1. Einleitung

Algorithms

We looked at Karatsuba Multiplication, the “Pasture Break” problem and starsearch.

The common theme here was recursion, also known as “divide and conquer” or subdivision of the task. By simplifying the task at hand, dividing it into multiple subtasks, you can reason more easily about complexity.

In Karatsuba, you look at multiplications of two numbers with $n/2$ digits, which each then is a multiplication of two $n/4$-digit numbers, etc… The same goes for starsearch, where we look at a smaller subset, before adding the last person back in.

Techniques

Telescoping

By plugging in the previous term into a recursive definition, we can more easily get a feel for it’s explicit form.

Ex: $M(n+1)=3\cdot M(n)$ turns into $M(n+1)=3\cdot (3\cdot M(n-1))$ and so on and so forth.

Keep in mind telescoping is not a proof technique, just for intuition!

Vollständige Induktion

We prove the I.A. (Induktionsanfang) for the first case. Then we prove the I.S. (Induktionsschritt) from $n\Rightarrow n+1$. Thus we conclude that the property $P$ holds for all $n_0$ to infinity.