Notes on binary search trees.

c-declarations
Joshua Potter 2024-08-03 13:31:33 -06:00
parent 0136ab29cf
commit 2c099dff15
16 changed files with 427 additions and 20 deletions

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@ -146,7 +146,15 @@
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title: "2024-08-03"
---
- [ ] Anki Flashcards
- [x] Anki Flashcards
- [x] KoL
- [ ] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
- [ ] Korean (Read 1 Story)
* Additional notes on binary search trees.

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@ -277,7 +277,7 @@ END%%
%%ANKI
Basic
Which hexadecimal digits are encoded in binary with a leading `1` bit?
Back: `8` through `F`
Back: `8` through `F`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1708631918825-->
@ -286,7 +286,7 @@ END%%
%%ANKI
Basic
Which hexadecimal digits are encoded in binary with a leading `0` bit?
Back: `0` through `7`
Back: `0` through `7`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1708631918829-->

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@ -13,6 +13,8 @@ A binary search tree (BST) is a [[trees#Binary Trees|binary tree]] satisfying th
> Let $x$ be a node in a binary search tree. If $y$ is a node in the left subtree of $x$, then $y.key \leq x.key$. If $y$ is a node in the right subtree of $x$, then $y.key \geq x.key$.
## Traversals
Consider an arbitrary node $x$ of some BST. Then:
* An **inorder** traversal visits $x$'s left child, then $x$, then $x$'s right child.
@ -190,6 +192,398 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
<!--ID: 1722342235784-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime of search?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303021-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for finding the minimum?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303023-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for finding the maximum?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303025-->
END%%
### Successors
The **successor** of a node in a binary search tree is the node whose value would appear immediately after in an in-order traversal.
%%ANKI
Basic
How do we define the successor of a BST node in terms of in-order traversals?
Back: As the node encountered immediately after in an in-order traversal.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623599-->
END%%
%%ANKI
Basic
Which node is the successor of $7$ in the following BST?
![[binary-search-tree.png]]
Back: $8$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623601-->
END%%
%%ANKI
Basic
Which node is the successor of $2$ in the following BST?
![[binary-search-tree.png]]
Back: $5$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623603-->
END%%
%%ANKI
Basic
Which node is the successor of $5$ in the following BST?
![[binary-search-tree.png]]
Back: Either $5$ or $6$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623604-->
END%%
%%ANKI
Basic
Which node(s) in a BST have no successor?
Back: The maximum.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623606-->
END%%
%%ANKI
Basic
What two cases are considered when finding the successor of a BST node?
Back: If the node has a right subtree or not.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623607-->
END%%
%%ANKI
Basic
Suppose a BST node has a right subtree. What is its successor?
Back: The minimum of the right subtree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623608-->
END%%
%%ANKI
Basic
Suppose a BST node does not have a right subtree. What is its successor?
Back: The first proper ancestor reached from the LHS.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623610-->
END%%
%%ANKI
Cloze
If a BST node has a right subtree, it's successor cannot have a {left} subtree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303026-->
END%%
%%ANKI
Basic
Suppose a BST node has a right subtree. *Why* can't its successor have a left subtree?
Back: Because then a node in that left subtree would be the actual successor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303028-->
END%%
%%ANKI
Basic
Suppose a BST node has a right subtree. *Why* can't its successor have a right subtree?
Back: N/A. It can.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303029-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for finding a node's successor?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303030-->
END%%
### Predecessors
The **predecessor** of a node in a binary search tree is the node whose value would appear immediately before in an in-order traversal.
%%ANKI
Basic
How do we define the predecessor of a BST node in terms of in-order traversals?
Back: As the node encountered immediately before in an in-order traversal.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623611-->
END%%
%%ANKI
Basic
Which node is the predecessor of $7$ in the following BST?
![[binary-search-tree.png]]
Back: $6$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623613-->
END%%
%%ANKI
Basic
Which node is the predecessor of $2$ in the following BST?
![[binary-search-tree.png]]
Back: N/A.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623614-->
END%%
%%ANKI
Basic
Which node is the predecessor of $5$ in the following BST?
![[binary-search-tree.png]]
Back: Either $2$ or $5$.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623615-->
END%%
%%ANKI
Basic
Which node(s) in a BST have no predecessor?
Back: The minimum.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623616-->
END%%
%%ANKI
Basic
What two cases are considered when finding the predecessor of a BST node?
Back: If the node has a left subtree or not.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623617-->
END%%
%%ANKI
Basic
Suppose a BST node has a left subtree. What is its predecessor?
Back: The maximum of the left subtree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623618-->
END%%
%%ANKI
Basic
Suppose a BST node does not have a left subtree. What is its predecessor?
Back: The first proper ancestor reached from the RHS.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722709623619-->
END%%
%%ANKI
Cloze
If a BST node has a left subtree, it's predecessor cannot have a {right} subtree.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303031-->
END%%
%%ANKI
Basic
Suppose a BST node has a left subtree. *Why* can't its successor have a left subtree?
Back: N/A. It can.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303033-->
END%%
%%ANKI
Basic
Suppose a BST node has a left subtree. *Why* can't its predecessor have a right subtree?
Back: Because then a node in that right subtree would be the actual predecessor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303034-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for finding a node's predecessor?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303035-->
END%%
## Deletions
Consider deleting node $z$ from a BST. There are three conceptual cases to consider corresponding to the number of children $z$ has:
* If $z$ has no children we can just replace $z$ with `NIL`.
* If $z$ has one child, we can replace $z$ with its child.
* If $z$ has two children, we swap $z$ with either its predecessor or successor, updating pointers as necessary to maintain the binary-search-tree property.
%%ANKI
Basic
Insert node $7.5$ into the following BST. Where is the new node located?
![[binary-search-tree.png]]
Back: As $8$'s left child.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303036-->
END%%
%%ANKI
Basic
Insert node $6.5$ into the following BST. Where is the new node located?
![[binary-search-tree.png]]
Back: As $7$'s left child.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303037-->
END%%
%%ANKI
Basic
Insert node $5.5$ into the following BST. Where is the new node located?
![[binary-search-tree.png]]
Back: As the lower $5$'s right child.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303038-->
END%%
%%ANKI
Basic
When does insertion into a BST modify the root node?
Back: When the tree being inserted into is empty.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303039-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for inserting a node?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303040-->
END%%
%%ANKI
Basic
How many cases are there to consider when deleting a node from a BST?
Back: Three.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303041-->
END%%
%%ANKI
Basic
What corresponds to the cases to consider when deleting a node from a BST?
Back: The number of children the node in question has.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303042-->
END%%
%%ANKI
Basic
Delete BST node $z$ with no children. Which node is $z$ replaced with?
Back: `NIL`
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303043-->
END%%
%%ANKI
Basic
Delete BST node $z$ with one child. Which node is $z$ replaced with?
Back: Its one child.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303044-->
END%%
%%ANKI
Basic
Delete BST node $z$ with two children. Which node is $z$ replaced with?
Back: Either its successor or predecessor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303045-->
END%%
%%ANKI
Basic
Delete BST node $z$ with two children. If replacing with its successor, what two subcases need to be considered?
Back: If $z$'s successor is its right child or not.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303046-->
END%%
%%ANKI
Basic
Delete BST node $z$ with two children. If replacing with its predecessor, what two subcases need to be considered?
Back: If $z$'s predessor is its left child or not.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303047-->
END%%
%%ANKI
Basic
In terms of the height $h$ of a BST, what is the runtime for deleting a node?
Back: $O(h)$
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303048-->
END%%
%%ANKI
Basic
Delete $z$ from the following BST. What does the resulting tree look like?
![[bst-right-child.png]]
Back:
![[bst-right-child-after.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303049-->
END%%
%%ANKI
Basic
Delete $z$ from the following BST. What does the resulting tree look like?
![[bst-left-child.png]]
Back:
![[bst-left-child-after.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303050-->
END%%
%%ANKI
Basic
Delete $z$ from the following BST. What does the resulting tree look like?
![[bst-right-succ.png]]
Back:
![[bst-right-succ-after.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303051-->
END%%
%%ANKI
Basic
Delete $z$ from the following BST. What does the resulting tree look like?
![[bst-deep-succ.png]]
Back:
![[bst-deep-succ-after.png]]
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713303052-->
END%%
%%ANKI
Basic
What arbitrary choice was implied in the deletion algorithm of the following BST?
![[bst-deep-succ.png]]
Back: To replace deleted nodes with their successor instead of predecessor.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
<!--ID: 1722713414719-->
END%%
## Bibliography
* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

