B-tree: find: Difference between revisions

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[[Category:Videos]]
{{#ev:youtube|https://www.youtube.com/watch?v=vbRZ8h6ROYc|500|right||frame}}
[[Category:B-Tree]]
[[Category:Algorithm]]
== General Information ==
== General Information ==
'''Algorithmic problem:''' [[Sorted sequence|Sorted sequence: find]]
'''Algorithmic problem:''' [[Sorted sequence#Find|Sorted sequence: find]]


'''Type of algorithm:''' loop
'''Type of algorithm:''' loop


'''Auxiliary data:''' A pointer '''''p''''' of type "pointer to a B-tree node".
'''Auxiliary data:''' A pointer <math>p</math> of type "pointer to a B-tree node of key type <math>\mathcal{K}</math>".


== Abstract View ==
== Abstract View ==
'''Invariant:''' Before and after each iteration:
'''Invariant:''' After <math>i\geq 0</math> iterations:
# '''''p''''' points to some node '''''N''''' of the B-tree and
# pointer <math>p</math> points to some node of the B-tree on height level <math>i</math> and
# the searched key is in the [[Directed tree|range]] of '''''N'''''.
# the searched key is in the [[Directed Tree#Ranges of Search Tree Nodes|range]] of that node.


'''Variant:''' '''''p''''' is redirected from the current node '''''N''''' to some child of the current node.
'''Variant:''' <math>i</math> is increased by <math>1</math>.


'''Break condition:'''
'''Break condition:'''
# '''''p''''' points to a leaf of the B-tree or (that is, inclusive-or)
# <math>p</math> points to a leaf of the B-tree or (that is, inclusive-or)
# the searched key is in the node to which '''''p''''' points.
# the searched key is in the node to which <math>p</math> points.
 
'''Remark:''' For example, the height of the subtree rooted at the node to which <math>p</math> points may be chosen as the induction parameter. For conciseness, the induction parameter is omitted in the following.


== Induction Basis ==
== Induction Basis ==
Line 25: Line 31:


== Induction Step ==
== Induction Step ==
'''Abstract view:'''
'''Abstract view:'''
# Let '''''N''''' denote the node to which '''''p''''' currently points.
# Let <math>N</math> denote the node to which <math>p</math> currently points.
# If the searched key is in '''''N''''', terminate the algorithm and return '''''true'''''.
# If the searched key is in <math>N</math>, terminate the algorithm and return '''true'''.
# Otherwise, if '''''N''''' is a leaf, terminate the algorithm and return '''''false'''''.
# Otherwise, if <math>N</math> is a leaf, terminate the algorithm and return '''false'''.
# Otherwise, let '''''p''''' point the child of '''''N''''' such that the searched key is in the [[Directed tree|range]] of that child
# Otherwise, let <math>p</math> point the child of <math>N</math> such that the searched key is in the [[Directed Tree#Ranges of Search Tree Nodes|range]] of that child.


'''Implementation:'''
'''Implementation:'''
# If '''''K''''' is one of the values <math>p.keys[1],\dots,p.keys[p.n]</math>, terminate the algorithm and return '''''true'''''.
# If <math>K</math> is one of the values <math>p</math>.keys<math>[1],\dots,p</math>.keys<math>[p.n]</math>, terminate the algorithm and return '''true'''.
# If <math>p.children[0] = void</math> (that is, the current node is a leaf), terminate the algorithm and return '''''false'''''.
# If <math>p</math>.children<math>[0] =</math> void (that is, the current node is a leaf), terminate the algorithm and return '''false'''.
# If <math>K < p.keys[1]</math> set <math>p := p.children[p.n]</math>.
# If <math>K < p</math>.keys<math>[1]</math> set <math>p := p</math>.children<math>[0]</math>.
# Otherwise, if <math>K > p.keys[p.n]</math set <math>p := p.children[p.n]</math>.
# Otherwise, if <math>K > p</math>.keys<math>[p.n]</math> set <math>p := p</math>.children<math>[p.n]</math>.
# Otherwise, there is exactly one <math>i \in \{1,\dots,p.n-1\}</math> such that <math>p.keys[i] < K < p.keys[i+1]</math>.
# Otherwise, there is exactly one <math>i \in \{1,\dots,p.n-1\}</math> such that <math>p</math>.keys<math>[i] < K < p</math>.keys<math>[i+1]</math>.
# Set <math>p := p.children[i]</math>.
# Set <math>p := p</math>.children<math>[i]</math>.


'''Correctness:''' Obvious.
'''Correctness:'''
Obvious.


== Complexity ==
== Complexity ==


'''Statement:''' The asymptotic complexity is in <math>\Theta(\log n)</math> in the worst case.
'''Statement:''' The asymptotic complexity is in <math>\Theta(T\cdot\log n)</math> in the worst case, where <math>T</math> is the complexity of the comparison and <math>n</math> the total number of keys in the B-tree.
 
