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Add notes on next lexicographic ordering.

c-declarations
Joshua Potter 2024-03-06 13:25:34 -07:00
parent ec45aad8c3
commit 1651c9d6c3
14 changed files with 696 additions and 113 deletions
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26
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"Basic": [ "Basic": [

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@ -0,0 +1,13 @@
---
title: "2024-03-06"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] Go (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)
* Added notes on how asymptotic notation relates to one another.

3
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@ -25,4 +25,5 @@ title: "2024-03-05"
* Q-114783 * Q-114783
* Q-26296 * Q-26296
* Leetcode Problems * Leetcode Problems
* [Next Permutation](https://leetcode.com/problems/next-permutation/) * [Next Permutation](https://leetcode.com/problems/next-permutation/)
* Purged all blockchain-related flashcards from Anki.

18
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@ -16,7 +16,7 @@ Aux. Memory | $O(1)$
%%ANKI %%ANKI
Basic Basic
What precondition must the input of binary search satisfy? What precondition must the input of `BINARY_SEARCH` satisfy?
Back: It must already be sorted. Back: It must already be sorted.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708781334247--> <!--ID: 1708781334247-->
@ -24,7 +24,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is the best case running time of binary search? What is the best case running time of `BINARY_SEARCH`?
Back: $\Omega(1)$ Back: $\Omega(1)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310004--> <!--ID: 1708117310004-->
@ -32,7 +32,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input does binary search perform best on? What input does `BINARY_SEARCH` perform best on?
Back: One in which the value being searched for is already in the middle. Back: One in which the value being searched for is already in the middle.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310011--> <!--ID: 1708117310011-->
@ -40,7 +40,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is the worst case running time of binary search? What is the worst case running time of `BINARY_SEARCH`?
Back: $O(\lg{n})$ Back: $O(\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310015--> <!--ID: 1708117310015-->
@ -48,7 +48,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input does binary search perform worst on? What input does `BINARY_SEARCH` perform worst on?
Back: One in which the value does not exist. Back: One in which the value does not exist.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310018--> <!--ID: 1708117310018-->
@ -56,7 +56,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is the typical output of binary search? What is the typical output of `BINARY_SEARCH`?
Back: The index of the element in the array being searched for, if found. Back: The index of the element in the array being searched for, if found.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708117310021--> <!--ID: 1708117310021-->
@ -106,7 +106,7 @@ int binary_search(const int needle, const int n, int A[static n]) {
%%ANKI %%ANKI
Basic Basic
In binary search, when could using floor for midpoint calculations yield different answers than ceiling? In `BINARY_SEARCH`, when could using floor for midpoint calculations yield different answers than ceiling?
Back: When there exist duplicate members. Back: When there exist duplicate members.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708174545522--> <!--ID: 1708174545522-->
@ -114,8 +114,8 @@ END%%
%%ANKI %%ANKI
Basic Basic
In binary search, what ensures left pointer `L` and right pointers `R` eventually satisfy `L > R`? In `BINARY_SEARCH`, what ensures left pointer `L` and right pointers `R` eventually satisfy `L > R`?
Back: The found midpoint is always excluded from the next binary search invocation. Back: The found midpoint is always excluded from the next `BINARY_SEARCH` invocation.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708174545527--> <!--ID: 1708174545527-->
END%% END%%

