Mumbai University > Information Technology > Sem 5 > Advanced Database Management System

Marks: 5M

Year: Dec 2014

• A multilevel index considers the index file, which we will now refer to as the first (or base) level of a multilevel index, as an ordered file with a distinct value for each K(i).
• Because a single-level index is an ordered file, we can create a primary index to the index itself;  In this case, the original index file is called the first-level index and the index to the index is called the second-level index.
• We can repeat the process, creating a third, fourth ... top level until all entries of the top level fit in one disk block
• A multi-level index can be created for any type of firstlevel index (primary, secondary, clustering) as long as the first-level index consists of more than one disk block
• However, insertion and deletion of new index entries is a severe problem because every level of the index is an ordered file

• Multilevel indexing improve the efficiency of searching an index file in following way:

  1) A multilevel index reduces the number of blocks accessed when searching for a record, given its indexing field value.

  2) To retain the benefits of using multilevel indexing while reducing index insertion and deletion problems,

  3) Designers adopted a multilevel index that leaves some space in each of its blocks for inserting new entries. This is called a dynamic multilevel index and is often implemented by using data structures called B-trees and B+-trees.

B+ Tree

A B+ tree is a balanced binary search tree that follows a multi-level index format. The leaf nodes of a B+ tree denote actual data pointers. B+ tree ensures that all leaf nodes remain at the same height, thus balanced. Additionally, the leaf nodes are linked using a link list; therefore, a B+ tree can support random access as well as sequential access.

Structure of B+ Tree

Every leaf node is at equal distance from the root node. A B+ tree is of the order n where nis fixed for every B+ tree.

Internal nodes

• Internal (non-leaf) nodes contain at least ⌈n/2⌉ pointers, except the root node.
• At most, an internal node can contain n pointers.

Leaf nodes

• Leaf nodes contain at least ⌈n/2⌉ record pointers and ⌈n/2⌉ key values.
• At most, a leaf node can contain n record pointers and n key values.
• Every leaf node contains one block pointer P to point to next leaf node and forms a linked list.

B+ Tree Insertion

• B+ trees are filled from bottom and each entry is done at the leaf node.
• If a leaf node overflows −
  • Split node into two parts.
  • Partition at i = ⌊(m+1)/2⌋.
  • First i entries are stored in one node.
  • Rest of the entries (i+1 onwards) are moved to a new node.
  • ith key is duplicated at the parent of the leaf.
• If a non-leaf node overflows −
  • Split node into two parts.
  • Partition the node at i = ⌈(m+1)/2⌉.
  • Entries up to i are kept in one node.
  • Rest of the entries are moved to a new node.

B+ Tree Deletion

• B+ tree entries are deleted at the leaf nodes.

• The target entry is searched and deleted.

  If it is an internal node, delete and replace with the entry from the left position.

• After deletion, underflow is tested,

• If distribution is not possible from left, then

  Distribute from the nodes right to it.

• If distribution is not possible from left or from right, then

  Merge the node with left and right to it.