B trees

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Sudan University of science and Technology College of graduate studies College of Computer Science and Information Technology Msc in Computer Science – Software Engineering Track

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B trees

  1. 1. Sudan University for Science and Technology College of graduate studies Msc in Computer Science B-Trees Space & time tradeoffsPresented byMohamed Zeinelabdeen Abdelgader
  2. 2. Outline  Space & time tradeoffs  B-tree  Definition  Search in B-Tree 2
  3. 3. Space & time tradeoffsTwo varieties of space & time algorithms: input enhancement — preprocess the input (or its part) to store some info to be used later in solving the problem  string searching algorithms prestructuring — preprocess the input to make accessing its elements easier  indexing schemes (e.g., B-trees) 3
  4. 4. B-tree The B-trees creators, R.Bayer and E. McCreight. The most common belief is that B stands for balanced, as all the leaf nodes are at the same level in the tree. B may also stand for Bayer, or for Boeing, because they were working for Boeing Scientific Research Labs at the time. 4
  5. 5. n[x] leaf[x] 3 FALSEDefinition Q T X keyi[x] i=1..n[x] A B-tree T is a rooted tree having the following properties:  1 Every node x has the following fields  n[x], the number of keys currently stored in node x  The n[x] keys themselves stored in nondecreasing order, so that key1[x] ≤ key2[x] ≤ … ≤ keyn[x] [x]  Leaf[x],a boolean value that is TRUE if x is a leaf and FALSE if x is an internal node. 5
  6. 6. n[x] leaf[x] 3 FALSE(cont.) ci[x] Q T X keyi[x] ... … … … i=1..n[x] i=1..n[x]+1 2 Each internal node x also contains n[x]+1 pointers c1[x], c2[x],…,cn[x]+1[x] to its children. leaf nodes have no children, so their ci fields are undefined 3 The keys keyi[x] separate the ranges of keys stored in each subtree:if ki is any key stored in the subtree with root ci[x], thenk1≤ key1 [x] ≤ k2 ≤ key2 [x] ≤… ≤ keyn[x] [x] ≤ kn[x]+1 6
  7. 7. n[x] leaf[x] 3 FALSE(cont.) ci[x] Q T X keyi[x] ... … … … i=1..n[x] i=1..n[x]+1 4 All leaves have the same depth, which is the tree’s height h. 5 There are lower and upper bounds on the number of keys a node can contain.These bounds can be expressed in terms of a fixed integer t≥2 called a minimum degree of the B-tree: 7
  8. 8. n[x] leaf[x] 3 FALSE(cont.) ci[x] Q T X keyi[x] ... … … … i=1..n[x] i=1..n[x]+1  Every node other than the root must have at least t-1 keys. Every internal node other than the root thus has at least t children, If the tree is nonempty, the root must have at least one key  Every node other than the root can contain at most 2t-1 keys, therefore, an internal node can have at most 2t children. we say that a node is full if it contains exactly 2t-1 keys 8
  9. 9. Search-overview  The search operation on a B-tree is analogous to a search on a binary tree.  Instead of choosing between a left and a right child as in a binary tree, a B-tree search must make an (n[x] +1)-way choice. The correct child is chosen by performing a linear search of the values in the node. 9
  10. 10. Search-exampleSearch for key R 10
  11. 11. Search analysis  After finding the value greater than or equal to the desired value, the child pointer to the immediate left of that value is followed. If all values are less than the desired value, the rightmost child pointer is followed.  Of course, the search can be terminated as soon as the desired node is found. 11
  12. 12. Thm: Let T be a B-tree with n>2 keys and t ≥ 2 of minimum degree. Then the height h of theB-tree is bounded above by
  13. 13. Algorithm B-TREE-SEARCH(x,k) i←1 while i ≤ n[x] and k > keyi[x] do i ← i + 1 if i ≤ n[x] and k = keyi[x] then return (x, i) if leaf[x] then return NIL else Disk-Read(ci[x]) return B-Tree-Search(ci[x], k) 13
  14. 14. Algorithm analysis Input size=Number of children of node. Basic operation= Disk-Read(ci[x]) Cases:  Best case : When find the key in the current node.  T(n) Є θ(1).  Average case : When the key is not find in the current node. 14
  15. 15. Algorithm analysis  The recurrence relation:  T(n)=T(n/t)+θ(1) where t>= 2  To solve the recurrence Use the master Theorem: T(n)=aT(n/b)+f(n) where f(n) Є θ (nd) , a>=1 b>=2 d>=0 If a<bd , T(n) Є θ( (nd) If a = bd , T(n) Є θ (nd log n) If a>bd , T(n) Є θ(nlogab) a=1 , b=t , d=0 T(n) Є θ(logn) 15
  16. 16. ReferencesIntroduction to The Design and Analysis Algorithms

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