331
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@ -332,7 +332,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What condition must $g(n)$ satisfy such that otherwise $\Theta(g(n))$ is empty? What condition must $g(n)$ satisfy such that $\Theta(g(n))$ is nonempty?
Back: $g(n)$ must be asymptotically nonnegative. Back: $g(n)$ must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708974221876--> <!--ID: 1708974221876-->
@ -397,6 +397,62 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1709053894064--> <!--ID: 1709053894064-->
END%% END%%
%%ANKI
Basic
What is the transitive property of $\Theta$-notation?
Back: $f(n) = \Theta(g(n))$ and $g(n) = \Theta(h(n))$ implies $f(n) = \Theta(h(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223294-->
END%%
%%ANKI
Basic
What is the reflexive property of $\Theta$-notation?
Back: $f(n) = \Theta(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223298-->
END%%
%%ANKI
Basic
What condition must $f(n)$ satisfy for equality $f(n) = \Theta(f(n))$ to hold?
Back: $f(n)$ must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223303-->
END%%
%%ANKI
Basic
*Why* must $f(n)$ be nonnegatively asymptotic for $f(n) = \Theta(f(n))$ to hold?
Back: Otherwise $\Theta(f(n))$ is the empty set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223309-->
END%%
%%ANKI
Basic
What is the symmetric property of $\Theta$-notation?
Back: $f(n) = \Theta(g(n))$ iff $g(n) = \Theta(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223316-->
END%%
%%ANKI
Basic
What is the transpose symmetric property of $\Theta$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223322-->
END%%
%%ANKI
Basic
$\Theta$-notation is likened to what comparison operator of real numbers?
Back: $=$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223329-->
END%%
## $O$-notation ## $O$-notation
![[big-o-notation.png]] ![[big-o-notation.png]]
@ -471,6 +527,7 @@ Basic
Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$? Using typical identifiers, what is the upper bound of $f(n)$ in $O(g(n))$?
Back: $cg(n)$ Back: $cg(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709750359817-->
END%% END%%
%%ANKI %%ANKI
@ -498,7 +555,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What condition must $g(n)$ satisfy such that otherwise $O(g(n))$ is empty? What condition must $g(n)$ satisfy such that $O(g(n))$ is nonempty?
Back: $g(n)$ must be asymptotically nonnegative. Back: $g(n)$ must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709053894093--> <!--ID: 1709053894093-->
@ -554,6 +611,62 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1709053894105--> <!--ID: 1709053894105-->
END%% END%%
%%ANKI
Basic
What is the transitive property of $O$-notation?
Back: $f(n) = O(g(n))$ and $g(n) = O(h(n))$ implies $f(n) = O(h(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223338-->
END%%
%%ANKI
Basic
What is the reflexive property of $O$-notation?
Back: $f(n) = O(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223346-->
END%%
%%ANKI
Basic
What condition must $f(n)$ satisfy for equality $f(n) = O(f(n))$ to hold?
Back: $f(n)$ must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223352-->
END%%
%%ANKI
Basic
*Why* must $f(n)$ be nonnegatively asymptotic for $f(n) = O(f(n))$ to hold?
Back: Otherwise $O(f(n))$ is the empty set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223358-->
END%%
%%ANKI
Basic
What is the symmetric property of $O$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223365-->
END%%
%%ANKI
Basic
What is the transpose symmetric property of $O$-notation?
Back: $f(n) = O(g(n))$ iff $g(n) = \Omega(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223372-->
END%%
%%ANKI
Basic
$O$-notation is likened to what comparison operator of real numbers?
Back: $\leq$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223381-->
END%%
## $o$-notation ## $o$-notation
$o$-notation refers to an upper bound that is not asymptotically tight. It is defined as set $$o(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}$$ $o$-notation refers to an upper bound that is not asymptotically tight. It is defined as set $$o(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq f(n) < cg(n) \}$$
@ -660,6 +773,77 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1709519002364--> <!--ID: 1709519002364-->
END%% END%%
%%ANKI
Basic
What condition must $g(n)$ satisfy such that $o(g(n))$ is nonempty?
Back: $g(n)$ must be asymptotically positive.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709750359822-->
END%%
%%ANKI
Basic
What is the transitive property of $o$-notation?
Back: $f(n) = o(g(n))$ and $g(n) = o(h(n))$ implies $f(n) = o(h(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223391-->
END%%
%%ANKI
Basic
What is the reflexive property of $o$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223399-->
END%%
%%ANKI
Basic
*Why* is there no reflexive property of $o$-notation?
Back: A function cannot be asymptotically smaller than itself.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223407-->
END%%
%%ANKI
Basic
What is the symmetric property of $o$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223417-->
END%%
%%ANKI
Basic
