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Intrusive containers and graph isomorphisms.

c-declarations
Joshua Potter 2024-05-12 12:30:30 -06:00
parent 5f8017ba3d
commit 48a4f9127f
7 changed files with 498 additions and 13 deletions
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"Basic": [

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- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Korean (Read 1 Story)
* Notes on linked lists and intrusive/non-intrusive data structures.
* Notes on graphs/isomorphisms.

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@ -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).

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---
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).

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@ -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).

123
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@ -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).

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