Giao trinh bai tap ds3sets

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Chapter 0
Introduction
Discrete Structures for Computing on September 11, 2014

Contents
Course description
Course outline
Document
Some applications

Huynh Tuong Nguyen, Tran Huong Lan
Faculty of Computer Science and Engineering
University of Technology - VNUHCM
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Contents

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

1 Course description

Course outline

Document

Contents
Course description
Course outline
Document
Some applications

2 Some applications

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Context

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Global
• 6 principal chapters on 45 hours for courses & exercises.
• 10 Labs (10%), 1 Assignment (10%)
• 2 evaluations: mid-exam (MCQ - 60 minutes - 40%) + final

exam (MCQ + writing - 120 minutes - 40%)

Contents
Course description
Course outline

Document
Some applications

Aims

The content of this subject is mainly a great part of logic, set
theory and graph theory.
This is the mathematical base for many topics of Computational
Science

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Subjects in general discrete mathematics course

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

☞ Logic
☞ Set theory
☞ Number theory
☞ Combinatorics: enumerative combinatorics, graph theory
☞ Algorithmics

Contents
Course description
Course outline
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Some applications

☞ Information theory
☞ Complexity theory
☞ Probability theory
☞ Proof
☞ Counting and Relations

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Topics relational to the course

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

1 Theoretical computer science
2 Information theory
3 Logic
4 Set theory
5 Combinatorics

Contents

6 Graph theory

Course description

7 Probability
8 Number theory

Course outline
Document
Some applications

9 Algebra
10 Calculus of finite differences, discrete calculus or discrete analysis
11 Geometry
12 Topology
13 Operations research: scheduling
14 Game theory, decision theory, utility theory, social choice theory
15 Discretization
16 Discrete analogues of continuous mathematics
17 . . .

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Introduction

Context

Huynh Tuong Nguyen,
Tran Huong Lan

Course outline
• Proof methods
• modular arithmetic over integers.

• induction, contradiction.

• Set theory
• relations, functions, cardinalities, relation, equivalence equation, partial order
• combinatorics: counting, principles of sum, multiplication, division, inclusion
and exclusion.

Contents
Course description
Course outline
Document
Some applications

• Graph theory
• directed, undirected, isomorphism
• weighted graphs, algorithm for finding shortest paths
• trees: features, binary trees, minimum spanning trees in connected and
weighted graphs

• flows network

• Probabilistics Modelling
• introductory random variables.

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Document

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Book

Contents
Course description

• Discrete mathematics and applications - Kenneth H. Rosen.
(Vietnamese translation - NXB KHKT 1997)

• Discrete mathematics - Richard Johnsonbaugh, Willey, 1997
• Discrete mathematics with algorithms - Micheal O. Albertson & Joan P.

Course outline
Document
Some applications

Hutchinson, Willey, 1998

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Introduction

Application

Huynh Tuong Nguyen,
Tran Huong Lan

•

it concerns a wide range of disciplines in various areas: science, technology,
business and commerce.

•

applied mathematicians are engaged in the creation, study and application of
advanced mathematical methods relevant to specific problems.

•

applied mathematics has assumed a much broader meaning and embraces such
diverse fields as communication theory, optimization, game theory and numerical
analysis.

•

today there is a remarkable variety of applications of mathematics in industry and
government, such as materials processing, design, medical diagnosis, development
of financial products, network management, weather prediction, etc.

Science

Engineers use technology, mathematics and
scientific knowledge to solve practical
problems. (wikipedia.org)

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Course description
Course outline
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Some applications

Engineering

Technology

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Introduction

Computing of algorithm complexity

Huynh Tuong Nguyen,
Tran Huong Lan

Know results
Size
n
10
102
103
104
105
106

O(log n)

3.10−9 s
7.10−9 s
10−8 s
1, 3.10−8 s
1, 7.10−8 s
2.10−8 s

Contents

Approximating of computational time
O(n)
O(n log n)
O(n2 )
O(2n )
10−8 s
3.10−8 s
10−7 s
10−6 s
10−7 s
7.10−7 s
10−5 s
4.1013 y
10−6 s
10−5 s
10−3 s
*
10−5 s
10−4 s
10−1 s
*

10−4 s
2.10−3 s
10s
*
10−3 s
2.10−2 s
17m
*

Course description

O(n!)
3.10−3 s
*
*
*
*
*

Course outline
Document
Some applications

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Mathematical model

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Solver
• Simplex, GLPK
• CPLEX, MPL
• Excel, Mathlab, etc.

Contents
Course description
Course outline
Document
Some applications

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Mathematical model

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Exercise

A bookseller A buys books from two publishers B, and C.
Publisher B offers a package of 5 mysteries and 5 romance novels
for $50, and publisher C offers a package of 5 mysteries and 10
romance novels for $150. The bookseller A wants to buy at least

2,500 mysteries and 3,500 romance novels, and he has promised C
(who has influence on the Senate Textbook Committee) that at
least 25% of the total number of books he purchases will come
from publisher C.
Question. How many packages should A order from each
publisher in order to minimize his cost and satisfy C ? What will
the novels cost him?

Contents
Course description
Course outline
Document
Some applications

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Mathematical model

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Solution

Let x be the number of packages from Publisher B, and let y be
the number of packages from C.
Problem: Minimize C = 50x + 150y subject to
• 5x + 5y ≥ 2.500

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Course outline
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Some applications

• 5x + 10y ≥ 3.500
• x − 4.5y ≤ 0
• x ≥ 0, y ≥ 0

Answer: Buy 484 packages from Publisher B and 108 from C for a
total cost of $40.400.

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Graph

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

• Shortest path problem
• Min cut and maximum flow
• Vehicle Routing Problem
Contents
Course description
Course outline

Document
Some applications

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Scheduling

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Contents
Course description
Course outline
Document
Some applications

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Introduction

Scheduling

Huynh Tuong Nguyen,
Tran Huong Lan

Exercise

Problem 1||Tmax .
Given 8 jobs with processing times and due dates as follows:
Job
J1
J2
J3
J4
J5
J6
J7
J8
pi

1

2

2

3

3

4

4

3

di

25

16

19

7

18

22

27

8

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Course description
Course outline
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Some applications

Let Ci be completion time of job Ji and let Ti = max(0, Ci − di ) its
tardiness.
Question. How to minimize Tmax = maxi Ti ? What is the minimum value of
Tmax ?

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Introduction

Timetabling

Huynh Tuong Nguyen,
Tran Huong Lan

Example
In the bipartite graph below, the vertices P1 , . . . , P6 represent workers and
edges J1 , . . . , J6 of jobs. An edge connects a worker with a job if the worker
has the necessary qualifications to occupy this job. Here, all the edges have an
unit weight 1, mean that Pi has the skill(competence) to operate Jj if there is
an edge between Pi and Jj .

Contents
Course description

P1

P2

P3

P4

P5

P6

Course outline
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J1

J2

J3

J4

J5

J6

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Game and simulation
Sally Salon Game

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Contents
Course description
Course outline

Document
Some applications

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Probabilistics Modelling

Introduction

Huynh Tuong Nguyen,
Tran Huong Lan

Calculating of Pi

Using a Monte-Carlo method to determine an approximate value
of π :
randomly draw a great number of points in a square of side 2, and
determine the ratio C/N where N is the total number of points,
and C the number of points whose distance to the center of the
square is ≤ 1).

Contents
Course description
Course outline
Document
Some applications

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