What is the transpose symmetric property of $o$-notation?
Back: $f(n) = o(g(n))$ iff $g(n) = \omega(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223426-->
END%%
%%ANKI
Basic
$o$-notation is likened to what comparison operator of real numbers?
Back: $<$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223435-->
END%%
%%ANKI
Cloze
{1:$\Omega$}-notation is to {2:$\geq$} whereas {2:$o$}-notation is to {1:$<$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223442-->
END%%
%%ANKI
Basic
How do we set theoretically say $f(n)$ is asymptotically smaller than $g(n)$?
Back: $f(n) = o(g(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223449-->
END%%
## $\Omega$-notation ## $\Omega$-notation
![[big-omega-notation.png]] ![[big-omega-notation.png]]
@ -762,7 +946,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What condition must $g(n)$ satisfy such that otherwise $\Omega(g(n))$ is empty? What condition must $g(n)$ satisfy such that $\Omega(g(n))$ is nonempty?
Back: $g(n)$ must be asymptotically nonnegative. Back: $g(n)$ must be asymptotically nonnegative.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709055157396--> <!--ID: 1709055157396-->
@ -816,6 +1000,76 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1709055157406--> <!--ID: 1709055157406-->
END%% END%%
%%ANKI
Basic
What is the transitive property of $\Omega$-notation?
Back: $f(n) = \Omega(g(n))$ and $g(n) = \Omega(h(n))$ implies $f(n) = \Omega(h(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223456-->
END%%
%%ANKI
Basic
What is the reflexive property of $\Omega$-notation?
Back: $f(n) = \Omega(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223462-->
END%%
%%ANKI
Basic
What condition must $f(n)$ satisfy for equality $f(n) = \Omega(f(n))$ to hold?
Back: $f(n)$ must be nonnegatively asymptotic.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223468-->
END%%
%%ANKI
Basic
*Why* must $f(n)$ be nonnegatively asymptotic for $f(n) = \Omega(f(n))$ to hold?
Back: Otherwise $\Omega(f(n))$ is the empty set.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223477-->
END%%
%%ANKI
Basic
What is the symmetric property of $\Omega$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223486-->
END%%
%%ANKI
Basic
What is the transpose symmetric property of $\Omega$-notation?
Back: $f(n) = \Omega(g(n))$ iff $g(n) = O(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223496-->
END%%
%%ANKI
Basic
$\Omega$-notation is likened to what comparison operator of real numbers?
Back: $\geq$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223503-->
END%%
%%ANKI
Cloze
{1:$\Theta$}-notation is to {2:$=$} whereas {2:$\Omega$}-notation is to {1:$\geq$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223511-->
END%%
%%ANKI
Cloze
{1:$O$}-notation is to {2:$\leq$} whereas {2:$\Omega$}-notation is to {1:$\geq$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223522-->
END%%
## $\omega$-notation ## $\omega$-notation
$\omega$-notation refers to a lower bound that is not asymptotically tight. It is defined as set $$\omega(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) \}$$ $\omega$-notation refers to a lower bound that is not asymptotically tight. It is defined as set $$\omega(g(n)) = \{ f(n) \mid \forall c > 0, \exists n_0 > 0, \forall n \geq n_0, 0 \leq cg(n) < f(n) \}$$
@ -922,6 +1176,77 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
<!--ID: 1709519002425--> <!--ID: 1709519002425-->
END%% END%%
%%ANKI
Basic
What condition must $g(n)$ satisfy such that $\omega(g(n))$ is nonempty?
Back: $g(n)$ must be asymptotically positive.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709750359826-->
END%%
%%ANKI
Basic
What is the transitive property of $\omega$-notation?
Back: $f(n) = \omega(g(n))$ and $g(n) = \omega(h(n))$ implies $f(n) = \omega(h(n))$.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223531-->
END%%
%%ANKI
Basic
What is the reflexive property of $\omega$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223537-->
END%%
%%ANKI
Basic
*Why* is there no reflexive property of $\omega$-notation?
Back: A function cannot be asymptotically larger than itself.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223544-->
END%%
%%ANKI
Basic
What is the symmetric property of $\omega$-notation?
Back: N/A
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223553-->
END%%
%%ANKI
Basic
What is the transpose symmetric property of $\omega$-notation?
Back: $f(n) = \omega(g(n))$ iff $g(n) = o(f(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223563-->
END%%
%%ANKI
Basic
$\omega$-notation is likened to what comparison operator of real numbers?
Back: $>$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223570-->
END%%
%%ANKI
Cloze
{1:$O$}-notation is to {2:$\leq$} whereas {2:$\omega$}-notation is to {1:$>$}.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223577-->
END%%
%%ANKI
Basic
How do we set theoretically say $f(n)$ is asymptotically larger than $g(n)$?
Back: $f(n) = \omega(g(n))$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1709752223586-->
END%%
## References ## References
* Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). * Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

