Class[1][23ed may] [algorithms]

ALGORITHMS, COMPUTER GRAPHICS,
AND MATHEMATICS FOR GAME
DEVELOPERS & COMPUTER
SCIENTISTS
Class[1]: Introduction to Algorithms, Big-O Notation
Presented by
Dr. Saajid Abuluaih
- May 2021 -

Algorithms, Computer Graphics, and Mathematics
for Game Developers and Computer Scientists
© 2021 arche1.co.jp | jaist.ac.jp
All rights reserved.
2
Agenda
One-Bite Wisdom
On Algorithms
Simple Algorithms
𝑖
Big O Notation
The Class Structure
On Pseudocode
Introduction to the Course
𝑔
𝑛
“The future is not just unwritten - it is unsearched”
-Bruce Sterling

3
BEFORE WE START,
I GOT ONE DISCLAIMER
TO ANNOUNCE
This course is not designed for totally beginners …
You need to have some basic knowledge on what Computer Science is all about
HOWEVER,
If you feel like you can’t understand something that you think it is important
Please let me know

P A G E 4
One-Bite Wisdom
Monty Hall Problem

5
• Let’s Make a Deal was a very famous game show back in the
70’s & 80’s
• Monty Hall is the host of the show
• There are three doors, one of them has a nice prize, and the
other two have gags (Usually a farm animal)
• You chose a door and then the host will open “Empty” another
• The host then will aske you whether to stick with or switch the
door you picked
MONTY HALL PROBLEM,
THE THREE DOORS GAME
References:
Monty Hall Problem - Wikipedia
Monty Hall Problem - Numberphile - YouTube
Monty Hall show - Let’s make a deal
Is there a good strategy to use in this case?

6
• Mathematicians say: in order to increase your chances to win
the nice prize, you must switch
MONTY HALL PROBLEM,
References:
Monty Hall show - Let’s make a deal
This strategy doesn’t guarantee that you will. But it will increase
your chances 13
1/3 2/3
1/3 2/3

7
Mathematicians say: in order to increase your chances of winning the nice prize, you must switch.
MONTY HALL PROBLEM,
References:
To explain it even better, consider the following scenario. You have six doors. You chose one. The host
open 4 empty doors. Will you prefer then to stick with your early door selection or will you switch.
1/6 5/6

WHAT THAT HAS TO DO WITH
OUR CLASS FOR TODAY?
NOTHING
it has nothing to do with today’s class

P A G E 9
Introduction
About the Course

This is a course in algorithms and 3D computer graphics that enables enrolled students to learn the theory
of computer graphics along with algorithms implementation for solving real-life problems, specifically that
relates to mathematics and physics of motion in 3D spaces. Ultimately, the course intends to deliver
knowledge that is sufficient to construct an interactive 3D application for simulating the movement of a
controllable entity.
The main topics this course intends to cove is:
10
INTRODUCTION,
WHAT THIS COURSE IS ALL ABOUT
- Algorithms:
- Big-O notation
- Definition, characteristics, usage,
design, optimization
- Programing useful algorithms for
popular problems
- Computer Graphic with Three.js:
- Scene creation:
- Light, camera
- Renderer, animation loop
- 3D Object creation:
- Geometry
- Material
- Mesh
- Mathematica and Physics
- Locations in 2D & 3D spaces
- Raycasting in 3D environments
- Vectors:
- Vector normal form
- Dot product
- Cross Product
- Intersections
- Matrices and Transformations
- Newtons Laws of Motions
Bonus:
- Programing languages: C#, JavaScript, TS

P A G E 11
The Class Structure
How the class lectures are going
to look like

The structure of delivering classes:
• One-bite wisdom 15 minutes (Knowledge Sharing Session)
• Algorithms (15 minutes)
• Lecture (45min ~ an hour)
• Quiz
• Homework
• Groups’ discussion 30 minutes (we need four groups for)
12
THE CLASS STRUCTURE,
WHAT THIS COURSE IS ALL ABOUT

P A G E 13
ON ALGORITHMS
WHAT WE WILL LEARN
ABOUT ALGORITHMS

Definition:
An algorithm is an unambiguous finite sequence of well-defined step-by-step computer-implementable
instructions, that takes data as an input instance (of a given problem) and produces an output that can be
called a solution to the problem.
But what is a problem:
A problem is a set of instances that all share the same characteristic
What is not an Algorithm:
We don’t call the implementation of an algorithm an algorithm. We call it “code”.
What makes an algorithm better than the other:
Long story short, “the performance” … the smaller number of steps needed to produce the solution the
better the algorithm is.
However!
Although, any type of instructions that handle inputs are considered “algorithms”; algorithmic thinking
always involve loops and conditions.
14
ON ALGORITHMS,
WHAT IS ALGORITHM?
References:
Algorithm
What exactly is an algorithm? Algorithms explained | BBC (Ideas)

A flowchart is the graphical or pictorial representation
of an algorithm with the help of different symbols,
shapes, and arrows to demonstrate a process or a
program.
15
ON ALGORITHMS,
DESIGNING AN ALGORITHM
References:
Explain Algorithm and Flowchart with Examples
Example 1: Print 1 to 20:
Algorithm:
• Step 1: Initialize X as 0,
• Step 2: Increment X by 1,
• Step 3: Print X,
• Step 4: If X is less than 20 then go back to
step 2.

