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B-Tree

B-Tree. Multiway search tree. A tree was defined as either an empty structure or a structure whose children are disjoint tree t 1 ,t 2 ,…t k . Each node of this kind of tree can have more than two children. This tree called a multiway tree of order m, or an m-way tree.

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B-Tree

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  1. B-Tree

  2. Multiway search tree • A tree was defined as either an empty structure or a structure whose children are disjoint tree t1,t2,…tk. Each node of this kind of tree can have more than two children. This tree called a multiway tree of order m, or an m-way tree. • A Multiway search tree of order m, or an m-way search tree, is a Multiway search tree in which • Each node has m children and m-1 keys • The keys in each node are in ascending order • The keys in the first i children are smaller the ith key • The keys in the lastm-i children are larger than the ith key • M-way search tree  m-way tree • Binary search tree  binary tree

  3. Introduction of B-tree • B-tree: proposed by Bayer and McCreight 1972 • A B-tree operates closely with secondary storage and can be tuned to reduce the impediments imposed by this storage • One important property of B-trees is the size of each node which can be made as large as the size of the block. (the basic unit of I/O operations associated with a disk is a block) • a B-tree of order t is a multiway search tree.

  4. Main Memory Secondary Memory ( RAM ) ( disks ) • B-Tree : • 性質： • 典型之 B-Tree 運用情況： 在 B-Tree 中若 node x 有 n[x] keys, 則 x 含有 n[x]+1 個 children. M D H Q T X B C J K L F G N P R S Y Z V W 資料量龐大無法全部存在 main memory. 處理 B-Tree 之 algorithm 只將部分之資料 copy 至 main memory.

  5. x key1 key2 C1[x] C3[x] C2[x] K1 K2 K3 • B-Tree T 之定義： T：rooted tree 並滿足下列： • 1. 對每一 node x 含下列之 fields • a. n[x]：node x 所含之 key 數. • b. 依遞增排列. • c. leaf[x] 為 true 若 x 為 leaf false 若 x 不為 leaf. • 2. 若 x 為一 internal node 則 x 有 n[x]+1 個 pointers C1[x] , C2[x] ,… , Cn[x]+1[x] 指向某 children. 若 x 為一 leaf. 其 Ci[x] 沒有定義. • 3.若 ki 為 Ci[x] 所指之 subtree 中任一 key 則

  6. 4. 每一 leaf 之高度相同.(all leaves are on the same level.) • 5. t：minimum degree of the B-tree. (a B-tree of order 2t) • a. node( 除 root ) 有 internal node ( 除 root ) 有 若 則 root 至少有一 key. • b. node( 至多 ) 有 internal node ( 除 root ) 有 a node is FULL, if it contains exactly 2t-1 keys.

  7. If , then for any n-key B-tree T of height h and minimum degree • Thm： Proof： 1 1 2 t-1 t-1 t t 2t t-1 t-1 t-1 t-1 t t t t 2t2

  8. A B-tree is not a binary tree because B-tree has many more than two children • B-trees may be formulated to store a set of elements or a bag of elements. (a given elements can occur many times in a bag but only once in a set) • A B-tree is balanced. • Every leaf in a B-tree has the same depth • 2-3-4 tree (discussed by Rudolf Bayer): a B-tree of order 4 (min degree=2)

  9. The elements in a B-tree node • Rule 1:the root may have as few as one elements (or even no elements if it also has no children); every other node has at least minimum elements • Rule 2: the maximum number of elements in a node is twice the value of minimum • Rule 3: the elements of each B-tree node are stored in a partially filled array, sorted from the smallest elements (at index 0) to the largest elements (at the final used position of the array) • Rule 4: the number of subtreesbelow a nonleaf node is always one more than the number of the elements in the node. • Rule 5: for any leaf node: (1) an element at index i is greater than all the elements in subtree number i of the node, and (b) an element at index i is less than all the elements in subtree number i+1 of the node.

