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Introduction CSCE 235 Introduction to Discrete Structures

Explore the fundamental principles of discrete mathematics and their application in problem-solving scenarios within computational settings. Learn to construct mathematical models and develop algorithms for efficient solutions, while ensuring correctness, termination, and optimality. Delve into tasks ranging from scheduling optimization to maximizing profits, gaining crucial insights into time and space complexity. Uncover the intrinsic role of rigorous mathematics in computer science, paving the way for advanced problem-solving techniques.

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Introduction CSCE 235 Introduction to Discrete Structures

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  1. Introduction CSCE 235 Introduction to Discrete Structures Fall 2008 Instructor: Berthe Y. Choueiry (Shu-we-ri) GTA: JieFeng http://cse.unl.edu/~choueiry/F08-235/ http://cse.unl.edu/~cse235/

  2. Outline • Introduction: syllabus, schedule, web, topics • Why Discrete Mathematics? • Basic preliminaries

  3. Introduction • Roll • Syllabus • Lectures: M/W/F 12:30—1:20 pm (Avery 109) • Recitations: Tue 5:30—6:20 pm (Avery 108) • Office hours: • Instructor: M/W 1:30—2:30 pm (Avery 123B) • TA: M/Thu 10:00—11:00 am (Student Res. Center) • Must have a cse account • Must use csehandin • Bonus points: report bugs • Web page

  4. Topics

  5. 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 • EdsgerDijkstra: “Computer Science is no more about computers than Astronomy is about telescopes.” • Computer Science is about problem solving.

  6. 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.

  7. 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.

  8. 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

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

  10. 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.

  11. 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 and space complexity

  12. The goal of this course • The goal of this course is to 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.

  13. Basic Preliminaries • A set is a collection of objects. • For example: • S = {s1,s2,s3,…,sn} is a finite set of n elements • S = {s1,s2,s3,…} is a infinite set of elements. • s1  S denotes that the object s1 is an element of the set S • s1  S denotes that the object s1 is not an element of the set S • LaTex • $S=\{s_1,s_2,s_3, \ldots,s_n\}$ • $s_i \in S$ • $si \notin S$

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