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Introduction Huynh Tuong Nguyen, Tran Huong Lan Chapter 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 Course description Course outline Document Contents Course description Course outline Document Some applications Some applications 0.2 Context Introduction Huynh Tuong Nguyen, Tran Huong Lan Global • principal chapters on 45 hours for courses & exercises • 10 Labs (10%), Assignment (10%) • 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 Theoretical computer science Information theory Logic Set theory Combinatorics Contents Graph theory Course description Probability Number theory Course outline Document Some applications 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 mysteries and romance novels for $50, and publisher C offers a package of 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 ≤ • x ≥ 0, y ≥ 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 jobs with processing times and due dates as follows: Job J1 J2 J3 J4 J5 J6 J7 J8 pi 2 3 4 di 25 16 19 18 22 27 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 ... exercises • 10 Labs (10%), Assignment (10%) • evaluations: mid-exam (MCQ - 60 minutes - 40%) + final exam (MCQ + writing - 120 minutes - 40%) Contents Course description Course outline Document... and applications - Kenneth H Rosen (Vietnamese translation - NXB KHKT 1997) • Discrete mathematics - Richard Johnsonbaugh, Willey, 1997 • Discrete mathematics with algorithms - Micheal O Albertson... 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