Computing (Fun)damentals - Stanford...

Computing (Fun)damentalsJULIA GONG && BEN NEWMAN05.02.2019SPLASH – FALL 2019

Who are we?

JULIA GONG BEN NEWMAN

Class of 2021

Mathematical and Computational Science

[email protected]

Class of 2021

Computer Science

[email protected]

mailto:[email protected]


the plan

• Boolean Algebra

• Logic Gates

• Computer Number Systems

• Bit Shifting

• Prefix, Infix, Postfix

• LISP Expressions

• Assembly Language

• Recursive Functions

• Graph Theory

• Data Structures

• Interpreting Programs

the plan

• Boolean Algebra

• Logic Gates


• Bit Shifting





• Graph Theory

• Data Structures


lower-level, systems

higher-level, algorithms

the plan

• Boolean Algebra

• Logic Gates


• Bit Shifting





• Graph Theory

• Data Structures


lower-level, systems

higher-level, algorithms

!!! WOW

CRASH

COURSE !!!

Why 0s and 1s?

Why 0s and 1s? …I will now

multiply

these

numbers…

Why 0s and 1s? …00101100

010110101

011101001

01010100…

Why 0s and 1s? …00101100

010110101

011101001

01010100…

To perform computations

quickly, we have to speak fluent computer!

Boolean Algebra

Variables: store values!𝑋 = 0 “false”

𝑌 = 1 “true”

Boolean Algebra

Variables: store values!

Common operations on variables:

• NOT

𝑋 = 0 “false”

𝑌 = 1 “true”

ത𝑋 = 1

Boolean Algebra



• NOT

• OR


𝑌 = 1 “true”

ത𝑋 = 1

𝑋 + 𝑌 = 1

Boolean Algebra



• NOT

• OR

• AND


𝑌 = 1 “true”

ത𝑋 = 1

𝑋 + 𝑌 = 1

𝑋𝑌 = 0

Boolean Algebra



• NOT

• OR

• AND

• XOR


𝑌 = 1 “true”

ത𝑋 = 1

𝑋 + 𝑌 = 1

𝑋𝑌 = 0

𝑋⊕ 𝑌 = 1

Boolean Algebra



• NOT

• OR

• AND

• XOR


𝑌 = 1 “true”

ത𝑋 = 1

𝑋 + 𝑌 = 1

𝑋𝑌 = 0

𝑋⊕ 𝑌 = 1

(also = ത𝑋𝑌 + 𝑋ത𝑌)

Boolean Algebra

Truth Tables! X Y ??

1 1 1

1 0 0

0 1 0

0 0 0

Meme credits: http://www.quickmeme.com/p/3vx72h

http://www.quickmeme.com/p/3vx72h

Boolean Algebra

Truth Tables! X Y XY

1 1 1

1 0 0

0 1 0

0 0 0



Boolean Algebra

Truth Tables!

De Morgan’s Laws!!

𝑋 + 𝑌 = ത𝑋ത𝑌

𝑋𝑌 = ത𝑋 + ത𝑌

X Y XY

1 1 1

1 0 0

0 1 0

0 0 0



Logic Gates (like Boolean Algebra, but visual!)

“Circuits”

input

variables

AND

ANDAND

OR

XOR

NOT

NOT

NOT

NOT

Circuit credits: https://drstienecker.com/tech-332/3-logic-circuits-boolean-algebra-and-truth-tables/

https://drstienecker.com/tech-332/3-logic-circuits-boolean-algebra-and-truth-tables/


“Circuits”

input

variables

AND

ANDAND

OR

XOR

NOT

NOT

NOT




“Circuits”

input

variables

AND

ANDAND

OR

XOR

NOT

NOT

NOT

NOT




“Circuits”

input

variables

AND

ANDAND

OR

XOR

NOT

NOT

NOT

NOT

output

variable




“Circuits”

input

variables

AND

ANDAND

OR

XOR

NOT

NOT

NOT

NOT

output

variable

𝑌 = ҧ𝐴𝐵𝐵𝐶 + (𝐵𝐵𝐶 ⊕ 𝐵𝐶)



Computer Number Systems

We’re used to base 10 (decimal)! But there’s…



Like base 2 (binary), 8 (octal), and 16 (hexadecimal)!

Meme credits: https://giphy.com/gifs/moments-part-11thnyggFkrmmc

https://giphy.com/gifs/moments-part-11thnyggFkrmmc



Like base 2 (binary), 8 (octal), and 16 (hexadecimal)!

Fun fact: colors can be written as hexadecimals (#FFFFFF = white!)

