449 lines
17 KiB
Markdown
449 lines
17 KiB
Markdown
---
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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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Why is a B-tree named the way it is?
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Back: There is no definitive answer.
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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: 1723289256280-->
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END%%
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%%ANKI
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Basic
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What was the motivation behind the development of the B-tree?
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Back: To find a data structure for efficient search that minimizes disk accesses.
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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: 1723289256283-->
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END%%
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%%ANKI
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Basic
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How is the order of a B-tree typically decided?
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Back: By choosing a value that best aligns with the size of a memory block.
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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: 1723289256285-->
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END%%
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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: A number equal to 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 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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Consider a B-tree of order $7$. How many children $c$ can each internal 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 internal 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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%%ANKI
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Basic
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What is the search runtime of a B-tree of order $m$ and height $h$?
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Back: $O(mh)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489725-->
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END%%
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%%ANKI
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Basic
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*Why* does searching a B-tree of order $m$ and height $h$ take $O(mh)$ time?
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Back: Each node may have $m - 1$ keys, and we may check $h$ nodes.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489726-->
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END%%
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%%ANKI
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Basic
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How many disk accesses are performed when searching a B-tree of order $m$ and height $h$?
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Back: $O(h)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489727-->
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END%%
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%%ANKI
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Basic
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*Why* does the number of disk accesses when searching a B-tree of height $h$ equal $O(h)$?
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Back: The size of each node presumably corresponds to a block of memory.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489728-->
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END%%
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%%ANKI
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Basic
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What is the search runtime of a B-tree of order $m$ containing $n$ keys?
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Back: $O(m\log_m{n})$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489729-->
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END%%
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%%ANKI
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Basic
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*Why* does searching a B-tree of order $m$ containing $n$ keys take $O(m\log_m{n})$ time?
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Back: Each node may have $m - 1$ keys, and we may check $\log_m{n}$ nodes.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489730-->
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END%%
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%%ANKI
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Basic
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How many disk accesses are performed when searching a B-tree of order $m$ containing $n$ keys?
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Back: $O(\log_m{n})$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489731-->
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END%%
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%%ANKI
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Basic
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*Why* does the number of disk accesses when searching a B-tree of order $m$ containing $n$ keys equal $O(\log_m{n})$?
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Back: The size of each node presumably corresponds to a block of memory.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489732-->
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END%%
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## Insertions
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A node of a B-tree of order $m$ is considered full when it has $m$ children (or equivalently $m - 1$ keys). Insertion operates analagously to a binary tree. If the node the key was inserted into then contains $m$ keys, split the node into two and place the median into the original parent node. This action may propagate upwards. If the root node becomes full, create a new root containing the median of the original root.
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%%ANKI
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Cloze
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A node in a B-tree of order $m$ is considered full when it has {$m - 1$} 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: 1723321489733-->
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END%%
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%%ANKI
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Cloze
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A node in a B-tree of order $m$ is considered full when it has {$m$} children.
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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: 1723321489734-->
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END%%
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%%ANKI
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Basic
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Which key(s) found in B-trees move levels during node splits?
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Back: The split node's median key.
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Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489735-->
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END%%
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%%ANKI
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Basic
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What does it mean for a B-tree split to be left-biased?
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Back: Prefer the median on the LHS.
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Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489736-->
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END%%
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%%ANKI
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Basic
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What does it mean for a B-tree split to be right-biased?
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Back: Prefer the median on the RHS.
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Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321615984-->
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END%%
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%%ANKI
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Basic
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Consider splitting a node in a B-tree of order $m$. How many keys are in the split nodes?
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Back: $\lfloor (m - 1) / 2 \rfloor$ and $\lceil (m - 1) / 2 \rceil$.
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Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489737-->
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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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Suppose the following B-tree node is full. What is the result after splitting?
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![[b-tree-full-node.png]]
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Back:
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![[b-tree-split-node.png]]
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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: 1723321489738-->
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END%%
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%%ANKI
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Basic
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Consider the following B-tree. What is the result of inserting `B`?
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![[b-tree-initial.png]]
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Back: Indeterminate. We do not know the order of the tree.
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489739-->
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END%%
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%%ANKI
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Basic
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Consider the following B-tree of order $6$. What is the result of inserting `B`?
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![[b-tree-initial.png]]
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Back:
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![[b-tree-inserted-b.png]]
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489740-->
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END%%
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%%ANKI
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Basic
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Consider the following B-tree of order $6$. What is the result of inserting `Q` (right biased)?
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![[b-tree-inserted-b.png]]
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Back:
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![[b-tree-inserted-q.png]]
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321489741-->
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END%%
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%%ANKI
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Basic
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What is the insertion runtime of a B-tree of order $m$ and height $h$?
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Back: $O(mh)$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321615987-->
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END%%
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%%ANKI
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Basic
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How many disk accesses are performed when inserting into a B-tree of order $m$ containing $n$ keys?
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Back: $O(\log_m{n})$
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Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
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<!--ID: 1723321615989-->
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END%%
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## B+ Tree
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The **B+ tree** is a B-tree with the following differences:
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* Internal nodes do not store values; that is, all values are stored in the leaf nodes.
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* Leaf nodes may include a pointer to the next leaf node to speed sequential access.
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%%ANKI
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Basic
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What is the *required* distinction between B-trees and B+ trees?
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Back: Values in B+ trees are only stored in leaf nodes.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926214-->
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END%%
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%%ANKI
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Basic
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In a B-tree, where can values be found?
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Back: In any node.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926220-->
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END%%
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%%ANKI
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Basic
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In a B+ tree, where can values be found?
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Back: In the leaf nodes.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926224-->
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END%%
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%%ANKI
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Basic
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What is the *optional* distinction between B-trees and B+ trees?
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Back: A B+ tree leaf node may include a pointer to the next leaf node.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926227-->
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END%%
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%%ANKI
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Basic
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How is a B+ tree defined in terms of B-trees?
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Back: As a B-tree in which all values must reside in the leaf nodes.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926231-->
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END%%
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%%ANKI
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Basic
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Why might a B+ tree implementation include pointers from leaf to leaf?
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Back: To speed up sequential access.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926235-->
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END%%
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%%ANKI
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Basic
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Which of B-trees and B+ trees likely have a higher order?
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Back: B+ trees.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926239-->
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END%%
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%%ANKI
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Basic
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Why do B+ trees typically have higher orders than B-trees?
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Back: Their internal nodes do not have values, leaving room for more keys.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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<!--ID: 1723325926244-->
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END%%
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%%ANKI
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Basic
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Which of B+ trees and B-trees are likely deeper?
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Back: B-trees.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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%%ANKI
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Basic
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Why are B+ trees typically shallower than B-trees?
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Back: Their internal nodes do not have values, leaving room for more keys.
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Reference: “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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## Bibliography
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* “B-Tree,” in _Wikipedia_, August 7, 2024, [https://en.wikipedia.org/w/index.php?title=B-tree](https://en.wikipedia.org/w/index.php?title=B-tree&oldid=1239132600).
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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). |