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\title{Theory I, Sheet 4 Solution}

\author{Eugen Sawin}

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\begin{document}

\maketitle

\section*{Exercise 4.1}

\begin{figure}[h]

\entrymodifiers={}

\xymatrix

{

& & *++[o][F-]{50} \ar@{-}[dl] \ar@{-}[dr] & & & & & & *++[o][F-]{50} \ar@{-}[dl] \ar@{-}[dr]\\

& *++[o][F-]{30} \ar@{-}[dl] \ar@{-}[dr] & & *++[o][F-]{60} \ar@{-}[dr] & \ar@{=>}[rr]^{delete(40)} & & & *++[o][F-]{30} \ar@{-}[dl] & & *++[o][F-]{60}\ar@{-}[dr]\\

*++[o][F-]{20} & & *++[o][F-]{40} & & *++[o][F-]{70} & & *++[o][F-]{20} & & & & *++[o][F-]{70}\\

}

\end{figure}

\begin{figure}[h]

\xymatrix

{

& & *++[o][F-]{50} \ar@{-}[dl] \ar@{-}[dr] & & & & & & *++[o][F-]{50} \ar@{-}[dl] \ar@{-}[dr]\\

& *++[o][F-]{30} \ar@{-}[dl] & & *++[o][F-]{60} \ar@{-}[dr] & \ar@{=>}[rr]^{delete(60)} & & & *++[o][F-]{30} \ar@{-}[dl] & & *++[o][F-]{70}\\

*++[o][F-]{20} & & & & *++[o][F-]{70} & & *++[o][F-]{20} & & & & \\

}

\end{figure}

There are no rotations required because the AVL-balance property is not violated.

\section*{Exercise 4.2}

We use the following Python function for the calculation of the hash code, which resembles the Java code presented in the lecture:

\begin{python}

def hash(string, m):

    return sum(map(ord, string)) % m   

\end{python}

We choose $m = 13$ and get following results:\\\\

\begin{tabular}{ l  c  l}

  \textbf{string} & \textbf{hash} & \textbf{collisions}\\ \hline 

  \emph{Hashing} & 4 & \\

  \emph{Exercise} & 5 & \\ 

  \emph{THEORYI} & 2 & \\

  \emph{god} & 2 & \emph{THEORYI}\\

  \emph{science} & 2 & \emph{THEORYI, god}\\

\end{tabular}\\\\

As you can see there are conflicts between \emph{god, science} and \emph{THEORYI}.

\section*{Exercise 4.3}

Let $U = \{0,...,N-1\}$, where $N = 49$ and $m = 35$. Let $a_i = 42i$ and $b_i = 28i$.\\

Let $H = \{h_i(k) = ((a_ik + b_i) \bmod N) \bmod m\}$ for $ i \in \{1,...,N(N-1)\}$ be a class of hash functions. We show that $H$ is \emph{not} universal by contradiction.\\\\

We assume $H$ is universal.

% \left\{ x \in X \mid x > \frac{1}{2}\right\} \]

\[\iff \frac{|\{h \in H \mid h(x) = h(y) \land x \neq y\}|}{|H|} \leq \frac{1}{m}\]

First we look at the mapping space of the inner modulo function for all pairs $(q, r)$ with \\$q = (a_ix + b_i) \bmod N$ and $r = (a_iy + b_i) \bmod N$ for all $x,y \in U$ and $x \neq y$.

\[q = r \iff (a_ix + b_i) \bmod N = (a_iy + b_i) \bmod N\]

\[\iff a_i(x - y) \bmod N = 0\]

\[\iff \exists_{c \in \mathbb{N}}: a_i(x - y) = cN\]

\[\implies a_i|N \land (x - y)|N\]

\[\iff \exists_{l \in \mathbb{N}, 0<l\leq7}: (x - y) = 7l\]

It follows that there are collisions for $(x - y) = 7l$ with $c = \frac{a_i \cdot 7l}{N} = \frac{42i \cdot 7l}{49} = 6il$\\

To determine the number of inner collisions, we calculate $\sum_{0 < l \leq 7} (48 - 7l)$.

\[\sum_{0 < l \leq 7} (48 - 7l) = 7(48 - \sum_{0 < l \leq 7}l) = 140\]

We know that the number of total collisions of a hash function is greater or equal to the number of collisions of its inner modulo mapping space.

\[ \implies \frac{140}{|H|} = \frac{140}{49 \cdot 48} > \frac{1}{35} \implies Contradiction!\]

Even within the mapping space of the inner modulo function we get more collisions than accepted for a universal class of hash functions; it follows that $H$ is \emph{not} universal. \qed

\end{document}