21 - Mathematical Introduction Induction II Discrete...

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Introduction Mathematical Induction II Discrete Mathematics Andrei Bulatov
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Discrete Mathematics – Mathematical Induction II 21-2 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
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Discrete Mathematics – Mathematical Induction II 21-3 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
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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 21-4 Greedy Algorithm
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Discrete Mathematics – Mathematical Induction II 21-5 Greedy Algorithm (cntd) Let P(n) be the proposition that if the greedy algorithm schedules n
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