Intrusive containers and graph isomorphisms.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -123,7 +123,8 @@
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notes/_journal/2024-05-12.md

@@ -7,3 +7,6 @@ title: "2024-05-12"
 - [ ] Sheet Music (10 min.)
 - [ ] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
+
+* Notes on linked lists and intrusive/non-intrusive data structures.
+* Notes on graphs/isomorphisms.

notes/data-structures/arrays.md

@@ -65,6 +65,36 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1715460959188-->
 END%%
 
+%%ANKI
+Cloze
+The {head} of an array refers to {its first element}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735208-->
+END%%
+
+%%ANKI
+Cloze
+The {tail} of an array refers to {all but its first element}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735211-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of prepending an element to a large enough array?
+Back: $O(n)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735214-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of finding the $k$th element of an array?
+Back: $O(1)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735217-->
+END%%
+
 ## Bibliography
 
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/data-structures/intrusive.md

@@ -0,0 +1,219 @@
+---
+title: Intrusive Containers
+TARGET DECK: Obsidian::STEM
+FILE TAGS: data_structure::intrusive
+tags:
+  - data_structure
+---
+
+## Overview
+
+An intrusive container is a data structure used to hold a collection of objects in which membership bookkeeping is stored in the objects themselves rather than a separate structure. For example, consider the following `struct`:
+
+```c
+struct Point {
+  float x, y;
+}
+```
+
+A non-intrusive implementation of a [[linked-lists|doubly linked list]] would look as follows:
+
+```c
+struct ListNode {
+  struct Point val;
+  struct ListNode *next, *prev;
+}
+```
+
+An intrusive implementation would instead modify `Point` like so:
+
+```c
+struct Point {
+  float x, y;
+  struct Point *next, *prev;
+}
+```
+
+%%ANKI
+Basic
+What is an intrusive container?
+Back: A collection of objects in which membership bookkeeping is stored in the objects themselves.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735123-->
+END%%
+
+%%ANKI
+Basic
+What is the opposite of an "intrusive container"?
+Back: A non-intrusive container.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735129-->
+END%%
+
+%%ANKI
+Basic
+Given the following `struct`, implement an intrusive doubly linked list.
+```c
+struct Point {
+  float x, y;
+}
+```
+Back:
+```c
+struct Point {
+  float x, y;
+  struct Point *next, *prev;
+}
+```
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: c17 data_structure::linked_list
+<!--ID: 1715534735133-->
+END%%
+
+%%ANKI
+Basic
+Is the following considered an intrusive container or non-intrusive container?
+```c
+struct Point {
+  float x, y;
+  struct Point *next, *prev;
+}
+```
+Back: Intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: c17 data_structure::linked_list
+<!--ID: 1715534821493-->
+END%%
+
+%%ANKI
+Basic
+Is the following considered an intrusive container or non-intrusive container?
+```c
+struct ListNode {
+  struct Point val;
+  struct ListNode *next, *prev;
+}
+```
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534821498-->
+END%%
+
+%%ANKI
+Basic
+Given the following `struct`, implement a non-intrusive doubly linked list.
+```c
+struct Point {
+  float x, y;
+}
+```
+Back:
+```c
+struct ListNode {
+  struct Point val;
+  struct ListNode *next, *prev;
+}
+```
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: c17 data_structure::linked_list
+<!--ID: 1715534735137-->
+END%%
+
+%%ANKI
+Basic
+Given the following `struct`, how many linked lists can a `Point` exist in?
+```c
+struct Point {
+  float x, y;
+  struct Point *next, *prev;
+}
+```
+Back: Zero or one.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: c17 data_structure::linked_list
+<!--ID: 1715534735140-->
+END%%
+
+%%ANKI
+Basic
+Given the following `struct`, how many linked lists can a `Point` exist in?
+```c
+struct ListNode {
+  struct Point *val;
+  struct ListNode *next, *prev;
+}
+```
+Back: Zero or more.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: c17 data_structure::linked_list
+<!--ID: 1715534735143-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers allow a `struct` to exist in an arbitrary number of linked lists?
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+Tags: data_structure::linked_list
+<!--ID: 1715534735146-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers perform more allocations?
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735149-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers perform more de-allocations?
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735152-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers has better spatial locality?
+Back: Intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735155-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers avoid modifying object definitions?
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735158-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers allow implicitly finding *all* containers an object is contained in?
+Back: Intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735161-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers cannot be used for opaque types?
+Back: Intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735164-->
+END%%
+
+%%ANKI
+Basic
+Which of intrusive or non-intrusive containers requires searching in each container for the same reference?
+Back: Non-intrusive.
+Reference: Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).
+<!--ID: 1715534735167-->
+END%%
+
+## Bibliography
+
+* Fuchsia. “Introduction to Fbl Intrusive Containers.” Accessed May 12, 2024. [https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction](https://fuchsia.dev/fuchsia-src/development/languages/c-cpp/fbl_containers_guide/introduction).