34
notes/algorithms/sorting/bubble-sort.md
View File

@ -22,7 +22,7 @@ Adaptive | Yes
%%ANKI %%ANKI
Basic Basic
Describe bubble sort in a single sentence. Describe `BUBBLE_SORT` in a single sentence.
Back: Repeatedly swap the smaller of adjacent records downward. Back: Repeatedly swap the smaller of adjacent records downward.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707589393196--> <!--ID: 1707589393196-->
@ -30,7 +30,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is bubble sort's best case runtime? What is `BUBBLE_SORT`'s best case runtime?
Back: $\Omega(n)$ Back: $\Omega(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634781--> <!--ID: 1707504634781-->
@ -38,7 +38,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How is it bubble sort achieves best case linear runtime? How is it `BUBBLE_SORT` achieves best case linear runtime?
Back: By terminating when no swaps occurred on a given iteration. Back: By terminating when no swaps occurred on a given iteration.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634782--> <!--ID: 1707504634782-->
@ -46,7 +46,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input value does bubble sort perform best on? What input value does `BUBBLE_SORT` perform best on?
Back: An already sorted array. Back: An already sorted array.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634784--> <!--ID: 1707504634784-->
@ -54,7 +54,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is bubble sort's worst case runtime? What is `BUBBLE_SORT`'s worst case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634785--> <!--ID: 1707504634785-->
@ -62,7 +62,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input value does bubble sort perform worst on? What input value does `BUBBLE_SORT` perform worst on?
Back: An array in reverse-sorted order. Back: An array in reverse-sorted order.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634787--> <!--ID: 1707504634787-->
@ -70,7 +70,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is bubble sort's average case runtime? What is `BUBBLE_SORT`'s average case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634788--> <!--ID: 1707504634788-->
@ -78,7 +78,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is bubble sort in place? Is `BUBBLE_SORT` in place?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634789--> <!--ID: 1707504634789-->
@ -86,7 +86,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is bubble sort stable? Is `BUBBLE_SORT` stable?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634791--> <!--ID: 1707504634791-->
@ -94,7 +94,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is bubble sort adaptive? Is `BUBBLE_SORT` adaptive?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634792--> <!--ID: 1707504634792-->
@ -125,7 +125,7 @@ void bubble_sort(const int n, int A[static n]) {
Basic Basic
What sorting algorithm does the following demonstrate? What sorting algorithm does the following demonstrate?
![[bubble-sort.gif]] ![[bubble-sort.gif]]
Back: Bubble sort. Back: `BUBBLE_SORT`
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634794--> <!--ID: 1707504634794-->
END%% END%%
@ -147,7 +147,7 @@ We prove $P$ maintains the requisite properties:
%%ANKI %%ANKI
Basic Basic
Given array `A[0..n-1]`, what is bubble sort's loop invariant? Given array `A[0..n-1]`, what is `BUBBLE_SORT`'s loop invariant?
Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`. Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634796--> <!--ID: 1707504634796-->
@ -155,7 +155,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is initialization of bubble sort's loop invariant? What is initialization of `BUBBLE_SORT`'s loop invariant?
Back: Sorting starts with an empty array which is trivially sorted. Back: Sorting starts with an empty array which is trivially sorted.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634797--> <!--ID: 1707504634797-->
@ -163,7 +163,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is maintenance of bubble sort's loop invariant? What is maintenance of `BUBBLE_SORT`'s loop invariant?
Back: Each iteration puts the next least element into the sorted subarray. Back: Each iteration puts the next least element into the sorted subarray.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634798--> <!--ID: 1707504634798-->
@ -171,7 +171,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How does bubble sort partition its input array? How does `BUBBLE_SORT` partition its input array?
Back: Back:
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
@ -182,7 +182,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which element will bubble sort move to `sorted`? Which element will `BUBBLE_SORT` move to `sorted`?
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
``` ```
@ -193,7 +193,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
Selection sort makes fewer {swaps} than bubble sort in the average case. Selection sort makes fewer {swaps} than `BUBBLE_SORT` in the average case.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634803--> <!--ID: 1707504634803-->
END%% END%%

38
notes/algorithms/sorting/insertion-sort.md
View File

@ -22,7 +22,7 @@ Adaptive | Yes
%%ANKI %%ANKI
Basic Basic
Describe insertion sort in a single sentence. Describe `INSERTION_SORT` in a single sentence.
Back: Repeatedly put the next record into a sorted array from right to left. Back: Repeatedly put the next record into a sorted array from right to left.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707589393194--> <!--ID: 1707589393194-->
@ -30,7 +30,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is insertion sort's best case runtime? What is `INSERTION_SORT`'s best case runtime?
Back: $\Omega(n)$ Back: $\Omega(n)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706925879541--> <!--ID: 1706925879541-->
@ -38,7 +38,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input value does insertion sort perform best on? What input value does `INSERTION_SORT` perform best on?
Back: An already sorted array. Back: An already sorted array.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706925921544--> <!--ID: 1706925921544-->
@ -46,7 +46,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is insertion sort's worst case runtime? What is `INSERTION_SORT`'s worst case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706926586947--> <!--ID: 1706926586947-->
@ -54,7 +54,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What input value does insertion sort perform worst on? What input value does `INSERTION_SORT` perform worst on?
Back: An array in reverse-sorted order. Back: An array in reverse-sorted order.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706926586951--> <!--ID: 1706926586951-->
@ -62,7 +62,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is insertion sort's average case runtime? What is `INSERTION_SORT`'s average case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707329732933--> <!--ID: 1707329732933-->
@ -70,7 +70,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is insertion sort in place? Is `INSERTION_SORT` in place?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706926586955--> <!--ID: 1706926586955-->
@ -78,7 +78,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is insertion sort stable? Is `INSERTION_SORT` stable?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706926586959--> <!--ID: 1706926586959-->
@ -86,7 +86,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is insertion sort adaptive? Is `INSERTION_SORT` adaptive?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634779--> <!--ID: 1707504634779-->
@ -109,7 +109,7 @@ void insertion_sort(const int n, int A[static n]) {
Basic Basic
What sorting algorithm does the following demonstrate? What sorting algorithm does the following demonstrate?
![[insertion-sort.gif]] ![[insertion-sort.gif]]
Back: Insertion sort. Back: `INSERTION_SORT`
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707400559085--> <!--ID: 1707400559085-->
END%% END%%
@ -131,14 +131,14 @@ We prove $P$ maintains the requisite properties:
%%ANKI %%ANKI
Basic Basic
Given array `A[0..n-1]`, what is insertion sort's loop invariant? Given array `A[0..n-1]`, what is `INSERTION_SORT`'s loop invariant?
Back: `A[0..i-1]` consists of the original `A[0..i-1]` elements but in sorted order. Back: `A[0..i-1]` consists of the original `A[0..i-1]` elements but in sorted order.
<!--ID: 1707332638371--> <!--ID: 1707332638371-->
END%% END%%
%%ANKI %%ANKI
Basic Basic
What is initialization of insertion sort's loop invariant? What is initialization of `INSERTION_SORT`'s loop invariant?
Back: Sorting starts with an singleton array which is trivially sorted. Back: Sorting starts with an singleton array which is trivially sorted.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707332638373--> <!--ID: 1707332638373-->
@ -146,7 +146,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is maintenance of insertion sort's loop invariant? What is maintenance of `INSERTION_SORT`'s loop invariant?
Back: Each iteration puts the current element into sorted order. Back: Each iteration puts the current element into sorted order.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707332638375--> <!--ID: 1707332638375-->
@ -154,7 +154,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
insertion sort makes fewer {comparisons} than selection sort in the average case. `INSERTION_SORT` makes fewer {comparisons} than `SELECTION_SORT` in the average case.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708002185982--> <!--ID: 1708002185982-->
END%% END%%
@ -167,7 +167,7 @@ If you repeat this process for every card in the deck, your left hand will event
%%ANKI %%ANKI
Basic Basic
What analogy does Cormen et al. use to explain insertion sort? What analogy does Cormen et al. use to explain `INSERTION_SORT`?
Back: Sorting a shuffled deck of playing cards. Back: Sorting a shuffled deck of playing cards.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706927594729--> <!--ID: 1706927594729-->
@ -175,7 +175,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What invariant does the left hand maintain in Cormen et al.'s insertion sort analogy? What invariant does the left hand maintain in Cormen et al.'s `INSERTION_SORT` analogy?
Back: It contains all drawn cards in sorted order. Back: It contains all drawn cards in sorted order.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1706927594732--> <!--ID: 1706927594732-->
@ -183,7 +183,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How does insertion sort partition its input array? How does `INSERTION_SORT` partition its input array?
Back: Back:
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
@ -194,7 +194,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How many comparisons does insertion sort typically perform with `x`? How many comparisons does `INSERTION_SORT` typically perform with `x`?
``` ```
[ sorted | x : unsorted ] [ sorted | x : unsorted ]
``` ```
@ -205,7 +205,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which element will insertion sort move to `sorted`? Which element will `INSERTION_SORT` move to `sorted`?
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
``` ```

