176 lines
6.2 KiB
Markdown
176 lines
6.2 KiB
Markdown
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---
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title: B-Tree
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TARGET DECK: Obsidian::STEM
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FILE TAGS: data_structure::b-tree
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tags:
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- b-tree
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- data_structure
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---
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## Overview
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A **B-tree of order $m$** is a tree that satisfies the following properties:
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* Every node has at most $m$ children.
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* Every node, except for the root, has at least $m / 2$ children.
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* All leaves appear on the same level.
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* A node with $k$ children contains $k - 1$ keys sorted in monotonically increasing order.
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The above is a modification of Knuth's definition in his "Art of Computer Programming" that defines leaves of the tree more consistently with how I use the term elsewhere. It also pulls in concepts from CLRS (such as keys needing to be sorted within nodes).
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%%ANKI
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Basic
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What hyperparameter is used to define a B-tree?
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Back: It's order, i.e. the maximum number of a children a node can have.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211541967-->
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END%%
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%%ANKI
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Basic
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In what direction do B-trees grow?
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Back: From bottom to top.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211541998-->
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END%%
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%%ANKI
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Basic
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Consider B-tree of order $m$. What does $m$ refer to?
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Back: The maximum number of children a node can have.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542004-->
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END%%
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%%ANKI
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Basic
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What is the maximum number of children a node in a B-tree have?
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Back: N/A. It depends on the tree's order.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542010-->
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END%%
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%%ANKI
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Basic
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What is the maximum number of children a node in a B-tree of order $m$ can have?
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Back: $m$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542016-->
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END%%
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%%ANKI
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Basic
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What is the minimum number of children a non-root node in a B-tree of order $m$ can have?
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Back: $\lceil m / 2 \rceil$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542022-->
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END%%
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%%ANKI
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Basic
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What is the maximum number of keys a node in a B-tree of order $m$ can have?
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Back: $m - 1$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542028-->
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END%%
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%%ANKI
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Basic
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What is the minimum number of keys a non-root node in a B-tree can have?
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Back: N/A. It depends on the tree's order.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542035-->
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END%%
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%%ANKI
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Basic
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What is the minimum number of keys a non-root node in a B-tree of order $m$ can have?
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Back: $\lceil m / 2 \rceil - 1$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542041-->
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END%%
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%%ANKI
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Basic
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A node in a B-tree of order $m$ has $k$ keys. How many children does it have?
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Back: $k + 1$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542046-->
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END%%
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%%ANKI
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Basic
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A node in a B-tree of order $m$ has $k$ children. How many keys does it have?
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Back: $k - 1$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542052-->
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END%%
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%%ANKI
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Basic
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*When* does a B-tree gain height?
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Back: When the root node is split.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542058-->
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END%%
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%%ANKI
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Basic
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Consider a B-tree of order $7$. How many children $c$ can each non-root node have?
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Back: $4 \leq c \leq 7$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542063-->
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END%%
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%%ANKI
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Basic
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Consider a B-tree of order $7$. How many children $c$ can the root have?
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Back: $0 \leq c \leq 7$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542069-->
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END%%
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%%ANKI
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Basic
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Consider a B-tree of order $7$. How many keys $k$ can each non-root node have?
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Back: $3 \leq k < 7$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542076-->
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END%%
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%%ANKI
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Basic
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Consider a B-tree of order $7$. How many keys $k$ can the root have?
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Back: $0 \leq k < 7$
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542082-->
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END%%
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%%ANKI
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Basic
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What instances exist of a B-tree of order $1$?
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Back: Just the empty tree.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542088-->
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END%%
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%%ANKI
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Basic
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*Why* can't we define a nonempty B-tree of order $1$?
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Back: Each node can have at most $1$ child, meaning each node contains $0$ keys.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542094-->
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END%%
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%%ANKI
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Basic
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How are keys arranged within a B-tree's nodes?
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Back: In monotonically increasing order.
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Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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<!--ID: 1723211542105-->
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END%%
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## Bibliography
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* Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Searching_, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
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* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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