\begin{frame}
    \frametitle{\problemtitle}
    \begin{block}{Problem}
        Find $n$ infinite binary strings $s_i$, which satisfy prophecies of the form
        \begin{itemize}
            \item $s_u[t] \neq s_v[t]$ 
            \item $t$ is the smallest index with the above property
        \end{itemize}
        Minimize the number of $1$'s in those strings.
    \end{block}
    \begin{itemize}
        \item<+-> Iterate over $t$ in decreasing order and process all prophecies with the current $t$
        \item<+-> The restrictions $s_u[t] \neq s_v[t]$ can be satisfied iff the graph formed by the edges $(u,v)$ is bipartite
        \item<+-> Greedily assign a $1$ to the smaller part of each component.
        \item<+-> Strings have to be identical for all further prophecies $\rightarrow$ merge each connected components into a single node
        \item<+-> Complexity: $\mathcal{O}(m\log m+n)$ for sorting and checking bipartiteness
    \end{itemize}
    % \solvestats
\end{frame}