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@ -328,7 +328,7 @@ END%%
%%ANKI
Basic
Let $h$ be a division method hash function. What does $h(10)$ evaluate to?
Back: To $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
Back: $10 \bmod{m}$, where $m$ is the number of slots in the hash table.
Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
Tags: hashing::static
<!--ID: 1720889385419-->

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@ -123,7 +123,7 @@ END%%
%%ANKI
Basic
RDF identifiers are represented using what?
RDF identifiers use what encoding?
Back: URIs.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
<!--ID: 1722191359882-->

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@ -67,7 +67,7 @@ where the `<authority>` is further composed as
Cloze
A URI has the following generic syntax:
{`<schema>`}`:`{`[//<authority>]`}`]`{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
{`<scheme>`}`:`{`[//<authority>]`}{`<path>`}{`[?<query>]`}{`[#<fragment>]`}.
Reference: “Uniform Resource Identifier.” In _Wikipedia_, July 22, 2024. [https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier](https://en.wikipedia.org/w/index.php?title=Uniform_Resource_Identifier&oldid=1235957234).
<!--ID: 1722211276499-->
END%%
@ -488,7 +488,7 @@ A **compact URI** (CURIE) is a denser representation of URIs. In its simplest fo
%%ANKI
Basic
URIs in RDF are usually represented in what condensed format?
Back: CURIE.
Back: CURIEs.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
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END%%
@ -496,7 +496,7 @@ END%%
%%ANKI
Basic
What is CURIE an acronym for?
Back: **C**ompact **URI**&.
Back: **C**ompact **URI**s.
Reference: Allemang, Dean, James A. Hendler, and Fabien L. Gandon. _Semantic Web for the Working Ontologist_. 3e ed. ACM Books 33. New York: Association for computing machinery, 2020.
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@ -647,7 +647,7 @@ END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ reflexive on $a$?
Back: N/A. We must ask if $R$ is reflexive on a set.
Back: N/A. We should ask if $R$ is reflexive on set $\{a\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%
@ -729,7 +729,7 @@ END%%
%%ANKI
Basic
Given $R = \{\langle a, a \rangle, \langle b, c \rangle\}$, is $R$ irreflexive on $a$?
Back: N/A. We must ask if $R$ is irreflexive on a set.
Back: N/A. We should ask if $R$ is irreflexive on set $\{a\}$.
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
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END%%