Iterations VS Recursion
The basic idea of using recursion and iteration is to repeat a sequence of instructions. Recursion is a
function call in which the function being called is the same as the one initiating the call, whereas iteration
occurs when a loop is continuously executed until a given condition is met.
16
On Algorithms,
Algorithms implementation
function Function1 (num) {
if (Base) {
return num;
}
return num + Function1(num - 1);
}
function Function1(num) {
var value = 1;
for (var i= num; i>=0; i--) {
value += i;
}
return total;
}
References:
Programming Loops vs Recursion - Computerphile

P A G E 17
Simple Algorithms
Let’s Brush our Coding Skills

Factorial of a positive integer 𝑛𝑢𝑚 is product of all 𝑣𝑎𝑙𝑢𝑒𝑠 from 𝑛𝑢𝑚 to 1. For example, the factorial of
𝑛𝑢𝑚 = 5 is 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120.
18
A SIMPLE ALGORITHM,
CALCULATING THE FACTORIAL VALUE OF A
POSITIVE INTEGER
function factorial(num) {
if (num === 1) {
return 1;
}
return num * factorial(num - 1);
}
function factorial(num) {
var total = 1;
for (var i= num; i>=1; i--) {
total = total * i;
}
return total;
}

P A G E 19
Quiz [1]
Answer This Question … IF YOU CAN

• Write the implementation of the following flowchart using your favorite
coding language. [Homework]
• What does the algorithm in the flowchart do?
• What is the output of the algorithm if N was 15?
20
Quiz [1]
What is the output of the following algorithm?
Start
Total = 0
i = 0
Read N
Is count < N
Print N
Total = Total + N
i = i + 1
End

P A G E 21
Big O Notation
Let’s Dive a Bit Into the Fundamentals

It is a mathematical notation that describes the limiting behavior of a function when the argument tends
towards a particular value or infinity. In another way, it shows how the code slows as the data grows (Big-
O notation used to measure space as well).
22
BIG-O NOTATION
What is Big-O?
References:
Big O: How Code Slows as Data Grows
Big O notation
Algorithms Notes for Professionals

We can not use the time to measure the performance of an algorithm. So, what to doe?
• Rather than counting seconds, we count the number of steps (operations) that we need to execute
every time to generate the outcome.
• Think of this simple algorithm that does nothing other than printout “hello world” how many steps
we need to execute?
23
BIG-O NOTATION
TIMING OUR ALGORITHMS
References:
• It is going to be 𝑂 1 .
• Big-O Notation is a way to formalize a fuzzy counting

What is the time complexity for the following algorithms:
24
BIG-O NOTATION
LET’S EXAMINE the number of steps needed in
algorithm THIS FUNCTION [P1]
References:
How about his one:
It is liner, 𝑂(𝑁)
It is liner, 𝑂(1)

What is the time complexity for the following function:
25
BIG-O NOTATION
LET’S EXAMINE the number of steps needed in
algorithm THIS FUNCTION [P2]
References:
What do you think this is?
Quadratic 𝑂(𝑁2
)
It is 𝑂(log(𝑁))

We can not use the time to measure the performance of an algorithm. So what to doe?
• 𝑂(1): Constant time
• 𝑂(log(𝑁)): Logarithmic running time
• 𝑂(𝑁): Linear running time
• 𝑂(𝑁2
): Quadratic. Polynomials running time.
• 𝑂(2𝑁
): Exponential running time
• 𝑂(! 𝑁): Factorial running time
26
BIG-O NOTATION
BIG-O COMPLEXITY MOST COMMON PATTERNS
References:

• How the code slows as the data grows (Big-O notation used to )
• What does the algorithm do?
• What is the output of the
• algorithm if N was 15?
27
Big-O Notation
What is the output of the following algorithm?
References:
Big O Cheatsheet

P A G E 28
Quiz [2]
Answer This Question … IF YOU CAN

• What is the Big-O notation for the following algorithm?
• Is there anyway to enhance it?
29
Quiz [1]
What is the time complexity of the following?
Start
Total = 0
i = 0
Read N
Is count < N
Print N
Total = Total + N
i = i + 1
End

P A G E 30
Pseudocode
Write A Code That Can Be Understood By
Human

31
Definition:
Pseudocode is an artificial and informal language that helps programmers develop algorithms. It is a "text-
based" step-by-step design tool.
Basic common pseudocode:
- Assignment
a ← expression
- Arithmetic operators and numeric procedures
a + b, a – b, a * b, a / b
- Relational and Boolean operators
a = b, a ≠ b, a > b, a < b, a ≥ b, a ≤ b
- Condition
Evaluates to true if condition is true; otherwise evaluates to false.
Evaluates to true if both condition1 and condition2 are true or condition3 is true;
- Iterations
Repeat n Times { < Your logic here > }
Repeat Until (condition) { < Your logic here > }
- Procedures
Procedure name (parameter1, parameter2, ...) { < Your logic here > Return (expression) }
On Algorithms,
Pseudocode
References:
PSEUDOCODE GUIDE

P A G E 32
HOMEWORK
Let’s Dive a Bit into Fundamentals

Find any simple problem, and design two different algorithms that lead to the same result:
• Use the flowcharts you learnt about today to present your algorithms
• What is the time complexity (Big-O notation) of the algorithms that you are going to design
• Write the pseudocode of your algorithms
Read Chapter 3 of [Algorithms: Notes for Professional] and:
• Summarize Big-Theta notation
• Summarize Big-Omega Notation
33
DEADLINE 27/5 3/6,
HOMEWORK

THANKS FOR YOUR
ATTENTION

Class[1][23ed may] [algorithms]

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