notes/data-structures/linked-lists.md

@@ -0,0 +1,108 @@
+---
+title: Linked Lists
+TARGET DECK: Obsidian::STEM
+FILE TAGS: data_structure::linked_list
+tags:
+  - data_structure
+  - linked_list
+---
+
+## Overview
+
+%%ANKI
+Basic
+What does it mean for a linked list to be singly linked?
+Back: Each entry in the list has a `next` pointer only.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735171-->
+END%%
+
+%%ANKI
+Basic
+What condition is used to check if a singly linked list is empty?
+Back: Its empty if the head of the list is `NIL`.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735173-->
+END%%
+
+%%ANKI
+Cloze
+The {head} of a linked list refers to {its first element}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735177-->
+END%%
+
+%%ANKI
+Cloze
+The {tail} of a linked list refers to {its last element}.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735180-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a singly linked list to be circular?
+Back: The tail points to the head.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735184-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of inserting an element at the front of a singly linked list?
+Back: $O(1)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735187-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of finding the $k$th element of a singly linked list?
+Back: $O(n)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735190-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a linked list to be doubly linked?
+Back: Each entry in the list have `next` and `prev` pointers.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735193-->
+END%%
+
+%%ANKI
+Basic
+What condition is used to check if a doubly linked list is empty?
+Back: Its empty if the head of the list is `NIL`.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735196-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for a doubly linked list to be circular?
+Back: The tail points to the head, and the head points to the tail.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735199-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of prepending an element to a doubly linked list?
+Back: $O(1)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735202-->
+END%%
+
+%%ANKI
+Basic
+What is the runtime of finding the $k$th element of a doubly linked list?
+Back: $O(n)$
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1715534735205-->
+END%%
+
+## Bibliography
+
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/set/graphs.md

@@ -1020,6 +1020,129 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1710807788538-->
 END%%
 
+## Isomorphisms
+
+An **isomorphism** between two graphs $G_1$ and $G_2$ is a bijection $f \colon V_1 \rightarrow V_2$ between the vertices of the graphs such that $(a, b)$ is an edge in $G_1$ if and only if $(f(a), f(b))$ is an edge in $G_2$. Here parenthesis are used to denote either ordered pairs (for directed graphs) or unordered pairs (for undirected graphs).
+
+We say $G_1$ and $G_2$ are **isomorphic**, denoted $G_1 \cong G_2$, if and only if there exists an isomorphism between $G_1$ and $G_2$.
+
+%%ANKI
+Basic
+What kind of mathematical object is an isomorphism between graphs?
+Back: A function.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560168-->
+END%%
+
+%%ANKI
+Basic
+What *kind* of function is an isomorphism between two graphs?
+Back: A bijective function.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560173-->
+END%%
+
+%%ANKI
+Basic
+What *is* an isomorphism between graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$?
+Back: A bijection $f \colon V_1 \rightarrow V_2$ such that $(a, b) \in E_1$ if and only if $(f(a), f(b)) \in E_2$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560176-->
+END%%
+
+%%ANKI
+Basic
+What is the domain of an isomorphism between graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$?
+Back: $V_1$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560179-->
+END%%
+
+%%ANKI
+Basic
+What is the codomain of an isomorphism between graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$?
+Back: $V_2$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560183-->
+END%%
+
+%%ANKI
+Basic
+What is the edge relation of isomorphism $f$ between graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$?
+Back: $(a, b) \in E_1$ if and only if $(f(a), f(b)) \in E_2$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560186-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for graphs $G_1$ and $G_2$ to be isomorphic?
+Back: There exists an isomorphism between them.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560190-->
+END%%
+
+%%ANKI
+Basic
+If two graphs are equal, are they isomorphic?
+Back: Yes.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560195-->
+END%%
+
+%%ANKI
+Basic
+If two graphs are isomorphic, are they equal?
+Back: Not necessarily.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560199-->
+END%%
+
+%%ANKI
+Basic
+Are the following two graphs equal?
+![[graph-isomorphic.png]]
+Back: No.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560203-->
+END%%
+
+%%ANKI
+Basic
+Are the following two graphs isomorphic?
+![[graph-isomorphic.png]]
+Back: Yes.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560207-->
+END%%
+
+%%ANKI
+Basic
+If the following graphs are isomorphic, what is the domain of the isomorphism?
+![[graph-isomorphic.png]]
+Back: $\{a, b, c\}$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560210-->
+END%%
+
+%%ANKI
+Basic
+If the following graphs are isomorphic, what is the codomain of the isomorphism?
+![[graph-isomorphic.png]]
+Back: $\{u, v, w\}$.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560214-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for two graphs to be equal?
+Back: Two graphs are equal if their vertex and edge sets are equal.
+Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
+<!--ID: 1715537560218-->
+END%%
+
 ## Bibliography
 
+* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
 * Thomas H. Cormen et al., _Introduction to Algorithms_, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).