24
notes/algorithms/sorting/merge-sort.md
View File

@ -22,7 +22,7 @@ Adaptive | -
%%ANKI %%ANKI
Basic Basic
What does the term "merge" in merge sort refer to? What does the term "merge" in `MERGE_SORT` refer to?
Back: The primary operation used to combine array halves. Back: The primary operation used to combine array halves.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467101--> <!--ID: 1708742467101-->
@ -30,7 +30,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is merge sort's best case runtime? What is `MERGE_SORT`'s best case runtime?
Back: $\Omega(n\lg{n})$ Back: $\Omega(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467109--> <!--ID: 1708742467109-->
@ -38,7 +38,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is merge sort's worst case runtime? What is `MERGE_SORT`'s worst case runtime?
Back: $O(n\lg{n})$ Back: $O(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467112--> <!--ID: 1708742467112-->
@ -46,7 +46,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is merge sort's average case runtime? What is `MERGE_SORT`'s average case runtime?
Back: $O(n\lg{n})$ Back: $O(n\lg{n})$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467115--> <!--ID: 1708742467115-->
@ -54,7 +54,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is the "divide" step of merge sort? What is the "divide" step of `MERGE_SORT`?
Back: Divide the input array into two subarrays of half size. Back: Divide the input array into two subarrays of half size.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467118--> <!--ID: 1708742467118-->
@ -62,15 +62,15 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is the "conquer" step of merge sort? What is the "conquer" step of `MERGE_SORT`?
Back: Call merge sort on the "divide"-step's two subarrays. Back: Call `MERGE_SORT` on the "divide"-step's two subarrays.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467121--> <!--ID: 1708742467121-->
END%% END%%
%%ANKI %%ANKI
Basic Basic
What is the "combine" step of merge sort? What is the "combine" step of `MERGE_SORT`?
Back: Merge the sorted subarrays produced by the "conquer" step. Back: Merge the sorted subarrays produced by the "conquer" step.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467125--> <!--ID: 1708742467125-->
@ -78,7 +78,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
When does merge sort "bottom out"? When does `MERGE_SORT` "bottom out"?
Back: When the sequence to be sorted has length 1 or less. Back: When the sequence to be sorted has length 1 or less.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467133--> <!--ID: 1708742467133-->
@ -128,7 +128,7 @@ void merge_sort(int i, int j, int *A) {
%%ANKI %%ANKI
Basic Basic
Where in merge sort's implementation are sentinels useful? Where in `MERGE_SORT`'s implementation are sentinels useful?
Back: As the last elements of the two arrays to combine. Back: As the last elements of the two arrays to combine.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467137--> <!--ID: 1708742467137-->
@ -136,7 +136,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What sentinel values are typically used in merge sort's "merge" operation? What sentinel values are typically used in `MERGE_SORT`'s "merge" operation?
Back: $\infty$ or the record type's equivalent. Back: $\infty$ or the record type's equivalent.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742467141--> <!--ID: 1708742467141-->
@ -146,7 +146,7 @@ END%%
Basic Basic
What sorting algorithm does the following demonstrate? What sorting algorithm does the following demonstrate?
![[merge-sort.gif]] ![[merge-sort.gif]]
Back: Merge sort. Back: `MERGE_SORT`
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708742590435--> <!--ID: 1708742590435-->
END%% END%%

