Introduction CSCE 235 Introduction to Discrete Structures Spring 2010 Instructor: Berthe Y. Choueiry (Shu-we-ri) GTA: Nobel Khandaker

IntroductionCSCE 235, Spring Outline Introduction: syllabus, schedule, web, topics Why Discrete Mathematics? Basic preliminaries

IntroductionCSCE 235, Spring Introduction Roll Syllabus Lectures: M/W/F 12:30—1:20 pm (Avery 119) Recitations: M 3:30—4:20 pm (Avery 110) Office hours: – Instructor: M/W 1:30—2:30 pm (Avery 360) – TA: Tue/Thu 9:00—10:00 am (Student Res. Center) Must have a cse account Must use cse handin Bonus points: report bugs Web page

IntroductionCSCE 235, Spring Topics TopicSections Propositional Logic1.1—1.2 Predicate Logic1.3—1.4 Proofs1.5—1.6 Sets21.—22 Functions2.3 Relations8.1,8.3—8.6 Algorithms3.1—3.3 Induction4.1—4.2 Counting5.1—5.2 Combinatorics5.3—5.5 Recursion7.1—7.2 PIE7.5 Graphs9.1—9.5 Trees10.1—10.3

IntroductionCSCE 235, Spring Why Discrete Mathematics? (I) Computers use discrete structures to represent and manipulate data. CSE 235 and CSE 310 are the basic building block for becoming a Computer Scientist Computer Science is not Programming Computer Science is not Software Engineering Edsger Dijkstra: “Computer Science is no more about computers than Astronomy is about telescopes.” Computer Science is about problem solving.

IntroductionCSCE 235, Spring Why Discrete Mathematics? (II) Mathematics is at the heart of problem solving Defining a problem requires mathematical rigor Use and analysis of models, data structures, algorithms requires a solid foundation of mathematics To justify why a particular way of solving a problem is correct or efficient (i.e., better than another way) requires analysis with a well- defined mathematical model.

IntroductionCSCE 235, Spring Problem Solving requires mathematical rigor Your boss is not going to ask you to solve – an MST (Minimal Spanning Tree) or – a TSP (Travelling Salesperson Problem) Rarely will you encounter a problem in an abstract setting However, he/she may ask you to build a rotation of the company’s delivery trucks to minimize fuel usage It is up to you to determine – a proper model for representing the problem and – a correct or efficient algorithm for solving it

IntroductionCSCE 235, Spring Scenario I A limo company has hired you/your company to write a computer program to automate the following tasks for a large event Task1: In the first scenario, businesses request – limos and drivers – for a fixed period of time, specifying a start data/time and end date/time and – a flat charge rate The program must generate a schedule that accommodates the maximum number of customers

IntroductionCSCE 235, Spring Scenario II Task 2: In the second scenario – the limo service allows customers to bid on a ride – so that the highest bidder get a limo when there aren’t enough limos available The program should make a schedule that – Is feasible (no limo is assigned to two or more customers at the same time) – While maximizing the total profit

IntroductionCSCE 235, Spring Scenario III Task 3: Here each customer – is allowed to specify a set of various times and – bid an amount for the entire event. – The limo service must choose to accept the entire set of times or reject it The program must again maximize the profit.

IntroductionCSCE 235, Spring What’s your job? Build a mathematical model for each scenario Develop an algorithm for solving each task Justify that your solutions work – Prove that your algorithms terminate.Termination – Prove that your algorithms find a solution when there is one.Completeness – Prove that the solution of your algorithms is correct Soundness – Prove that your algorithms find the best solution (i.e., maximize profit).Optimality (of the solution) – Prove that your algorithms finish before the end of life on earth.Efficiency, time & space complexity

IntroductionCSCE 235, Spring The goal of this course Give you the foundations that you will use to eventually solve these problems. – Task1 is easily (i.e., efficiently) solved by a greedy algorithm – Task2 can also be (almost) easily solved, but requires a more involved technique, dynamic programming – Task3 is not efficiently solvable (it is NP-hard) by any known technique. It is believed today that to guarantee an optimal solution, one needs to look at all (exponentially many) possibilities

IntroductionCSCE 235, Spring Basic Preliminaries A set is a collection of objects. For example: – S = {s 1,s 2,s 3,…,s n } is a finite set of n elements – S = {s 1,s 2,s 3,…} is a infinite set of elements. s 1  S denotes that the object s 1 is an element of the set S s 1  S denotes that the object s 1 is not an element of the set S LaTex – $S=\{s_1,s_2,s_3, \ldots,s_n\}$ – $s_i \in S$ – $si \notin S$