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

1. Sudan University for Science and Technology
College of graduate studies
Msc in Computer Science
B-Trees
Space & time tradeoffs
Presented by
Mohamed Zeinelabdeen Abdelgader

2. Outline
 Space & time tradeoffs
 B-tree
 Definition
 Search in B-Tree
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3. Space & time tradeoffs
Two 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)
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4. B-tree
 The B-tree's 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. n[x] leaf[x]
3 FALSE
Definition 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. 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], then
k1≤ key1 [x] ≤ k2 ≤ key2 [x] ≤… ≤ keyn[x] [x] ≤
kn[x]+1
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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:
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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
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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.
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10. Search-example
Search for key R
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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.
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12. Thm: Let T be a B-tree with n>2 keys and t ≥ 2 of minimum degree
. Then the height h of the
B-tree is bounded above by

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)
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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.
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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. References
Introduction to The Design and Analysis
Algorithms