28
notes/algorithms/sorting/selection-sort.md
View File

@ -23,7 +23,7 @@ Adaptive | No
%%ANKI %%ANKI
Basic Basic
Describe selection sort in a single sentence. Describe `SELECTION_SORT` in a single sentence.
Back: Repeatedly put the smallest unsorted record at the end of a sorted array. Back: Repeatedly put the smallest unsorted record at the end of a sorted array.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707589393190--> <!--ID: 1707589393190-->
@ -31,7 +31,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is selection sort's best case runtime? What is `SELECTION_SORT`'s best case runtime?
Back: $\Omega(n^2)$ Back: $\Omega(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773323--> <!--ID: 1707398773323-->
@ -39,7 +39,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is selection sort's worst case runtime? What is `SELECTION_SORT`'s worst case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773326--> <!--ID: 1707398773326-->
@ -47,7 +47,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is selection sort's average case runtime? What is `SELECTION_SORT`'s average case runtime?
Back: $O(n^2)$ Back: $O(n^2)$
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773327--> <!--ID: 1707398773327-->
@ -55,7 +55,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is selection sort in place? Is `SELECTION_SORT` in place?
Back: Yes Back: Yes
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773328--> <!--ID: 1707398773328-->
@ -63,7 +63,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is selection sort stable? Is `SELECTION_SORT` stable?
Back: No Back: No
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773330--> <!--ID: 1707398773330-->
@ -71,7 +71,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Is selection sort adaptive? Is `SELECTION_SORT` adaptive?
Back: No Back: No
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707504634778--> <!--ID: 1707504634778-->
@ -101,7 +101,7 @@ void selection_sort(const int n, int A[static n]) {
Basic Basic
What sorting algorithm does the following demonstrate? What sorting algorithm does the following demonstrate?
![[selection-sort.gif]] ![[selection-sort.gif]]
Back: Selection sort. Back: `SELECTION_SORT`
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707400943836--> <!--ID: 1707400943836-->
END%% END%%
@ -123,7 +123,7 @@ We prove $P$ maintains the requisite properties:
%%ANKI %%ANKI
Basic Basic
Given array `A[0..n-1]`, what is selection sort's loop invariant? Given array `A[0..n-1]`, what is `SELECTION_SORT`'s loop invariant?
Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`. Back: `A[0..i-1]` is a sorted array of the `i` least elements of `A`.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773331--> <!--ID: 1707398773331-->
@ -131,7 +131,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is initialization of selection sort's loop invariant? What is initialization of `SELECTION_SORT`'s loop invariant?
Back: Sorting starts with an empty array which is trivially sorted. Back: Sorting starts with an empty array which is trivially sorted.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773333--> <!--ID: 1707398773333-->
@ -139,7 +139,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What is maintenance of selection sort's loop invariant? What is maintenance of `SELECTION_SORT`'s loop invariant?
Back: Each iteration puts the next least element into the sorted subarray. Back: Each iteration puts the next least element into the sorted subarray.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1707398773334--> <!--ID: 1707398773334-->
@ -147,7 +147,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
How does selection sort partition its input array? How does `SELECTION_SORT` partition its input array?
Back: Back:
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
@ -158,7 +158,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Which element will selection sort move to `sorted`? Which element will `SELECTION_SORT` move to `sorted`?
``` ```
[ sorted | unsorted ] [ sorted | unsorted ]
``` ```
@ -169,7 +169,7 @@ END%%
%%ANKI %%ANKI
Cloze Cloze
Selection sort makes fewer {swaps} than insertion sort in the average case. `SELECTION_SORT` makes fewer {swaps} than `INSERTION_SORT` in the average case.
Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009). Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
<!--ID: 1708002177782--> <!--ID: 1708002177782-->
END%% END%%

