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\usepackage{amsmath}
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\title{Algoritmen en Datastructuren - assignment 2}
\author{Camil Staps\\\small{s4498062}}
\begin{document}
\maketitle
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\begin{enumerate}
\item \begin{enumerate}
\item Claim: $T(n) \in\pazocal O\left(n^2\right)$.
\begin{proof}
With induction over $n$ we prove $T(n) \leq 3\cdot\frac{n(n+1)}{2}$ for all $n\in\mathbb N$.
\textbf{Induction basis}: for $n=0$ we have $T(n) \leq 3\cdot n = 0 = 3\cdot\frac{n(n+1)}{2}$.
\textbf{Inductive step}: let's assume $T(k) \leq 3\cdot\frac{k(k+1)}{2}$ for some $k\in\mathbb N$ (IH). Then for $k+1$:
\begin{align*}
T(k+1) &\leq T(k) + 3(k+1)\\
&\leq 3\cdot\frac{k(k+1)}{2} + 3k + 1 \qquad\text{(IH)}\\
&= 1\frac12\left(k^2+3k\right) + 1\\
&\leq 1\frac12\left(k^2+3k+2\right)\\
&= 3\cdot\frac{(k+1)(k+2)}{2}.
\end{align*}
So we see that if $T(k)\leq3\cdot\frac{k(k+1)}{2}$ for some $k\in\mathbb N$, also $T(k+1)\leq3\cdot\frac{(k+1)(k+2)}{2}$. From the principle of induction now follows that for all $n\in\mathbb N$ we have $T(n) \leq 3\cdot\frac{n(n+1)}{2}$.
Now since $3\cdot\frac{n(n+1)}{2}$ has order $n^2$, also $T(n)\in\pazocal O\left(n^2\right)$.
\end{proof}
\item Claim: $T(n) \in\pazocal O\left(\ln n\right)$.
\begin{proof}
With induction over $n$ we prove $T(n) \leq 42\log_2 n$ for all $n \in\mathbb N, n>1$.
\textbf{Induction basis}: for $n=2$ we have $T(n) = T\left(\frac n2\right)+42 = 42 \leq 42\log_2 n$.
\textbf{Inductive step}: let's assume for some $k\in\mathbb N$ that $T(k') \leq 42\log_2k'$ for all $k'\in\{n\in\mathbb N\mid n>1\}, k'<k$ (IH). Then for $k$:
\begin{align*}
T(k) &\leq T\left(\tfrac k2\right) + 42\\
&\leq 42\log_2\left(\tfrac k2\right) + 42 \qquad\text{(IH)}\\
&= 42\left(\log_2k - \log_22\right) + 42\\
&= 42\log_2k.
\end{align*}
From the principle of induction now follows that for all $n\in\mathbb N$ we have $T(n)\leq 42\log_2n$.
Now since $42\log_2n \in\pazocal O(\ln n)$, also $T(n)\in\pazocal O(\ln n)$.
\end{proof}
\item Claim: $T(n) \in\pazocal O(\ln n)$.
\begin{proof}
With induction over $n$ we prove there exists some $c\in\mathbb N$ s.t. $T(n)\leq c\cdot\log_3n$ for all $n\in\mathbb{N},n>4$.
\textbf{Induction basis}: for $n=5$ we have $T(n) = 3T(\frac n3)+13 = 13 \leq c\log_3n$ with $c=13$ for example.
\textbf{Inductive step}: let's assume for some $k\in\mathbb N$ that $T(k') \leq c_{k'}\log_3k'$ for all $k'\in\{n\in\mathbb N\mid n>4\}$, $k'<k$ (IH). Then for $k$:
\begin{align*}
T(k) &\leq 3T\left(\tfrac k3\right) + 13\\
&\leq 3\cdot c\log_3\left(\tfrac k3\right) + 13 \qquad\text{with $c=c_{\left(\tfrac k3\right)}$}\\
&= 3c\cdot(\log_3k - 1) + 13\\
&= 3c\cdot\log_3k + 13 - 3c \qquad\text{$3c>13$ for $c>4$, therefore:}\\
&\leq 3c\cdot\log_3k\\
&= c'\cdot\log_3k \qquad\text{with $c'=3c$}.
