of wasted space

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 Internal node of a B+ tree with q −1 search values
 Leaf node of a B+ tree with q − 1 search values and q − 1
data pointers

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Ptr1 Key1 Ptr2 Key2 Ptri Keyi …….. Keyn-1 Ptrn
X XX
X<Key1 Keyi-1<X<Keyi X>Keyn-1
Tree Pointer
Tree Pointer
Tree Pointer

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 A dynamic index structure that adjusts gracefully to
inserts and deletes
 A balanced tree
 Leaf pages are not allocated sequentially. They are
linked together through pointers (a doubly linked list)

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 One solution is to prune from the tree all the branches
that do not lead to any key
 The resulting tree is called a trie (short for reTRIEvaL and
pronounced ‘try’)
 The number of steps needed to search a trie is
proportional to the number of characters in a key

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 Splay trees are a form of a BST. A splay tree maintains a
balance without any explicit balance condition such as
color
 Instead, ‘splay operations’, which involve rotations, are
performed within the tree every time an access is made

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 If we use a BST or even an AVL tree, then the records of the newly
admitted patient’s records will go to a leaf position, far from the
root, and the access will be slower
 Instead, we want to keep the records that are newly inserted or
frequently accessed very near to the root, while the inactive
records far off, in the leaf positions
 However, we do not want to rebuild the tree into the desired
shape. Instead, we need to make a tree a self-adjusting data
structure that automatically changes its shape to bring the
records closer to the root as they are used frequently, allowing
inactive records to drift slowly down towards the leaves. Such
trees are called as splay trees

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 A red-black tree is a BST with one extra bit of storage per
node: its colour, which can either be red or black

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 Every node is either red or black
 All the external nodes (leaf nodes) are black
 The rank in a tree goes from zero upto the maximum rank which
occurs at the root. The rank of two consecutive nodes differs by
utmost 1. Each leaf node has a rank 0
 If a node is red, then both its children are black. In other words,
consecutive red nodes are disallowed. This means every red
node is followed by a black node; on the other hand, a black
node may be followed by a black or a red node
 This implies that utmost 50% of the nodes on any path from
external node to root are red

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 The number of black nodes on any path from but not
including the node x to leaf is called as black height of the
node x, denoted as bh(x)
 Every simple path from the root to a leaf contains the
same number of black nodes
 In addition, every simple path from a node to a
descendent leaf contains the same number of black nodes
 If a black node has a rank r, then its parent has the rank r
+ 1
 If a red node has a rank r, then its parent will have the
rank r as well

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 A KD-tree is a data structure used in computer science
during orthogonal range searching, for instance, to find
the set of points that fall into a given rectangle

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An AA tree is a balanced BST with the following properties:
 Every node is colored either red or black
 The root is black
 If a node is red, both of its children are black
 Every path from a node to a null reference has the same
number of black nodes
 Left children may not be red

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 They eliminate half the reconstructing cases
 They simplify deletion by removing an annoying case
 If an internal node has only one child, that child must be a
red child
 We can always replace a node with the smallest child in the
right subtree; it will either be a leaf node or have a red child
 AA tree, balanced BST, supports efficient operations, since
most operations only have to traverse one or two root-to-
leaf paths

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 In each node of AA tree, we store a level. The level is
defined by the following rules:
 If a node is a leaf, its level is one
 If a node is red, its level is the level of its parent
 If a node is black, its level is one less than the level of
its parent
 Here, the level is the number of left links to a null
reference

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 A horizontal link is a connection between a node and a
child with equal levels
 The properties of such horizontal links are as follows:
 Horizontal links are right references
 There cannot be two consecutives horizontal links
 Nodes at level two or higher must have two children
 If a node has no right horizontal link, its two children
are at the same level

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
 A node of a BST has only one key value entry stored in it. A multiway
tree has many key values stored in each node and thus each node
may have multiple subtrees
 Different indexing techniques are used to search a record in O(1)
time. The index is a pair of key value and address. It is an indirect
addressing that imposes order on a file without rearranging the file
 Indexing techniques are classified as Hashed indexing, Tree
indexing, B-tree, B+ tree, Trie tree
 Splay trees are self-adjusting trees

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