33
notes/combinatorics/combinations.md
View File

@ -13,29 +13,26 @@ A $k$-**combination** of $n$ objects is an unordered "choice" of $k$ objects fro
```c ```c
void combinations_aux( void combinations_aux(
const int i, const int n, const size_t n, int A[static n],
const int j, const int k, const size_t k, int stack[static k],
int A[static n], const size_t i
int res[static k]
) { ) {
if (j == k) { if (n < k) {
for (int m = 0; m < k; ++m) {
printf("%d ", A[res[m]]);
}
printf("\n");
return
} else if (n - i < k - j) {
return; return;
} }
res[j] = A[i]; if (k == 0) {
combinations_aux(i + 1, n, j + 1, k, A, res); print_array(i, stack);
combinations_aux(i + 1, n, j, k, A, res); return;
}
stack[i] = A[0];
combinations_aux(n - 1, A + 1, k - 1, stack, i + 1);
combinations_aux(n - 1, A + 1, k, stack, i);
} }
void combinations(const int n, const int k, int A[static n]) { void combinations(const size_t n, const size_t k, int A[static n]) {
int *res = malloc(sizeof(int) * k); int *stack = calloc(k, sizeof(int));
combinations_aux(0, n, 0, k, A, res); combinations_aux(n, A, k, stack, 0);
free(res); free(stack);
} }
``` ```

2
notes/combinatorics/inclusion-exclusion.md
View File

@ -53,7 +53,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Given finite sets $A$ and $B$, what is $|A \cup B \cup C|$? Given finite sets $A$, $B$, and $C$, what is $|A \cup B \cup C|$?
Back: $|A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|$ Back: $|A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|$
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). 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: 1708438356480--> <!--ID: 1708438356480-->

