Notes on B-tree insertions.

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@@ -186,7 +186,12 @@
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     "non-complete-tree.png",
     "relation-ordering-example.png",
-    "infinite-cartesian-product.png"
+    "infinite-cartesian-product.png",
+    "b-tree-full-node.png",
+    "b-tree-split-node.png",
+    "b-tree-initial.png",
+    "b-tree-inserted-b.png",
+    "b-tree-inserted-q.png"
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notes/_journal/2024-08-10.md

@@ -0,0 +1,12 @@
+---
+title: "2024-08-10"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Read through Chapter 3 of "Modern C".
+* Notes on B-tree insertions.

notes/data-structures/b-tree.md

@@ -18,6 +18,30 @@ A **B-tree of order $m$** is a tree that satisfies the following properties:
 
 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).
+<!--ID: 1723289256280-->
+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).
+<!--ID: 1723289256283-->
+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).
+<!--ID: 1723289256285-->
+END%%
+
 %%ANKI
 Basic
 What hyperparameter is used to define a B-tree?
@@ -106,14 +130,6 @@ Reference: Donald Ervin Knuth, _Art of Computer Programming, 3: Sorting and Sear
 <!--ID: 1723211542052-->
 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).
-<!--ID: 1723211542058-->
-END%%
-
 %%ANKI
 Basic
 Consider a B-tree of order $7$. How many children $c$ can each non-root node have?
@@ -172,26 +188,179 @@ END%%
 
 %%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).
-<!--ID: 1723289256280-->
+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).
+<!--ID: 1723321489725-->
 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).
-<!--ID: 1723289256283-->
+*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).
+<!--ID: 1723321489726-->
 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.
+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).
+<!--ID: 1723321489727-->
+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).
+<!--ID: 1723321489728-->
+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).
+<!--ID: 1723321489729-->
+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).
+<!--ID: 1723321489730-->
+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).
+<!--ID: 1723321489731-->
+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).
+<!--ID: 1723321489732-->
+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).
-<!--ID: 1723289256285-->
+<!--ID: 1723321489733-->
+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).
+<!--ID: 1723321489734-->
+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).
+<!--ID: 1723321489735-->
+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).
+<!--ID: 1723321489736-->
+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).
+<!--ID: 1723321615984-->
+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).
+<!--ID: 1723321489737-->
+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).
+<!--ID: 1723211542058-->
+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).
+<!--ID: 1723321489738-->
+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).
+<!--ID: 1723321489739-->
+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).
+<!--ID: 1723321489740-->
+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).
+<!--ID: 1723321489741-->
+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).
+<!--ID: 1723321615987-->
+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).
+<!--ID: 1723321615989-->
 END%%
 
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