Meme credits: https://giphy.com/gifs/moments-part-11thnyggFkrmmc

https://giphy.com/gifs/moments-part-11thnyggFkrmmc


4 2 10

= 4 * 10 + 2 * 1

= 4 * 101 + 2 * 100


4 2 16

= 4 * 16 + 2 * 1

= 4 * 161 + 2 * 160

= 6610


4 2 8

= 4 * 8 + 2 * 1

= 4 * 81 + 2 * 80

= 3410


16110 = X8?

84 83 82 81 80


0 0 816110 = X8?

84 83 82 81 80


0 0 2 8

16110 = 2*82

16110 = X8?

84 83 82 81 80


0 0 2 4 8

16110 = 2*82 + 4*81

16110 = X8?

84 83 82 81 80


0 0 2 4 1 8

16110 = 2*82 + 4*81 + 1*80

16110 = X8?

84 83 82 81 80


0 0 2 4 1 8

16110 = 2*82 + 4*81 + 1*80

16110 = X8?

84 83 82 81 80

Always go from big to small!


0 0 2 4 1 8

16110 = 2*82 + 4*81 + 1*80

16110 = X8?

84 83 82 81 80

Always go from big to small!

That’s how things fit ☺Insightful picture credits: https://www.akindjourney.com/articles/big-rocks

https://www.akindjourney.com/articles/big-rocks

Bit Shifting

Essentially operations on bits to go between numbers (really fast in a computer!)

Bit Shifting


12 << 2 = 1 * 2 * 2 = 1 * 22 = 4 = 1002

11 << 2 = 3 * 2 * 2 = 3 * 22 = 12 = 11002

Bit Shifting


Example:

(LSHIFT-2(LCIRC-2(RSHIFT-1 10100)))

Bit Shifting


Example:


10100

Bit Shifting


Example:


1010

Bit Shifting


Example:


01010

Bit Shifting


Example:


1010

Circuit credits: http://clipart-library.com/clipart/angel-wings-clip-art-33.htm

http://clipart-library.com/clipart/angel-wings-clip-art-33.htm

Bit Shifting


Example:


10100

Bit Shifting


Example:


0100

Circuit credits: http://clipart-library.com/clipart/angel-wings-clip-art-33.htm

http://clipart-library.com/clipart/angel-wings-clip-art-33.htm

Bit Shifting


Example:


01001

Bit Shifting


Example:


1001

Bit Shifting


Example:


001

Bit Shifting


Example:


00100

Bit Shifting


Example:


00100YAY!

Prefix, Infix, Postfix


INFIX – what we’re used to!

X * (Y + Z)


PREFIX (no parentheses!)

* X + Y Z

INFIX

X * (Y + Z)



* X + Y Z

INFIX

X * (Y + Z)

Y + Z



* X + Y Z

INFIX

X * (Y + Z)

X * (Y + Z)



* X + Y Z

INFIX

X * (Y + Z)

POSTFIX (also no parentheses!)

X Y Z + *

Recursive Functions

Recursive Functions Hope someone down

there knows the

solution!


there knows the

solution!

Hope someone down

there knows the

solution!


there knows the

solution!

Hope someone down

there knows the

solution!

Hope someone down

there knows the

solution!

Recursive Functions

Recursive Functions

DONE

Recursive Functions

DONE

I know what to do with

this already completed

thing!

Recursive Functions

Recursive Functions

DONE

Recursive Functions

DO

NE

DO

NE

Recursive Functions

DONE

Recursive Functions

Recursive FunctionsJust finished that off without

knowing what you all did.

Recursive Functions

PLOT TWIST

Just finished that off without


Recursive Functions

They’re all the

same person,

just at different

times!

Just finished that off without


Recursive Functions

1st recursive call2nd call3rdlast

Recursive Functions

How did we know to stop here?


Recursive Functions

RECUR-

SIVE

CASE

RECUR-

SIVE

CASE

RECUR-

SIVE

CASE

How did we know to stop here? The answer was basic!

BASE

CASE


Recursive Functions

RECURSIVE SUBSTRUCTURE

Strategy: Ask yourself, “what subproblem should I solve so that, if I knew the answer to that problem, I could solve the whole problem?”

Recursive Functions

Base case = the simplest “duh” case you know you can solve!

Usually just returning 0, 1, a concrete sum, an empty list, empty string, or something very simple!

Recursive case = oh dear, I have no idea how to solve this entire problem, but let me solve part of it and then get my friends to solve the rest, and then I can do my part.

