Sunday, July 10, 2011

Blogging my way through CLRS section 3.1 [part 5]

Part 4 here.
I wrote an entire blog post explaining the answers to 2.3 but Blogger decided to eat it. I don't want to redo those answers so here is 3.1:
For now on I will title my posts with the section number as well to help Google.

Question 3.1-1: Let $f(n)$ and $g(n)$be asymptotically non-negative functions. Using the basic definition of $\theta$-notation, prove that $\max(f(n) , g(n)) \in \theta(f(n) + g(n))$ .

CLRS defines $\theta$ as $\theta(g(n))= \{ f(n) :$ there exists some positive constants $c_1, c_2$, and $n_0,$ such that $0 \leq c_1g(n) \leq f(n) \leq c_2g(n)$ for all $n \geq n_0\}$ Essentially we must prove that there exists some $c_1$ and $c_2$ such that $c_1 \times (f(n) + g(n)) \leq \max(f(n), g(n)) \leq c_2 \times (f(n) + g(n))$ There are a variety of ways to do this but I will choose the easiest way I could think of. Based on the above equation we know that $\max(f(n), g(n)) \leq f(n) + g(n)$ (as f(n) and g(n) must both me non-negative) and we further know that $\max(f(n), g(n))$ can't be more than twice f(n)+g(n). What we have then are the following inequalities: $$\max(f(n), g(n)) \leq c_1 \times (f(n) + g(n))$$ and $$c_2 \times (f(n) + g(n)) \leq 2 \times \max(f(n), g(n))$$ Solving for $c_1$ we get 1 and for $c_2$ we get $\frac {1} {2}$

Question 3.1-2: Show for any real constants $a$ and $b$ where $b \gt 0$ that $(n+a)^b \in \theta(n^b)$
Because $a$ is a constant and the definition of $\theta$ is true after some $n_0$ adding $a$ to $n$ does not affect the definition and we simplify to $n^b \in \theta(n^b)$ which is trivially true

Question 3.1-3: Explain why the statement "The running time of $A$ is at least $O(n^2)$," is meaningless.

I'm a little uncertain of this answer but I think this is what CLRS is getting at when we say a function $f(n)$ has a running time of $O(g(n))$ what we really mean is that $f(n)$ has an asymptotic upper bound of $g(n)$. This means that $f(n) \leq g(n)$ after some $n_0$. To say a function has a running time of at least g(n) seems to be saying that $f(n) \leq g(n) \And f(n) \geq g(n)$ which is a contradiction.

Question 3.1-4: Is $2^{n+1} = O(2^n)$? Is $2^{2n} = O(2^n)$?

$2^{n+1} = 2 \times 2^n$. which means that $2^{n+1} \leq c_1 \times 2^n$ after $n_0$ so we have our answer that $2^{n+1} \in o(2^n)$ Alternatively we could say that the two functions only differ by a constant coefficient and therefore the answer is yes.

There is no constant such that $2^{2n} = c \times 2^n$ and thefore $2^{2n} \notin O(2^n)$

Question 3.1-5: Prove that for any two functions $f(n)$ and $g(n)$, we have $f(n) \in \theta(g(n)) \iff f(n) \in O(g(n)) \And f(n) \in \Omega(g(n))$

This is an "if an only if" problem so we must prove this in two parts:

Firstly, if $f(n) \in O(g(n))$ then there exists some $c_1$ and $n_0$ such that $f(n) \leq c_1 \times g(n)$ after some $n_0$. Further if $f(n) \in Omega(g(n))$ then there exists some $c_2$ and $n_0$ such that $f(n) \geq c_2 \times g(n)$ after some $n_0$.

If we combine the above two statements (which come from the definitions of $\Omega$ and O) than we know that there exists some $c_1, c_2, and n_0,$ such that $c_1g(n) \leq f(n) \leq c_2g(n)$ for all $n \geq n_0\}$

We could do the same thing backward for the other direction: If $f(n) \in \theta(g(n))$ then we could split the above inequality and show that each of the individual statements are true.

Question 3.1-6: Prove that the running time of an algorithm is $\theta(g(n)) \iff$ its worst-case running time is $O(g(n))$ and its best case running time $\Omega(g(n))$.
I'm going to try for an intuitive proof here instead of a mathematical one. If the worst case is asymptotically bound above in the worst case by a certain function and is asymptotically bound from below in the best case which means that the function is tightly bound by both those functions. f(n) never goes below some constant times g(n) and never goes above some constant times g(n). This is what we get from the above definition of $\theta(g(n)))$ A mathematical follows from question 3.1-5.

Question 3.1-7: Prove that $o(g(n)) \cap \omega(g(n)) = \varnothing$

little o and little omega are defined as follows: \[o(g(n)) = \{ f(n) : \forall c > 0 \exists n_0 \text{such that } 0 \leq f(n) \leq c \times g(n) \forall n \gt n_0\] and \[\omega(g(n)) = \{ f(n) : \forall c > 0 \exists n_0 \text{such that } 0 \leq c \times g(n) \leq f(n) \forall n \gt n_0\]

In other words

$$f(n) \in o(g(n)) \iff \lim_{n \to \infty} \frac {f(n)} {g(n)} = 0$$ and $$f(n) \in \omega(g(n)) \iff \lim_{n \to \infty} \frac {f(n)} {g(n)} = \infty$$

It is obvious that these can not be true at the same time. This would require that $0 = \infty$