B+ trees

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By eTech(mr.erfan@gmail.com)[edit]

In computer science, a B+ tree (BplusTree) is a type of tree which represents sorted data in a way that allows for efficient insertion, retrieval and removal of records, each of which is identified by a key. It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a "block" or "node"). In a B+ tree, in contrast to a B-tree, all records are stored at the leaf level of the tree; only keys are stored in interior nodes.

The primary value of a B+ tree is in storing data for efficient retrieval in a block-oriented storage context. This is primarily because unlike binary search trees, B+ trees have very high fanout (typically on the order of 100 or more), which reduces the number of I/O operations required to find an element in the tree.

ReiserFS, NSS, XFS, JFS2, and NTFS filesystems all use this type of tree for metadata indexing. Relational databases such as PostgreSQL and MySQL also often use this type of tree for table indices.

Contents [hide] 1 Details 1.1 Search 1.2 Insertion 2 Characteristics 3 Implementation 4 History 5 See also 6 External links

[edit] Details The order of a B+ tree measures the capacity of nodes (i.e. the number of children nodes) in the tree. It is defined as a number d such that , where m is the number of children in each node. For example, if the order of a B+ tree is 7, each internal node (except for the root) may have between 4 and 7 children; the root may have between 2 and 7.

[edit] Search The algorithm to perform a search for a record r follows pointers to the correct child of each node until a leaf is reached. Then, the leaf is scanned until the correct record is found (or until failure).

function search(record r)
  u := root
  while (u is not a leaf) do
    choose the correct pointer in the node
    move to the first node following the pointer
    u := current node
  scan u for r

This pseudocode assumes that no repetition is allowed.

[edit] Insertion do a search to determine what bucket the new record should go in if the bucket is not full, add the record. otherwise, split the bucket. allocate new leaf and move half the bucket's elements to the new bucket insert the new leaf's smallest key and address into the parent. if the parent is full, split it also now add the middle key to the parent node repeat until a parent is found that need not split if the root splits, create a new root which has one key and two pointers.

[edit] Characteristics For a b-order B+ tree with h levels of index:

The maximum number of records stored is n = bh The minimum number of keys is 2(b / 2)h − 1 The space required to store the tree is O(n) Inserting a record requires O(logbn) operations in the worst case Finding a record requires O(logbn) operations in the worst case Removing a (previously located) record requires O(logbn) operations in the worst case Performing a range query with k elements occurring within the range requires O(logbn + k) operations in the worst case.

[edit] Implementation The leaves (the bottom-most index blocks) of the B+ tree are often linked to one another in a linked list; this makes range queries simpler and more efficient (though the aforementioned upper bound can be achieved even without this addition). This does not substantially increase space consumption or maintenance on the tree.

If a storage system has a block size of B bytes, and the keys to be stored have a size of k, arguably the most efficient B+ tree is one where b = (B / k) − 1. Although theoretically the one-off is unnecessary, in practice there is often a little extra space taken up by the index blocks (for example, the linked list references in the leaf blocks). Having an index block which is slightly larger than the storage system's actual block represents a significant performance decrease; therefore erring on the side of caution is preferable.

If nodes of the B+ tree are organised as arrays of elements, then it may take a considerable time to insert or delete an element as half of the array will need to be shifted on average. To overcome this problem elements inside a node can be organized in a binary tree or a B+ tree instead of an array.

B+ trees can also be used for data stored in RAM. In this case a reasonable choice for block size would be the size of processor's cache line. However some studies have proved that a block size few times larger than processor's cache line can deliver better performance if cache prefetching is used.

Space efficiency of B+ trees can be improved by using some compression techniques. One possibility is to use delta encoding to compress keys stored into each block. For internal blocks, space saving can be achieved by either compressing keys or pointers. For string keys, space can be saved by using the following technique: Normally the ith entry of an internal block contains the first key of block i+1. Instead of storing the full key, we could store the shortest prefix of the first key of block i+1 that is strictly greater (in lexicographic order) than last key of block i. There is also a simple way to compress pointers: if we suppose that some consecutive blocks i, i+1...i+k are stored contiguously, then it will suffice to store only a pointer to the first block and the count of consecutive blocks.

All the above compression techniques have some drawbacks. First, a full block must be decompressed to extract a single element. One technique to overcome this problem is to divide each block into sub-blocks and compress them separately. In this case searching or inserting an element will only need to decompress or compress a sub-block instead of a full block. Another drawback of compression techniques is that the number of stored elements may vary considerably from a block to another depending on how well the elements are compressed inside each block.

[edit] History The B tree was first described in the paper Organization and Maintenance of Large Ordered Indices. Acta Informatica 1: 173–189 (1972) by Rudolf Bayer and Edward M. McCreight. There is no single paper introducing the B+ tree concept. Instead, the notion of maintaining all data in leaf nodes is repeatedly brought up as an interesting variant. An early survey of B trees also covering B+ trees is Douglas Comer: "The Ubiquitous B-Tree", ACM Computing Surveys 11(2): 121–137 (1979). Comer notes that the B+ tree was used in IBM's VSAM data access software and he refers to an IBM published article from 1973.