B+-Tree Index Files
o B+-tree indices are an alternative to indexed-sequential files.
o Disadvantage of indexed-sequential files: performance degrades as file grows, since many overflow blocks get created. Periodic reorganization of entire file is required.
o 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. o Disadvantage of B+-trees: extra insertion and deletion overhead, space overhead.
o Advantages of B+-trees outweigh disadvantages, and they are used extensively.
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
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.
o Ki are the search-key values
o 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.
Leaf Nodes in B+-Trees
o Properties of a leaf node:
For i = 1, 2, . . ., n–1, pointer Pi either points to a filerecord with search-key value Ki, or to a bucket ofpointers to file records, each record having search-keyvalue Ki. Only need bucket structure if search-key doesnot form a primary key. If Li, Lj are leaf nodes and i < j, Li‘s search-key values are less than Lj‘s search-key values. Pn points to next leaf node in search-key order.
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:
o Al l the search-keys in the subtree to which P1 points are less than K1.
o 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 Km–1. Example of a B+-tree
B+-tree for account file (n = 3) B+-tree for account file (n - 5)
o Leaf nodes must have between 2 and 4 v alues ( (n–1)/2 and n –1, with n = 5).
o Non-leaf nodes other than root must have between 3 and 5 children ( (n/2 and n with n =5). o Root must have at least 2 children.
Observations about B+-trees
o Since the inter-node connections are done by pointers, ―logically‖ close blocks need not be ―physically‖ close.
o Th e non-leaf levels of the B+-tree form a hierarchy of sparse indices.
o The B+-tree contains a relatively small number of levels (logarithmic in the size of the main file), thus searches can be conducted efficiently.
o Insertions and deletions to the main file can be handled efficiently, as the index can be restructured in logarithmic time.
Queries on B+-Trees
Find all records with a search-key value of k. o Start with the root node
Examine the node for the smallest search-key value > k.
If such a value exists, assume it is Kj. Then follow Pi to the child node.
Otherwise k Km–1, where there are m pointers in the node. Then follow Pm to the child node. o If the node reached by following the pointer above is not a leaf node, repeat the above procedure on the node, and follow the corresponding pointer.
o Eventually reach a leaf node. If for some i, key Ki = k follow pointer Pi to the desired record or bucket. Else no record with search-key value k exists.
Result of splitting node containing Brighton and Downtown on inserting NOTES Clearview
B+-Tree before and after insertion of ―Clearview‖ 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
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
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.
Otherwise, 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
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 whichh as 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.
Examples of B+-Tree Deletion Before and after deleting ―Downtown‖
o The removal of the leaf node containing ―Downtown‖ did not result in its parent having too little
pointers. So the cascaded deletions stopped with the deleted leaf node‘s parent.
B+-Tree File Organization
o Index file degradation problem is solved by using B+-Tree indices. Data file degradation problem is solved by using B+-Tree File Organization.
o The leaf nodes in a B+-tree file organization store records, instead of pointers.
o 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.
Leaf nodes are still required to be half full.
Insertion and deletion are handled in the same way as insertion and deletion of entries in a B+-tree index
Example of B+-tree File Organization
o Good space utilization is important since records use more space than pointers.
o To improve space utilization, involve more sibling nodes in redistribution during splits and merges.
Involving 2 siblings in redistribution (to avoid split / merge where possible) results in each node having at least entries
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