  10. 66 88 Subtree number 1 Subtree number 2 Subtree number 0 Each element in subtree number 1 is between 66 and 88. Each element in subtree number 2 is greater than 88 Each element in subtree number 0 is less than 66

  11. convention： • Root of the B-tree is always in main memory. • Any nodes that are passed as parameters must already have had a DISK_READ operation performed on them. • Operations： • Searching a B-Tree. • Creating an empty B-tree. • Splitting a node in a B-tree. • Inserting a key into a B-tree. • Deleting a key from a B-tree.

  12. B-Tree-Search(x,k)： • Algorithm： • Total CPU time： B-Tree-Search(x,k) { }

  13. B-Tree-Created(T)： • Algorithm： • time： B-Tree-Create(T) { }

  14. P Q R T U V T3 T4 T1 T2 T7 T8 T5 T6 • B-Tree-Split-Child(x,i,y)： • Splitting a node in a B-Tree： Keyi-1[x] Keyi [x] x t=4 … N S W … … N W … y=Ci[x] z=Ci+1[x] y=Ci[x] P Q R S T U V full T3 T4 T7 T8 T1 T2 T5 T6 Splitting a full node y ( have 2t-1 keys ) around its median key keyt[y] into 2 nodes having (t-1) keys each.

  15. B-Tree-Split-Child(x,I,y) • Algorithm： { }

  16. A D F L N P T3 T4 T1 T2 T7 T8 T5 T6 • B-Tree-Insert(T,k)： • Insert a key in a B-Tree： Root[T] t=4 S Root[T] H r r A D F H L N P T3 T4 T7 T8 T1 T2 T5 T6

  17. Algorithm： B-Tree-Insert(T,k) { { B-Tree-Split-Child(S,l,r) B-Tree-Insert-Nonfull(S,k) } }

  18. B-Tree-Insert-Nonfull(x,k) { • B-Tree-Insert-Nonfull(x,k)： • Algorithm： { { B-Tree-Insert-Nonfull(Ci[x],k) } }

  19. t=3 (a) Initial tree • Example：Inserting keys into a B-Tree. G M P X A C D E J K N O R S T U V Y Z (b) B inserted G M P X G M P T X A B C D E J K N O R S T U V Y Z R S U V (c) Q inserted G M P T X A B C D E J K N O Q R S U V Y Z

  20. (d) L insert P G M T X A B C D E J K L N O Q R S U V Y Z (e) F insert P C G M T X A B J K L N O Q R S U V Y Z D E F

  21. ( x has t keys ) x • Deleting a key from a B-Tree： 1. K is in x and x is a leaf： 2. K is in x and x is an internal node： a. b. c. if both y,z has t-1 keys. Merge y,z and k into y. Recursively delete k from y. K delete k from x. x Recursively delete k’ and replace k by k’ in x. K y k’ x K z k’ x K z y t+1 t+1 2t-1 K

  22. x K Ci[x] 3. If K is not in internal node x： a. If Ci[x] has only t-1 keys but has a sibling with t keys b. If Ci[x] and all of Ci[x]’s siblings have t-1 keys, merge ci with one sibling. k is in this subtree. x • Move a key from x down to Ci[x]. • Move a key from Ci[x]’s sibling to x. • Move an appropiate child to Ci[x] from its sibling. K Ci[x] t-1 t x 0 Ci[x] t-1 keys t-1 keys 2t-1 keys 0

  23. t=3 (a) Initial tree P • Example：Deleting a key from a B-Tree. C G M T X A B J K L N O Q R S U V Y Z D E F (b) F delete：case 1 P C G M T X A B J K L N O Q R S U V Y Z D E (c) M delete：case 2a P C G L T X A B J K N O Q R S U V Y Z D E

  24. (d) G deleted：case 2c P C L T X A B N O D E J H Q R S U V Y Z (e) D deleted：case 3b C L P T X N O Q R S U V Y Z A B E J K (e’) tree shrinks in height C L P T X N O Q R S U V Y Z A B E J K (f) B delete：case 3a E L P T X N O Q R S U V Y Z A C J K

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