\end{align*}
So under the assumption of the IH for $k'<k$ we see there exists a $c\in\mathbb N$ s.t. $T(k) \leq c\cdot\log_3 k$.
From the principle of induction this now follows for any $k\in\{n\in\mathbb N\mid n>4\}$.
Now since $c\cdot\log_3n \in\pazocal O(\ln n)$ we also have $T(n)\in\pazocal O(\ln n)$.
\end{proof}
\item Claim: $T(n) \in\pazocal O(n\ln n)$.
\begin{proof}
With induction over $n$ we prove there exists some $c\in\mathbb N^+$ s.t. $(T(n)\leq c\cdot n\cdot\log_4n$ for all $n\in\mathbb N,n>4$.
\textbf{Induction basis}: for $n=5$ we have $T(n) = 4T(\frac n4)+n = n \leq c\cdot n\log_4n$ with $c=1$ for example.
\textbf{Inductive step}: let's assume for some $k\in\mathbb N$ that $T(k') \leq c_{k'}\cdot k'\cdot\log_4k'$ for all $k'\in\{n\in\mathbb N\mid n>4\}$, $k'<k$ (IH). Then for $k$:
\begin{align*}
T(k) &\leq 4T\left(\tfrac k4\right) + k\\
&\leq 4\cdot c\cdot \tfrac k4\cdot\log_4\tfrac k4 + k\\
&= c\cdot k\cdot(\log_4k - 1) + k\\
&= c\cdot k\cdot\log_4k + (1-c)k \qquad\text{and since $c\in\mathbb N^+$\dots}\\
&\leq c\cdot k\cdot\log_4k.
\end{align*}
So under the assumption of the IH for $k'<k$ we see there exists a $c\in\mathbb N$ s.t. $T(k)\leq c\cdot k\cdot\log_4 k$.
From the principle of induction this now follows for any $k\in\{n\in\mathbb N\mid n>4\}$.
Now since $c\cdot n\cdot\log_4n \in\pazocal O(n\ln n)$ we also have $T(n)\in\pazocal O(n \ln n)$.
\end{proof}
\end{enumerate}
\item \begin{enumerate}
\item The algorithm is performs equally well on sorted and unsorted lists. However, it performs better on lists with size (close to) $2^n$ for some $n\in\mathbb N$, since it needs a minimum amount of \texttt{merge} calls in that case. The best case would be a list of size $2^n$ for some $n\in\mathbb N$, the worst case a list of size in between $2^n$ and $2^{n+1}$, that is, $1\frac12\cdot2^n$ for some $n\in\mathbb N$.
\item Dividing takes constant time, i.e. $\Theta(1)$. Conquering takes $T(\ceil{\frac n2})+T(\floor{\frac n2})$. Composing takes linear time, $\Theta(n)$ in the \texttt{merge} function and once again linear time in the final \texttt{for} loop in \texttt{mergesort}. That gives:
$$T(n) \leq T\left(\ceil*{\tfrac n2}\right) + T\left(\floor*{\tfrac n2}\right) + c, \qquad c\in\mathbb N$$
\item This is essentially the same as 2.1d, so this is $\pazocal{O}(n\ln n)$.
\item As said in 2.2a, the worst and best case complexity is the same, so we also have $\Omega(n\ln n)$.
\end{enumerate}
\item \begin{enumerate}
\item \begin{proof}
Since $f\in\pazocal O(h)$, there exists a $c_f\in\mathbb N$ and $n_{0,f}\in\mathbb N$ s.t. for all $n>n_0$ we have $c_f\cdot h(n)\geq f(n)$. Similarly we have $c_g$ and $n_{0,g}$ with respect to $g$.
