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Data Structures in Java Lecture 5: Sequences and Series, Proofs - - PowerPoint PPT Presentation

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Data Structures in Java Lecture 5: Sequences and Series, Proofs 9/23/2015 Daniel Bauer 1 Algorithms An algorithm is a clearly specified set of simple instructions to be followed to solve a problem. Algorithm Analysis Questions:

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Data Structures in Java

Lecture 5: Sequences and Series, Proofs

9/23/2015

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Daniel Bauer

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Algorithms

  • An algorithm is a clearly specified set of simple

instructions to be followed to solve a problem.

  • Algorithm Analysis — Questions:
  • Does the algorithm terminate?
  • Does the algorithm solve the problem? (correctness)
  • What resources does the algorithm use?
  • Time / Space

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Contents

  • 1. Sequences and Series
  • 2. Proofs

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Sequences

  • What are these sequences?
  • 0, 2, 4, 6, 8, 10, …
  • 2, 4, 8, 16, 32, 64, …
  • 1, 1/2, 1/4, 1/8, 1/16, …

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Sequences

  • What are these sequences?
  • 0, 2, 4, 6, 8, 10, …
  • 2, 4, 8, 16, 32, 64, …
  • 1, 1/2, 1/4, 1/8, 1/16, …

Arithmetic Sequence

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Sequences

  • What are these sequences?
  • 0, 2, 4, 6, 8, 10, …
  • 2, 4, 8, 16, 32, 64, …
  • 1, 1/2, 1/4, 1/8, 1/16, …

Arithmetic Sequence Geometric Sequence

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Arithmetic Series

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Arithmetic Series

  • Arithmetic Sequence of length N, with start term


a and common difference d. 


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Arithmetic Series

  • Arithmetic Sequence of length N, with start term


a and common difference d. 


  • Series: The sum of all elements of a sequence.


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Sum-Formulas for Arithmetic Series

  • In particular (for a=1 and d=1):

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Geometric Series

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Geometric Series

  • Geometric Sequence with start term s and 


common ratio A. 


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Geometric Series

  • Geometric Sequence with start term s and 


common ratio A. 


  • Geometric Series:


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Geometric Series

  • Geometric Sequence with start term s and 


common ratio A. 


  • Geometric Series:

  • Often 0 < A < 1 or A = 2

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Sum-Formulas for Finite Geometric Series

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Sum-Formulas for Finite Geometric Series

  • In particular, if s=1

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Sum-Formulas for Finite Geometric Series

  • In particular, if s=1
  • In Computer Science we often have A = 2

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Sum-Formulas for Infinite Geometric Series

  • nly if 0<A<1

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Sum-Formulas for Infinite Geometric Series

  • In particular, if s=1

and

  • nly if 0<A<1

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Sum-Formulas for Infinite Geometric Series

  • In particular, if s=1
  • For instance,

and

  • nly if 0<A<1

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Analyzing the Towers of Hanoi Recurrence

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base case: geometric series

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The End of The World

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Legend says that, at the beginning of time, priests were given a puzzle with 64 golden disks. Once they finish moving all the disks according to the rules, the wold is said to end. If the priests move the disks at a rate of 1 disk/second, how long will it take to solve the puzzle?

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The End of The World

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Legend says that, at the beginning of time, priests were given a puzzle with 64 golden disks. Once they finish moving all the disks according to the rules, the wold is said to end. If the priests move the disks at a rate of 1 disk/second, how long will it take to solve the puzzle?

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The End of The World

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Legend says that, at the beginning of time, priests were given a puzzle with 64 golden disks. Once they finish moving all the disks according to the rules, the wold is said to end. If the priests move the disks at a rate of 1 disk/second, how long will it take to solve the puzzle? seconds minutes days years

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Contents

  • 1. Sequences and Series
  • 2. Proofs

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Types of Proofs

  • Proof by Induction
  • Proof by Contradiction
  • Proof by Counterexample

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Proofs by Induction

  • We are proving a theorem T. (“this property holds for all

cases.”).

  • Step 1: Base case. We know that T holds (trivially) for some

small value.

  • Step 2: Inductive step:
  • Inductive Hypothesis: Assume T holds for all cases up to

some limit k.

  • Show that T also holds for k+1.
  • This proves that T holds for any k.

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  • For the Fibonacci numbers, we prove that 





  • Base case: 

  • Inductive step:
  • Assume the theorem holds for i = 1,2,…,k
  • We need to show that 


Proof by Induction - Example

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Proof by Induction - Inductive Step

  • We know that and by the

inductive hypothesis:


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Proof by Counter-Example

  • We are proving that theorem T is false. (“this

property does not hold for all cases.”).

  • It is sufficient to show that there is a case for which

T does not hold.

  • Example:
  • Show that is false.
  • There are i for which ,e.g.

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Proof by Contradiction

  • We want to proof that T is true.
  • Step 1: Assume T is false.
  • Step 2: Show that this assumption leads to a

contradiction.

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Proof by Contradiction - Example

  • We want to proof that there is an infinite number of primes.
  • Assume the number of primes is finite and the largest

prime it Pk .

  • Let the sequence of all primes be P1, P2,,…,Pk





  • Since N>Pk, N cannot be prime, so it must have a

factorization into primes. Such a factorization cannot exist: dividing N by any P1, P2,,…,Pk will always leave a remainder of 1.

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