![Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
15
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](https://image.slidesharecdn.com/13indexingmtrees-150824112416-lva1-app6892/85/13-Indexing-MTrees-Data-Structures-using-C-by-Varsha-Patil-15-320.jpg)
![Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil
19
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](https://image.slidesharecdn.com/13indexingmtrees-150824112416-lva1-app6892/85/13-Indexing-MTrees-Data-Structures-using-C-by-Varsha-Patil-19-320.jpg)
13. Indexing MTrees - Data Structures using C++ by Varsha Patil
- 1. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 1
- 2. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 2 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
- 3. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 3 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
- 4. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 4 The file organization breaks down into two more aspects: Directory—for collection of indices File organization—for the physical organization of records
- 5. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 5 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
- 6. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 6 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
- 7. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 7 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
- 8. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 8 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
- 9. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 9 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
- 10. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 10 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
- 11. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 11 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
- 12. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 12 The operations related to hashed indexes are the same as those for hash tables
- 13. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 13 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
- 14. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 14 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
- 15. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 15 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
- 16. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 16 Ptr1 Key1 Ptr2 Key2 Ptri Keyi …….. Keyn-1 Ptrn X XX X<Key1 Keyi-1<X<Keyi X>Keyn-1
- 17. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 17 Search a node Insertion of a key into a B-tree Deletion from a B-tree
- 18. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 18 B+ trees are internal data structures That is, the nodes contain whatever information is associated with the key as well as the key values
- 19. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 19 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
- 20. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 20 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
- 21. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 21 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
- 22. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 22 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)
- 23. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 23 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
- 24. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 24 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
- 25. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 25 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
- 26. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 26 A red-black tree is a BST with one extra bit of storage per node: its colour, which can either be red or black
- 27. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 27 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
- 28. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 28 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
- 29. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 29 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
- 30. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 30 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
- 31. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 31 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
- 32. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 32 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
- 33. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 33 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
- 34. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 34 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
- 35. Oxford University Press © 2012Data Structures Using C++ by Dr Varsha Patil 35