Day 13, Mat385

Last time

Next time

• Announcements
  • Our first of two midterm exams is next week, and will cover chapters 1-3 (the sections we've covered, at any rate). On Monday we'll have a review. You'll be allowed a one-page (front and back) sheet for the exam.

  • Quiz today is over 3.2: recurrence relations

  • There are student tutors at NKU who may be able to help:
    • Learning Plus: Mitch Playforth
    • Of those in the Math/Stat lab (MP457), Sydney, Nghia and William are all CS majors so they may be able to help. Here are their hours.

• Last time:
  • We analyzed two algorithms: binary search, and the Euclidean algorithm.
    • Binary search: requires a sorted list, so we added a mergesort first, which was fairly expensive: $$n{\log }_2(n)-n+1$$

      If one is doing a one-off search, then use sequentialsearch -- requiring $n$ comparisons in the worst case. But if one is doing multiple searches of the list, then it may pay to sort it first:

      

    • Euclid's GCD (Greatest Common Divisor)

      has a ratio of Fibonacci numbers as the worst case, which we discovered with this handout.

  • Marked up
    • highlights from 3.3

• Today:
  • We'll finish off 3.3: Analysis of Algorithms,
    • with a quick example illustrating a linear, first-order, non-constant coefficient recurrence relation (#12, p. 196): $$\begin{array}{ll}S(1)=2& \\ S(n)=3(n+1)S(n-1)& n\ge 2\end{array}$$
    • a wrap-up of Euclid's GCD (Greatest Common Divisor), whose upper bound calculation will lead us to a divide-and-conquer recurrence, with worst case $$E(n)\le 2{\log }_2(n).$$

      Last time we used this handout to discover that the worst-case scenario is successive Fibonacci numbers, e.g. $$gcd(89,55)$$ Using the fact that $$F(n)=\text{Round}\left(\frac{{\varphi }^n}{\sqrt{(}5}\right)$$ for all $n\ge 1$, we can get a tighter bound on $E(n)$: $$E(n)\le \text{Round}[{\log }_{\varphi }\left(\sqrt{5}a\right)]-2$$ where $\varphi$ is the golden mean or ratio: $$\varphi =\frac{1+\sqrt{5}}{2}$$

      

  • We'll get a start on Section 4.1 (Sets).

  • Then have a quiz....

• The Curious History of Venn Diagrams: A look at the curious history of Venn diagrams and how they blend logic with geometry (Scientific American)

• Some Fun, Motivating Examples from Propositional and Predicate Calculus