--- title: B-Tree TARGET DECK: Obsidian::STEM FILE TAGS: data_structure::b-tree tags: - 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 determined? 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 can 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 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 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 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%% ## Bibliography * 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).