21.Mathematical Induction II.pdf midterm #2

21.Mathematical Induction II.pdf midterm #2 - Mathematical...

Info icon This preview shows pages 1–6. Sign up to view the full content.

Mathematical Induction II Discrete Mathematics Andrei Bulatov
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Discrete Mathematics – Mathematical Induction II 22-2 Principle of Mathematical Induction Principle of mathematical induction : To prove that a statement that assert that some property P(n) is true for all positive integers n, we complete two steps Basis step : We verify that P(1) is true. Inductive step : We show that the conditional statement P(k) P(k + 1) is true for all positive integers k To prove the conditional statement, we assume that P(k) is true (it is called inductive hypothesis ) and show that under this assumption P(k + 1) is also true
Image of page 2
Discrete Mathematics – Mathematical Induction II 22-3 Analysis of Algorithms Consider the following problem There is a group of proposed talks to be given. We want to schedule as many talks as possible in the main lecture room. Let be the talks, talk begins at time and ends at time . (No two lectures can proceed at the same time, but a lecture can begin at the same time another one ends.) We assume that . m t t t , , , 2 1 K j t j b j e m e e e K 2 1 9:00 10:00 11:00 12:00 1 t 2 t 3 t 5 t 4 t 6 t 7 t 8 t
Image of page 3

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Greedy algorithm: At every step choose a talk with the earliest ending time among all those talks that begin after all talks already scheduled end. We prove that the greedy algorithm is optimal in the sense that it always schedules the most talks possible in the main lecture hall. 9:00 10:00 11:00 12:00 1 t 2 t 3 t 5 t 4 t 6 t 7 t 8 t Discrete Mathematics – Mathematical Induction II 22-4 Greedy Algorithm
Image of page 4
Discrete Mathematics – Mathematical Induction II 22-5 Greedy Algorithm (cntd) Let P(n) be the proposition that if the greedy algorithm schedules n talks, then it is not possible to schedule more than n talks. Basis step. Suppose that the greedy algorithm has scheduled only one talk, . This means that every other talk starts before , and ends after . Hence, at time each of the remaining talks needs to use the lecture hall. No two talks can be scheduled because of that.
Image of page 5

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern