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B-Trees. Katherine Gurdziel 252a-ba. Outline. What are b-trees? How does the algorithm work? Insertion Deletion Complexity What are b-trees used for? LEDA for AVL and Red Black trees. Description. B-trees are balanced search trees with one root B-tree nodes can have multiple children

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B trees

B-Trees

Katherine Gurdziel

252a-ba


Outline
Outline

  • What are b-trees?

  • How does the algorithm work?

    • Insertion

    • Deletion

  • Complexity

  • What are b-trees used for?

  • LEDA for AVL and Red Black trees


Description
Description

  • B-trees are balanced search trees with one root

  • B-tree nodes can have multiple children

  • Every leaf has the same depth - the trees height

  • There are lower and upper bounds on the number of elements a leaf can contain


Example b tree
Example B-Tree

  • Bounds on nodes can be expressed with a fixed integer t

  • Every node other than the root must have at least t-1 keys, internal node has at least t children

  • Every node can have at most 2t-1 children


Insertion
Insertion

T = 3 so a node can hold at most 5 keys.



Complexity
Complexity

  • If n >= 1, then for any n-key B-tree T of height h and minimum degree t >= 2:

    h <= logt (n+1)/2

  • Like red-black trees the height of B-trees grows as O(lg n) but for B-trees the base of the logarithm can be many times larger.

  • B-trees save a factor of about lg t over red-black trees in the number of nodes examined for most tree operations.


Complexity of procedures
Complexity of Procedures

  • Searching a B-tree is similar to searching a binary search tree except that a multiway branching decision is made at each node according to the number of the node’s children. O(t logt n)

    It is the same for insertion and deletion.

  • Splitting a node O(t logt n)


Accessing secondary storage
Accessing Secondary Storage

  • Primary memory storage is more expensive than secondary, so secondary storage for computers often exceeds the amount of primary by several orders of magnitude.

  • The access time for a page may be large while the time to read the information once it is accessed is small.

  • System can only keep a limited amount of information in main memory at any time.

  • The algorithm for B-trees is well suited to efficiently access secondary storage.


Suitability of b trees
Suitability of B-trees

  • A large branching factor reduces the height of the tree and the number of disk accesses required to find any page

  • A B-tree can copy selected pages from disk to and from memory as needed so the size of the tree is not dependent on the size of main memory

  • The number of disk accesses required on a B-tree is proportional to the tree’s height.

  • For example, a B-tree with a branching factor of 1001 and height 2 stores over one billion pages and since the root is kept permanently in main memory only two disk accesses are required to find any page



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