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title TARGET DECK FILE TAGS tags
B-Tree Obsidian::STEM data_structure::b-tree
b-tree
data_structure

Overview

A B-tree of order m is a tree that satisfies the following properties:

  • Every node has at most m children.
  • Every node, except for the root, has at least m / 2 children.
  • All leaves appear on the same level.
  • A node with k children contains k - 1 keys sorted in monotonically increasing order.

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).

%%ANKI Basic Why is a B-tree named the way it is? Back: There is no definitive answer. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What was the motivation behind the development of the B-tree? Back: To find a data structure for efficient search that minimizes disk accesses. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic How is the order of a B-tree typically decided? Back: By choosing a value that best aligns with the size of a memory block. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What hyperparameter is used to define a B-tree? Back: It's order, i.e. the maximum number of a children a node can have. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic In what direction do B-trees grow? Back: From bottom to top. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider B-tree of order m. What does m refer to? Back: The maximum number of children a node can have. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the maximum number of children a node in a B-tree have? Back: N/A. It depends on the tree's order. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the maximum number of children a node in a B-tree of order m can have? Back: m Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the minimum number of children a non-root node in a B-tree of order m can have? Back: \lceil m / 2 \rceil Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the maximum number of keys a node in a B-tree of order m can have? Back: m - 1 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the minimum number of keys a non-root node in a B-tree of order m can have? Back: \lceil m / 2 \rceil - 1 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic A node in a B-tree of order m has k keys. How many children does it have? Back: k + 1 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic A node in a B-tree of order m has k children. How many keys does it have? Back: k - 1 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider a B-tree of order 7. How many children c can each internal non-root node have? Back: 4 \leq c \leq 7 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider a B-tree of order 7. How many children c can the root have? Back: 0 \leq c \leq 7 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider a B-tree of order 7. How many keys k can each internal non-root node have? Back: 3 \leq k < 7 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider a B-tree of order 7. How many keys k can the root have? Back: 0 \leq k < 7 Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What instances exist of a B-tree of order 1? Back: Just the empty tree. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Why can't we define a nonempty B-tree of order 1? Back: Each node can have at most 1 child, meaning each node contains 0 keys. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic How are keys arranged within a B-tree's nodes? Back: In monotonically increasing order. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic What is the search runtime of a B-tree of order m and height h? Back: O(mh) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why does searching a B-tree of order m and height h take O(mh) time? Back: Each node may have m - 1 keys, and we may check h nodes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many disk accesses are performed when searching a B-tree of order m and height h? Back: O(h) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why does the number of disk accesses when searching a B-tree of height h equal O(h)? Back: The size of each node presumably corresponds to a block of memory. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the search runtime of a B-tree of order m containing n keys? Back: O(m\log_m{n}) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why does searching a B-tree of order m containing n keys take O(m\log_m{n}) time? Back: Each node may have m - 1 keys, and we may check \log_m{n} nodes. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many disk accesses are performed when searching a B-tree of order m containing n keys? Back: O(\log_m{n}) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Why does the number of disk accesses when searching a B-tree of order m containing n keys equal O(\log_m{n})? Back: The size of each node presumably corresponds to a block of memory. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

Insertions

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.

%%ANKI Cloze A node in a B-tree of order m is considered full when it has {m - 1} keys. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Cloze A node in a B-tree of order m is considered full when it has {m} children. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Which key(s) found in B-trees move levels during node splits? Back: The split node's median key. Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a B-tree split to be left-biased? Back: Prefer the median on the LHS. Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What does it mean for a B-tree split to be right-biased? Back: Prefer the median on the RHS. Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider splitting a node in a B-tree of order m. How many keys are in the split nodes? Back: \lfloor (m - 1) / 2 \rfloor and \lceil (m - 1) / 2 \rceil. Reference: Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic When does a B-tree gain height? Back: When the root node is split. Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Suppose the following B-tree node is full. What is the result after splitting? !b-tree-full-node.png Back: !b-tree-split-node.png Reference: Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).

END%%

%%ANKI Basic Consider the following B-tree. What is the result of inserting B? !b-tree-initial.png Back: Indeterminate. We do not know the order of the tree. Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider the following B-tree of order 6. What is the result of inserting B? !b-tree-initial.png Back: !b-tree-inserted-b.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic Consider the following B-tree of order 6. What is the result of inserting Q (right biased)? !b-tree-inserted-b.png Back: !b-tree-inserted-q.png Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic What is the insertion runtime of a B-tree of order m and height h? Back: O(mh) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

%%ANKI Basic How many disk accesses are performed when inserting into a B-tree of order m containing n keys? Back: O(\log_m{n}) Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

END%%

B+ Tree

The B+ tree is a B-tree with the following differences:

  • Internal nodes do not store values; that is, all values are stored in the leaf nodes.
  • Leaf nodes may include a pointer to the next leaf node to speed sequential access.

%%ANKI Basic What is the required distinction between B-trees and B+ trees? Back: Values in B+ trees are only stored in leaf nodes. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic In a B-tree, where can values be found? Back: In any node. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic In a B+ tree, where can values be found? Back: In the leaf nodes. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic What is the optional distinction between B-trees and B+ trees? Back: A B+ tree leaf node may include a pointer to the next leaf node. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic How is a B+ tree defined in terms of B-trees? Back: As a B-tree in which all values must reside in the leaf nodes. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic Why might a B+ tree implementation include pointers from leaf to leaf? Back: To speed up sequential access. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic Which of B-trees and B+ trees likely have a higher order? Back: B+ trees. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic Why do B+ trees typically have higher orders than B-trees? Back: Their internal nodes do not have values, leaving room for more keys. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic Which of B+ trees and B-trees are likely deeper? Back: B-trees. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

%%ANKI Basic Why are B+ trees typically shallower than B-trees? Back: Their internal nodes do not have values, leaving room for more keys. Reference: “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.

END%%

Bibliography

  • “B-Tree,” in Wikipedia, August 7, 2024, https://en.wikipedia.org/w/index.php?title=B-tree.
  • Donald Ervin Knuth, Art of Computer Programming, 3: Sorting and Searching, 2. ed., 34. (Reading, Mass: Addison-Wesley, 1995).
  • Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).