BTree

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B-Tree
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Definition

• The order of B-tree m
• Max. of descendants that a page can have
• Every page except the root and leaves
• has at least m/2 descendents, ?m/2?
• The root has at least two descendants.
• All the leaves appear on the same level.
• The leaf level forms a complete, ordered index.
• Insert a new key split (or redistribution)
• Delete a key merge or redistribution

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Variations

• Page (Node) structure
• max. m keys and max. m descendants
• max. m keys and max. (m1) descendants

m K1 D1 K2 D2 Km-1 Dm-1 Km Dm
lt K1
Km-1 lt k lt Km
m D0 K1 D1 K2 D2 Km-1 Dm-1 Km Dm
lt K1
Km-1 lt k lt Km
Km ltk
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Variations

• no Key duplication in B-tree

m D0 K1 D1 K2 D2 Km-1 Dm-1 Km Dm
lt K1
Km-1 lt k lt Km
Km lt k
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Variations

• ltKey, Record-refgt
• ltkey, refgt order? ???
• Key-only
• record ref. only in the leaf nodes

m D0 ltK1,R1gt D1 ltKm-1,Rm-1gt Dm-1
ltKm,Rmgt Dm
Km-1 lt k lt Km
Km lt k
lt K1
Data File
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variations
m D0 K0 D1 K1 D2 Km-2 Dm-1 Km-1
lt K0
Km-2 lt k lt Km-1
root
m R0 K0 R1 Km-2 Rm-1 Km-1
internal
internal
leaf
leaf
leaf
Data file
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Worst-Case Analysis

• In worst-case
• root has only 2 descendants
• minimal breadth of all the other nodes is ?m/2?
• let d be the height of a B-tree

Level 1 2 3 d
Minimum of Descendants 2 2 ?m/2? 2
?m/2?2 2 ?m/2?d-1
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Worst-case Analysis

• the upper bound for the depth of a B-tree with N
  keys
• N gt 2 ?m/2?d-1
• d lt 1 log ?m/2? (N/2)
• 1,000,000 keys, order 512 B-tree
• d lt 1 log?512/2? (1000000/2)
• d lt 3.37
• a order 512 B-tree with 1,000,000 keys has a
  depth of no more than 3 levels

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Insertion Rules

• B-tree
• max. (m-1) keys and max. m descendants
• key-only record ref. only in the leaf nodes
• duplicate keys
• Insert a key k to a node with n keys in a order m
  B-tree
• If n lt (m-1), simply insert k
• update the parent node if k is the new largest
  key of the node
• If n (m-1), (overfull)
• split the node (with m keys ) to 2 nodes
• update the parent node
• insert a new key for the new node
• update the old key (in the parent) for the old
  node if k is the new largest key of the node

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Create a B-tree

• Key sequence
• C S D T A M P I B W N G U R K E H O L J Y Q Z F X
  V
• order 4 B-tree

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Deletion Rules

• Delete a key k from a node with n keys in a order
  m B-tree
• If n gt ?m/2? , simply delete k
• update the parent node if k was the largest key
  of the node
• If n ?m/2? and (underfull)
• one of the siblings has less than ?m/2? keys,
  then
• merge the node with the siblings ( s (n-1) gt
  ?m/2? )
• delete a key from the parent node
• update the parent node if k was the largest key
  of the two nodes before merge
• one of the siblings has extra keys, then
• redistribute all the keys of the node and the
  sibling
• update the old keys (in the parent) for the two
  nodes
• sibling at the same level with the same parent

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Delete keys

• C, P, H

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Merge and Redistribute
Delete C - only can merge
Delete W - only can redistribute
Delete M - Both are possible
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Ways to redistribute

• By moving only one key from sibling
• Dividing the keys as evenly as possible
• Redistribution During insertion
• place the overflowing keys into another nodes
  (siblings) if possible
• Space utilization
• only splitting 67
• mixed over 86
• B-tree
• Every page except for the root has at least
  ?(2m-1)/3?
• How to manage (split) the initial root

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Indexed Sequential File Access and B-Tree
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ISAM

• Indexed Sequential Access Method
• Applications that need both
• sequential access of the file, and
• indexed access

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B-Tree

• Data File (Sequence Set)
• A (doubly) Linked List of Blocks
• Records in a Block are sorted by the key values
• The order of the records are not necessarily
  physically sequential throughout the file
• Maintaining Sequence Set
• split, merge, redistribute
• overflow, underflow when a block be less than
  half full
• B-Tree (Index Set)
• The leaf node (block) contains the representative
  keys (separator) of blocks in the sequence set
  and block addresses

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Shortest separator
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Prefix B-Tree

• Index Set
• contains shortest separators or prefixes of the
  keys rather than copies of the actual keys
• Variable-length
• the maximum of descendants can vary