Recursive Functions



Recursive case = oh dear, I have no idea how to solve this entire problem, but let me solve part of it and then get my friends to solve the rest, and then I can do my part. my later self

Recursive Functions



Recursive case = oh dear, I have no idea how to solve this entire problem, but let me solve part of it and then get my friends to solve the rest, and then I can do my part. #lifelessons my later self

Recursive Functions: Fibonacci Sequence

fibonacci | 1, 1, 2, 3, 5, 8, 13, 21, …


fibonacci | 1, 1, 2, 3, 5, 8, 13, 21, …

n | 0, 1, 2, 3, 4, 5, 6, 7, …


fibonacci(n) | 1, 1, 2, 3, 5, 8, 13, 21, …

n | 0, 1, 2, 3, 4, 5, 6, 7, …


fibonacci(n) | 1, 1, 2, 3, 5, 8, 13, 21, …

n | 0, 1, 2, 3, 4, 5, 6, 7, …

def fibonacci(n):

if n == 0 or n == 1:

return 1


fibonacci(n) | 1, 1, 2, 3, 5, 8, 13, 21, …

n | 0, 1, 2, 3, 4, 5, 6, 7, …

def fibonacci(n):

return 1

BASE

CASE


fibonacci(n) | 1, 1, 2, 3, 5, 8, 13, 21, …

n | 0, 1, 2, 3, 4, 5, 6, 7, …

def fibonacci(n):

if n == 0 or n == 1:

return 1

return fibonacci(n-1) + fibonacci(n-2)

RECUR-

SIVE

CASE

Hope someone down

there knows the

solution!

Recursive Functions: Factorial

0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

…


0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

…

def factorial(n): ?? ?? RECUR-

SIVE

CASE

Hope you all can help

me figure out the

solution!

BASE

CASE


def factorial(n):

if ??:

??

else:

??

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE


def factorial(n):

if n == 0 or n == 1:

??

else:

??

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE


def factorial(n):

if n == 0 or n == 1:

return 1

else:

??

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE


def factorial(n):

if n == 0 or n == 1:

return 1

else:

return n * factorial(n – 1)

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE


0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE

3!


0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE

3!

2!


0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4 * 3 * 2 * 1

RECUR-

SIVE

CASE


me figure out the

solution!

BASE

CASE

3!

2!

1!

Recursive Functions: Tower of Hanoi

move(disk, source, dest, spare):

disk 1

disk 2

disk 3

source spare destination

(initial state)



source spare destination

disk 1

disk 2

disk 3

(the goal)



if disk == 1:

move disk 1 from source to dest

else if disk > 1:

move(disk – 1, source, spare, dest)

move disk n from source to dest

move(disk – 1, spare, dest, source)

Graph Theory

Modeling systems with individual units and relationships between them

Graph Theory

Modeling systems with individual units and relationships between them

Graph G = (V, E)

V = vertices (nodes)

E = edges (relationship between nodes)

Graph Theory

Modeling systems with individual units and relationships between them• Social media!

nodes: people

edge relation:

“is friends with”

Meme credits: https://www.researchgate.net/figure/Graph-theory-analysis-in-social-network-Image-courtesy-of-5_fig1_321846685

https://www.researchgate.net/figure/Graph-theory-analysis-in-social-network-Image-courtesy-of-5_fig1_321846685

Graph Theory


• The internet!

Meme credits: https://mathinsight.org/network_introduction

nodes: webpages

edge relation:

“links to”

(in 2004)

https://mathinsight.org/network_introduction

Graph Theory


• The internet!

• Maps (cities and roads)!

nodes: cities

edge relation:

“can be reached from”

Graph Theory


• The internet!


• Relationships between words!

nodes: words

edge relation:

“appears in similar contexts as”

Graph Theory


• The internet!



• Task lists and dependencies!

nodes: tasks

edge relation:

“must be completed before”

Graph Theory


• The internet!




• Job requests on machines!

nodes: job requests

edge relation:

“has higher priority than”

Graph Theory


• The internet!




• Job requests on machines!

Graphs can be used for LOTS of things…• Finding shortest paths, longest paths, least costly paths

• Strongly connected components

• Bipartite graphs

• …and soooo much more!

Graph Theory

• Graphs can be represented in many ways! One way is an adjacency matrix:

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

AB

CD

E

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

AB

CD

E

Arrows = directed graph

No arrows = undirected graph

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

AB

CD

E

“cycle”

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

Fun fact: this matrix has all paths

of length 1!

If you square this matrix, you get

all paths of length 2! If you

multiply it by itself n times, you

get all paths of length n!

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

And everything along the

diagonal denotes all cycles of

that length!

(“paths of length n from a node

to itself”)

Graph Theory


A

B

C

D

E

A B C D E0 1 0 1 0

0 0 1 1 0

1 0 1 1 1

1 0 0 0 0

0 0 0 1 0

We only have one cycle of length 1, which makes sense!