Then let $c=c_f + c_g, n_0 = \text{max}(n_{0,f},n_{0,g})$. By the above we know $(c_f + c_g)\cdot h(n) \geq f(n) + g(n)$ for all $n>n_0$. It follows from substitution that $c\cdot h(n)\geq f(n) + g(n)$, and therefore $f + g \in\pazocal O(h)$.
\end{proof}
\item \begin{proof}
For $n\in\mathbb N$ we know $n^m\geq n^k$ (since $k\leq m$). We can thus choose $c=1,n_0=0$ to find that for all $n>n_0$ it holds that $n^k \leq c\cdot n^m$. Therefore, $n^k\in\pazocal O\left(n^m\right)$.
\end{proof}
\item \begin{proof}
From (b) we know that all terms ($c_kn^k$, $c_{k-1}n^{k-1}$, etc.) are in $\pazocal O\left(n^k\right)$. From (a) it then follows that their sum is also in $\pazocal O\left(n^k\right)$.
\end{proof}
\end{enumerate}
\item \begin{enumerate}
\item \begin{minted}[tabsize=0,linenos,xleftmargin=20pt,fontsize=\footnotesize]{c}
int findzerosum(float* a, float* b, unsigned int n, float* out) {
unsigned int ai, bi; // counters
bi = n - 1; // start b at the max. element
for (ai = 0; ai < n; ai++) { // start a at the min. element
while (a[ai] + b[bi] > 0) bi--; // as long as b > -a, choose lower b
if (a[ai] + b[bi] == 0) { // if a = -b, return a, b and success
out[0] = a[ai];
out[1] = b[bi];
return 0;
} // otherwise, choose higher a
}
return -1; // if all as checked but nothing found, return failure
}
\end{minted}
The worst case is that we need the highest element from $a$ and the lowest element from $b$, meaning we need to iterate both lists fully. In that case we need $2n+c$ operations for some $n,c\in\mathbb N$: $2n$ for iterating and checking the elements, and $c$ for returning the correct elements.
This shows we have a worst-case complexity of $\pazocal O(n)$. The best case is to find the right elements directly; in this case we need constant time.
\item We first rewrite our answer from \textit{(a)} to support other `goals' than zero:
\begin{minted}[tabsize=0,linenos,xleftmargin=20pt,fontsize=\footnotesize]{c}
int findsum(float* a, float* b, unsigned int n, float goal, float* out) {
unsigned int ai, bi;
bi = n - 1;
for (ai = 0; ai < n; ai++) {
while (a[ai] + b[bi] > goal) bi--; // this line changed
if (a[ai] + b[bi] == goal) { // this line changed
out[0] = a[ai];
out[1] = b[bi];
return 0;
}
}
return -1;
}
\end{minted}
%stopzone
Then for arrays $a,b,c$ we iterate over $a$ and try to find elements in $b$ and $c$ that fit using the algorithm above:
\begin{minted}[tabsize=0,linenos,firstnumber=last,xleftmargin=20pt,fontsize=\footnotesize]{c}
int findzerosum3(float* a, float* b, float* c, unsigned int n, float* out) {
unsigned int ai; // counter
for (ai = 0; ai < n; ai++) { // for every a in [a]
if (0 == findsum(b, c, n, -a[ai], out)) { // if we can find elements in b and c that fit
out[2] = a[ai]; // store values and return success
return 0;
} // otherwise, choose next a
}
return -1; // when [a] is exhausted, return failure
}
\end{minted}
Note that this would also work when $a$ is not sorted. The worst-case complexity of \texttt{findsum} is, as discussed above, in $\pazocal O(n)$. The worst-case complexity of \texttt{findzerosum3} is than $\pazocal O(n^2)$, because \texttt{findsum} is executed at most $n$ times, and the rest of the algorithm uses constant time.
\end{enumerate}
\end{enumerate}
\end{document}