'''Proof:''' Follows immediately from the the [[B-tree#Depth of a B-tree|maximum height]] of the B-tree.


'''Proof:''' Follows immediately from the fact that the height of B-tree with '''''n''''' nodes is in <math>\Theta(\log n)</math> (cf. the remark clause of the [[B-Trees]] page).
== Pseudocode ==
<code>
B-TREE-FIND(''x,k'')
1 ''i'' = 1
2 '''while''' i &le; ''x.n'' and ''k'' &gt; ''x.key<sub>i</sub>''
3        ''i'' = ''i'' + 1
4 '''if''' ''i'' &le; ''x.n'' and ''k'' == ''x.key<sub>i</sub>''
5        '''return''' (''x.i'')
6 '''elseif''' '' x.leaf''
7        '''return''' NIL
8 '''else''' DISK-READ(''x.c<sub>i</sub>'')
9        '''return''' B-TREE-FIND(''x.c<sub>i</sub>,k'')
</code>

Latest revision as of 13:46, 3 March 2017

General Information

Algorithmic problem: Sorted sequence: find

Type of algorithm: loop

Auxiliary data: A pointer [math]\displaystyle{ p }[/math] of type "pointer to a B-tree node of key type [math]\displaystyle{ \mathcal{K} }[/math]".

Abstract View

Invariant: After [math]\displaystyle{ i\geq 0 }[/math] iterations:

  1. pointer [math]\displaystyle{ p }[/math] points to some node of the B-tree on height level [math]\displaystyle{ i }[/math] and
  2. the searched key is in the range of that node.

Variant: [math]\displaystyle{ i }[/math] is increased by [math]\displaystyle{ 1 }[/math].

Break condition:

  1. [math]\displaystyle{ p }[/math] points to a leaf of the B-tree or (that is, inclusive-or)
  2. the searched key is in the node to which [math]\displaystyle{ p }[/math] points.

Remark: For example, the height of the subtree rooted at the node to which [math]\displaystyle{ p }[/math] points may be chosen as the induction parameter. For conciseness, the induction parameter is omitted in the following.

Induction Basis

Abstract view: p is initialized so as to point to the root of the B-tree.

Implementation: Obvious.

Proof: Obvious.

Induction Step

Abstract view:

  1. Let [math]\displaystyle{ N }[/math] denote the node to which [math]\displaystyle{ p }[/math] currently points.
  2. If the searched key is in [math]\displaystyle{ N }[/math], terminate the algorithm and return true.
  3. Otherwise, if [math]\displaystyle{ N }[/math] is a leaf, terminate the algorithm and return false.
  4. Otherwise, let [math]\displaystyle{ p }[/math] point the child of [math]\displaystyle{ N }[/math] such that the searched key is in the range of that child.

Implementation:

  1. If [math]\displaystyle{ K }[/math] is one of the values [math]\displaystyle{ p }[/math].keys[math]\displaystyle{ [1],\dots,p }[/math].keys[math]\displaystyle{ [p.n] }[/math], terminate the algorithm and return true.
  2. If [math]\displaystyle{ p }[/math].children[math]\displaystyle{ [0] = }[/math] void (that is, the current node is a leaf), terminate the algorithm and return false.
  3. If [math]\displaystyle{ K \lt p }[/math].keys[math]\displaystyle{ [1] }[/math] set [math]\displaystyle{ p := p }[/math].children[math]\displaystyle{ [0] }[/math].
  4. Otherwise, if [math]\displaystyle{ K \gt p }[/math].keys[math]\displaystyle{ [p.n] }[/math] set [math]\displaystyle{ p := p }[/math].children[math]\displaystyle{ [p.n] }[/math].
  5. Otherwise, there is exactly one [math]\displaystyle{ i \in \{1,\dots,p.n-1\} }[/math] such that [math]\displaystyle{ p }[/math].keys[math]\displaystyle{ [i] \lt K \lt p }[/math].keys[math]\displaystyle{ [i+1] }[/math].
  6. Set [math]\displaystyle{ p := p }[/math].children[math]\displaystyle{ [i] }[/math].

Correctness: Obvious.

Complexity

Statement: The asymptotic complexity is in [math]\displaystyle{ \Theta(T\cdot\log n) }[/math] in the worst case, where [math]\displaystyle{ T }[/math] is the complexity of the comparison and [math]\displaystyle{ n }[/math] the total number of keys in the B-tree.

Proof: Follows immediately from the the maximum height of the B-tree.

Pseudocode

B-TREE-FIND(x,k)
1 i = 1
2 while i ≤ x.n and k > x.keyi 
3        i = i + 1
4 if ix.n and k == x.keyi 
5        return (x.i)
6 elseif  x.leaf
7        return NIL
8 else DISK-READ(x.ci) 
9        return B-TREE-FIND(x.ci,k)