AB

CD

E

Simple Data Structures

Stacks: first in, last out

Queues: last in, last out


Stacks: first in, last out (like a stack of plates at the cafeteria)

Queues: last in, last out



Queues: last in, last out (like the carpool line or grocery check-out)

1 “push”

Credits to the site that gave me a hand: http://www.clipartpanda.com/clipart_images/light-hand-print-clip-art-at-58133828

http://www.clipartpanda.com/clipart_images/light-hand-print-clip-art-at-58133828




1“push”


2





1

“push”


23





1

“push”


234





1

“push”


2345





1

“pop”


234

5





1

“pop”


234





1

etc.


234






on the other hand…






1

“enqueue”






1

“enqueue”

2






1

“enqueue”

2 3






1

“enqueue”

2 3 4






1

“enqueue”

2 3 4 5






1

“dequeue”

2 3 4 5






“dequeue”

2 3 4 5



• LOTS of other cool stuff, including:

• Lists

• Sets

• Maps

• Hash sets, hash maps, hash tables

• Binary search trees

• Red-black trees

• Tries



• Lists

• Sets

• Maps



• Red-black trees

• Tries

“panda”, “blueberry”, “cat”



• Lists

• Sets

• Maps



• Red-black trees

• Tries

1, 2, 3, 4, 5, 6, 7, 8, 9, 100



• Lists

• Sets

• Maps



• Red-black trees

• Tries

1, 2, 3, 4, 5, 6, 7, 8, 9, 100

Order matters!

Index from 0 to 9, or 1 to 10



• Lists

• Sets

• Maps



• Red-black trees

• Tries

1, 2, 3, 4, 5, 6, 7, 8, 9, 100

Order matters!

Index from 0 to 9, or 1 to 10

Can be implemented as

stack or queue!



• Lists

• Sets

• Maps



• Red-black trees

• Tries

a

bc

d



• Lists

• Sets

• Maps



• Red-black trees

• Tries

a

bc

d

THERE’S NO ORDER!!



• Lists

• Sets

• Maps



• Red-black trees

• Tries

a

bc

d

THERE’S NO ORDER!!

And all elements are unique

(no duplicates) :’)Meme credits: https://www.memesmonkey.com/topic/unique#&gid=1&pid=7

https://www.memesmonkey.com/topic/unique#&gid=1&pid=7



• Lists

• Sets

• Maps



• Red-black trees

• Tries

“username”,

“password”,

“name“,

“fav color”



• Lists

• Sets

• Maps



• Red-black trees

• Tries

value

“username”,

“password”,

“name“,

“fav color”

“cs.is.cool”,

“cantguessthis”,

“brobot“,

“steel gray”



• Lists

• Sets

• Maps (dictionaries)



• Red-black trees

• Tries

value

“username”,

“password”,

“name“,

“fav color”

“cs.is.cool”,

“cantguessthis”,

“brobot“,

“steel gray”



• Lists

• Sets




• Red-black trees

• Tries

Long story, but TL;DR:

Use randomness to get

elements quickly!



• Lists

• Sets




• Red-black trees

• Tries

root

1



• Lists

• Sets




• Red-black trees

• Tries

root

1

0 5



• Lists

• Sets




• Red-black trees

• Tries

root

1

0 5

-2 3½ 7



• Lists

• Sets




• Red-black trees

• Tries

< root

root

1

0 5

-2 3½ 7



• Lists

• Sets




• Red-black trees

• Tries

< root > root

root

1

0 5

-2 3½ 7



• Lists

• Sets




• Red-black trees

• Tries

< root > root

root

1

0 5

-2 3½ 7

works for all nodes!



• Lists

• Sets




• Red-black trees

• Tries

root

1

0 5

-2 3½ 7

2

where to put??



• Lists

• Sets




• Red-black trees

• Tries

root

1

0 5

-2 3½ 7

2



• Lists

• Sets




• Red-black trees

• Tries“self-balancing” binary search tree

with cool properties



• Lists

• Sets




• Red-black trees

• TriesLike binary search tree, but 3

branches per node

PHEW! And there’s so much more…


For some fun ACSL (American Computer Science League)-inspired questions to put the skills you learned to use:

https://web.stanford.edu/~jxgong/splash/computing-fundamentals/

http://acsl.org/

https://web.stanford.edu/~jxgong/splash/computing-fundamentals/


But hopefully that gets you excited for the world of CS!!!

Hope you had fun, found this interesting, and are inspired to dig deeper into this awesome field of study ☺


But hopefully that gets you excited for the world of CS!!!

Hope you had fun, found this interesting, and are inspired to dig deeper into this awesome field of study ☺

Feel free to ping us with any questions, comments, concerns, good jokes, general advice questions, college advice questions, or anything that’s on your mind!!

[email protected] + [email protected]



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