L13_AdvancedInduction_print

L13_AdvancedInduction_print - 1 COMP170 Discrete...

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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 4.5, pp. 189-193 Advanced Induction Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 Note: We have skipped section 4.4 of the textbook because the material it contains, especially the Master Theorem , will be taught in later classes, e.g., COMP271 3 More Advanced Induction • Induction, as we’ve seen it so far, was about defining a statement p ( n ) , and then proving • In “practice”, in some real induction proofs, p ( n ) might not be fullly defined before we start the proof and will only be fully described during the description of the proof • In some cases it also helps to use a stronger induction hypothesis than the “natural” one. p ( n- 1) ⇒ p ( n ) or ( p (1) ∧ p (2) ∧ ··· ∧ p ( n- 1)) ⇒ p ( n ) 4 If T ( n ) ≤ 2 T ( n/ 2) + cn for some constant c , then T ( n ) = O ( n log n ) . If T ( n ) ≤ T ( n/ 3) + cn for some constant c , then T ( n ) = O ( n ) . If T ( n ) ≤ 4 T ( n/ 2) + cn for some constant c , then T ( n ) = O ( n 2 ) . We will illustrate these concepts with three example proofs: Example 1 Example 3 Example 2 Examples 1 & 2 will illustrate how to derive the induction statement p ( n ) while proving p ( n ) Example 3 will illustrate what is meant by using a stronger induction hypothesis. 5 As before we will assume that n is a power of 2 From definition of big O we need to show that A naive induction proof would assume that (*) T ( n ) ≤ kn log n was true for n = 2 i- 1 and then prove that (*) was also true for n = 2 i if T ( n ) ≤ 2 T ( n/ 2) + cn for some constant c , then T ( n ) = O ( n log n ) . ∃ n , k such that ∀ n > n , T ( n ) ≤ kn log n Our problem is that we do not know what k is so we can’t prove (*) Example 1: 6 We want to prove that if, for all n = 2 i , T ( n ) ≤ 2 T ( n/ 2) + cn for some constant c , ⇒ ∀ n > n , T ( n ) ≤ kn log n Our proof will be by induction,...
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L13_AdvancedInduction_print - 1 COMP170 Discrete...

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