B-tree: Difference between revisions

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'''Implementation Invariant:'''
'''Implementation Invariant:'''
# There is a number <math>M\in\mathbb{N}</math>, <math>M>1</math>, which is specific for the B-tree object and is constant throughout the life time of the B-tree object. This value <math>M</math> is called the '''order''' of the B-tree.
# There is a number <math>M\in\mathbb{N}</math>, <math>M>1</math>, which is specific for the B-tree object and is constant throughout the life time of the B-tree object. This value <math>M</math> is called the '''order''' of the B-tree.
# A B-tree of order <math>M</math> is a [[Multi-way search tree|multi-way search tree]] with <math>2M-1</math> slots for keys and, consequently, <math>2M</math> slots for children pointers, in each node.
# A B-tree of order <math>M</math> is a [[Directed Tree#Multi-way Search Tree|multi-way search tree]] with <math>2M-1</math> slots for keys and, consequently, <math>2M</math> slots for children pointers, in each node.
## The slots for keys are denoted keys<math>[1],\ldots,</math>keys<math>[2M\!-\!1]</math>.
## The slots for keys are denoted keys<math>[1],\ldots,</math>keys<math>[2M\!-\!1]</math>.
## The slots for children are denoted children<math>[1],\ldots,</math>children<math>[2M]</math>.
## The slots for children are denoted children<math>[1],\ldots,</math>children<math>[2M]</math>.
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# The filled key slots are the ones at positions <math>1,\ldots,n</math>; the filled children slots are the ones at positions <math>0,\ldots,n</math> (except for leaves, of course, where no child slot is filled at all).
# The filled key slots are the ones at positions <math>1,\ldots,n</math>; the filled children slots are the ones at positions <math>0,\ldots,n</math> (except for leaves, of course, where no child slot is filled at all).
# The keys appear in ascending order in a B-tree node, that is, keys<math>[i]<</math> keys<math>[i\!+\!1]</math> for all <math>i\in\{1,\ldots,n-1\}</math>.
# The keys appear in ascending order in a B-tree node, that is, keys<math>[i]<</math> keys<math>[i\!+\!1]</math> for all <math>i\in\{1,\ldots,n-1\}</math>.
# The children pointers appear in the order as used in the definition of [[Multi-way search tree|multi-way search trees]]. In other words, for , the [[Directed Tree#Range|range]] of the node to which points is
# The children pointers appear in the order as used in the definition of [[Directed Tree#Multi-way Search Tree|multi-way search trees]]. In other words, for <math>i\in\{0,\ldots,n\}</math>, the [[Directed Tree#Range|range]] of the node to which childrenƒmulti points is
## for ;
## for ;
## for ;
## for ;
## for .
## for .
# All leaves of a B-tree are on the same height level.
# All leaves of a B-tree are on the same height level.

Revision as of 11:06, 26 May 2015


Abstract data structure: Sorted sequence

Implementation Invariant:

  1. There is a number [math]\displaystyle{ M\in\mathbb{N} }[/math], [math]\displaystyle{ M\gt 1 }[/math], which is specific for the B-tree object and is constant throughout the life time of the B-tree object. This value [math]\displaystyle{ M }[/math] is called the order of the B-tree.
  2. A B-tree of order [math]\displaystyle{ M }[/math] is a multi-way search tree with [math]\displaystyle{ 2M-1 }[/math] slots for keys and, consequently, [math]\displaystyle{ 2M }[/math] slots for children pointers, in each node.
    1. The slots for keys are denoted keys[math]\displaystyle{ [1],\ldots, }[/math]keys[math]\displaystyle{ [2M\!-\!1] }[/math].
    2. The slots for children are denoted children[math]\displaystyle{ [1],\ldots, }[/math]children[math]\displaystyle{ [2M] }[/math].
  3. The attribute [math]\displaystyle{ n }[/math] of a node stores the number of filled key slots in this node.
  4. In each tree node except for the root, it is [math]\displaystyle{ n\geq M-1 }[/math]; for the root, it is [math]\displaystyle{ n\geq 1 }[/math].
  5. The filled key slots are the ones at positions [math]\displaystyle{ 1,\ldots,n }[/math]; the filled children slots are the ones at positions [math]\displaystyle{ 0,\ldots,n }[/math] (except for leaves, of course, where no child slot is filled at all).
  6. The keys appear in ascending order in a B-tree node, that is, keys[math]\displaystyle{ [i]\lt }[/math] keys[math]\displaystyle{ [i\!+\!1] }[/math] for all [math]\displaystyle{ i\in\{1,\ldots,n-1\} }[/math].
  7. The children pointers appear in the order as used in the definition of multi-way search trees. In other words, for [math]\displaystyle{ i\in\{0,\ldots,n\} }[/math], the range of the node to which childrenƒmulti points is
    1. for ;
    2. for ;
    3. for .
  8. All leaves of a B-tree are on the same height level.