246
notes/combinatorics/permutations.md
View File

@ -43,7 +43,7 @@ void permutations(const size_t n, int A[static n]) {
} }
``` ```
The above approach prints out all permutations of an array. The above approach prints out all permutations of an array (assuming distinct values).
%%ANKI %%ANKI
Basic Basic
@ -166,6 +166,250 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
<!--ID: 1708366788613--> <!--ID: 1708366788613-->
END%% END%%
### Lexicographic Ordering
We can find the next lexicographic ordering of an array via a procedure of "pivot", "swap", and "reverse". The function `void next(const size_t n, int A[static n])` defined in [[#Overview]] shows the details, taking in a permutation and producing the next, lexicographically speaking. To prove correctness, consider the following:
```
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
```
Here the RHS side is the longest increasing sequence we could find, from right to left. That is, $a_{i+1} > a_{i+2} > \cdots > a_n$. Denote $a_i$ as the pivot. Next, swap the smallest element in the RHS greater than $a_i$, say $a_j$, with $a_i$. This produces
```
[ a₁ a₂ ... aⱼ | aᵢ₊₁ aᵢ₊₂ ... aᵢ ... aₙ ]
```
Notice the RHS remains in sorted order. Since $a_j$ was the next smallest element, reversing the reverse-sorted RHS produces the next permutation, lexicographically speaking:
```
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
```
Eventually the swapped $a_j$ will be the largest in the RHS ensuring that the breakpoint will eventually move one more position leftward.
%%ANKI
Basic
What algorithm does `NEXT_LEXICO_ARRAY` refer to?
Back: The finding of the next lexicographic ordering of an array.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610310-->
END%%
%%ANKI
Basic
What does array `A`'s next lexicographic ordering refer to?
Back: The permutation that follows `A` in a sorted list of all distinct permutations of `A`.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610316-->
END%%
%%ANKI
Basic
How does lexicographic ordering of arrays relate to permutations of an array?
Back: Each lexicographic ordering corresponds to a permutation.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610319-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: How many invocations guarantee all permutations of `A[1:n]`?
Back: $n!$
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610322-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: When does `< n!` iterations yield all permutations of `A[1:n]`?
Back: When `A` contains duplicates.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610326-->
END%%
%%ANKI
Basic
What is `NEXT_LEXICO_ARRAY`'s worst-case running time?
Back: $O(n)$
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610329-->
END%%
%%ANKI
Basic
What is `NEXT_LEXICO_ARRAY`'s best-case running time?
Back: $\Omega(n)$
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610332-->
END%%
%%ANKI
Basic
What is `NEXT_LEXICO_ARRAY`'s auxiliary memory usage?
Back: $O(1)$
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610335-->
END%%
%%ANKI
Basic
What is the next lexicographic ordering of `[ 1 3 2 4 ]`?
Back: `[ 1 3 4 2 ]`
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610344-->
END%%
%%ANKI
Basic
What is the next lexicographic ordering of `[ 2 1 4 3 ]`?
Back: `[ 2 3 1 4 ]`
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610349-->
END%%
%%ANKI
Basic
What is the next lexicographic ordering of `[ 4 3 2 1 ]`?
Back: N/A
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610357-->
END%%
%%ANKI
Basic
What is the output of `NEXT_LEXICO_ARRAY([ 1 2 3 4 ])`?
Back: `[ 1 2 4 3 ]`
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610364-->
END%%
%%ANKI
Basic
What is the output of `NEXT_LEXICO_ARRAY([ 4 3 2 1 ])`?
Back: `[ 1 2 3 4 ]`
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610371-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: Which element will be the pivot of `[ 1 2 3 4 ]`?
Back: `3`
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610377-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: Which element will be the pivot of `[ 4 3 2 1 ]`?
Back: N/A
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756677668-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What property does the RHS of `A[1:n]` exhibit before swapping?
```
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
```
Back: Values are in decreasing order.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610384-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What property does the LHS of `A[1:n]` exhibit before swapping?
```
[ a₁ a₂ ... aᵢ | aᵢ₊₁ aᵢ₊₂ ... aₙ ]
```
Back: N/A
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610393-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What property does the RHS of `A[1:n]` exhibit after swapping?
```
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
```
Back: Values are in decreasing order.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
<!--ID: 1709756610399-->
END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What property does the LHS of `A[1:n]` exhibit after swapping?
```
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
```
Back: N/A
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
What is the first step taken in the `NEXT_LEXICO_ARRAY` algorithm?
Back: Finding the pivot element.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What does the "pivot" refer to?
Back: The element preceding the longest increasing subarray from right-to-left.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Cloze
The `NEXT_LEXICO_ARRAY` algorithm can be summed up as "{pivot}", "{swap}", and "{reverse}".
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: *Which* element is swapped with the pivot?
Back: The smallest element to its right that is greater than it.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What is done after swapping the pivot element?
Back: Reverse the subarray to the right of where the pivot element was originally located.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What step follows swapping `A[1:n]`'s pivot element?
```
[ a₁ a₂ ... aⱼ | aₙ ... aᵢ ... aᵢ₊₂ aᵢ₊₁ ]
```
Back: Reverse the elements to the right of $a_j$.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
%%ANKI
Basic
`NEXT_LEXICO_ARRAY`: What invariant is maintained after swapping the pivot?
Back: The elements to the right of the original pivot remain in non-increasing order.
Reference: [https://leetcode.com/problems/next-permutation/description/](https://leetcode.com/problems/next-permutation/description/)
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END%%
## Falling Factorials ## Falling Factorials
If we generalize to choosing $k \leq n$ elements of $k$ objects, we can calculate the $k$-permutation of $n$. This is denoted as $(n)_k$, sometimes called the **falling factorial**. $$(n)_k = \frac{n!}{(n - k)!}$$ If we generalize to choosing $k \leq n$ elements of $k$ objects, we can calculate the $k$-permutation of $n$. This is denoted as $(n)_k$, sometimes called the **falling factorial**. $$(n)_k = \frac{n!}{(n - k)!}$$

11
notes/encoding/integer.md
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@ -549,7 +549,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
Let $x$ be a $w$-bit two's-complement number. What is it's complement? Let $x$ be a $w$-bit two's-complement number. What is it's complement?
Back: The number $y$ such that $x + y = 2^w$. Back: $2^w - x$
Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561). Reference: “Two’s-Complement.” In *Wikipedia*, January 9, 2024. [https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561](https://en.wikipedia.org/w/index.php?title=Two%27s_complement&oldid=1194543561).
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END%% END%%
@ -634,10 +634,11 @@ Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs
END%% END%%
%%ANKI %%ANKI
Cloze Basic
Subtracting $x$ from {a number of all `1` bits} is equivalent to {`~x`}. Subtracting `x` from all `1` bits is equivalent to what bitwise operation?.
Back: `~x`
Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html). Reference: Finley, Thomas. “Two’s Complement,” April 2000. [https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html](https://www.cs.cornell.edu/~tomf/notes/cps104/twoscomp.html).
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END%% END%%
## Casting ## Casting
@ -964,7 +965,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What hardware-level advantage does two's-complement introduce over other signed encodings? What hardware-level advantage does two's-complement introduce over other signed encodings?
Back: The same circuits can be used for unsigned and two's-complement addition. Back: The same circuits can be used for unsigned and two's-complement arithmetic.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016. Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
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END%% END%%

2
notes/posix/signals.md
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@ -239,7 +239,7 @@ END%%
%%ANKI %%ANKI
Basic Basic
What distinguishes `abort()` from `exit()`? What distinguishes `abort()` from `exit()`?
Back: The former sends a `SIGABRT` signal. The latter closes the application normally. Back: The latter gracefully closes the application.
Reference: N/A Reference: N/A
Tags: c17 Tags: c17
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