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

0.1

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

0.2

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

0.3

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

Document

Some applications

☞ Information theory

☞ Complexity theory

☞ Probability theory

☞ Proof

☞ Counting and Relations

0.4

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 . . .

0.5

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.

0.6

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

0.7

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)

Contents

Course description

Course outline

Document

Some applications

Engineering

Technology

0.8

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

0.9

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

0.10

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

0.11

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

Contents

Course description

Course outline

Document

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.

0.12

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

0.13

Scheduling

Introduction

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

Course description

Course outline

Document

Some applications

0.14

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

Contents

Course description

Course outline

Document

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 ?

0.15

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

Document

Some applications

J1

J2

J3

J4

J5

J6

0.16

Game and simulation

Sally Salon Game

Introduction

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

Course description

Course outline

Document

Some applications

0.17

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

0.18

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

0.1

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

0.2

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

0.3

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

Document

Some applications

☞ Information theory

☞ Complexity theory

☞ Probability theory

☞ Proof

☞ Counting and Relations

0.4

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 . . .

0.5

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.

0.6

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

0.7

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)

Contents

Course description

Course outline

Document

Some applications

Engineering

Technology

0.8

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

0.9

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

0.10

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

0.11

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

Contents

Course description

Course outline

Document

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.

0.12

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

0.13

Scheduling

Introduction

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

Course description

Course outline

Document

Some applications

0.14

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

Contents

Course description

Course outline

Document

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 ?

0.15

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

Document

Some applications

J1

J2

J3

J4

J5

J6

0.16

Game and simulation

Sally Salon Game

Introduction

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

Course description

Course outline

Document

Some applications

0.17

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

0.18

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