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\documentclass{article}
\input{structure.tex}

\title{CENG567: Homework \#2}
\author{Yiğit Sever}
\date{\today}

\addbibresource{mylib.bib}

%----------------------------------------------------------------------------------------

\begin{document}

\maketitle

\section{Checking Consistency of Judgements}%
\label{sec:checking_consistency_of_judgements}

Given the collection of $n$ butterflies and a potential judgement between every pair (or not if the judgement is ambiguous), we have a graph $G = (V, E)$ with $n = |V|$ vertices and $|E| = m \le \frac{n(n-1)}{2}$ edges, with every edge $(i, j) \in E$ labelled either \enquote{same} or \enquote{different}. At the end of our algorithm, the vertices should be \emph{consistently} labelled as either \texttt{A} and \texttt{B} or our algorithm should be able to prove that $G$ cannot be labelled as so.

A modified graph traversal using either BFS or DFS (since a node can be discovered multiple times in both of them) will work. Here we will modify the graph traversal given on \emph{page 42} on our \nth{3} lecture slides that uses BFS. The input of the algorithm is a node $s \in E$. If, due to ambiguous (i.e.\ missing) nodes, the graph is not connected, the algorithm should be run until every connected component is discovered.
% TODO: run until every cc is discovered is handwavey <15-11-20, yigit> %

{\centering
    \begin{minipage}{.7\linewidth}
        \begin{algorithm}[H]
            \SetKwProg{Fn}{function}{}{end}
            \DontPrintSemicolon{}
            \SetAlgoLongEnd{}
            \Fn{consistency\_check(s: node)}{
                \KwData{$K$ = data structure of discovered nodes}
                \KwResult{boolean = whether $G$ is consistent or not}
                \textbf{label} $s$ as \texttt{A} \tcc*[r]{the opposite label is B}
                put $s$ in $K$\;
                \While{$K$ is not empty}{
                    take a node $v$ from $K$\;
                    \If{$v$ is not marked \enquote{explored}}{
                        mark $v$ \enquote{explored}\;
                        \For{each edge $(v, w)$ incident to $v$}{
                            \uIf{$w$ is labelled}{
                                \If{the label of $w$ is not consistent with the label of $v$ with respect to the judgement $(v,w$)}{
                                    terminate the algorithm; $G$ is \textbf{inconsistent}\;
                                }
                            }
                            \Else{
                                \uIf{$(v,w)$ is labelled \enquote{same}}{
                                    \textbf{label} $w$ with the label of $v$\;
                                }
                                \Else(\tcc*[f]{$(v,w)$ is labelled \enquote{different}}){
                                    \textbf{label} $w$ with the opposite label of $v$\;
                                }
                            }
                            put $w$ in $K$\;
                        }
                    }
                }
                the spanned connected component is \textbf{consistent}
            }
            \caption{Modified graph traversal algorithm so solve judgement consistency checking problem}%
            \label{alg:question_1}
        \end{algorithm}
    \end{minipage}
    \par
}

With the assumption that accessing the labels $(u, w)$ takes $\mathcal{O}(1)$ time this algorithm has the same running time as BFS; $\mathcal{O}(m+n)$.

\section{Reachability}%
\label{sec:reachability}

First, compute all strongly connected components (SCCs) of $G$ by using~\parencite{tarjanDepthFirst1972} per \emph{page 72} of the \nth{3} lecture notes in $\mathcal{O}(E+V)$ time. Instead of labelling the SCCs with the root node, we will initially label all nodes of the SCC $F'$ with the $min(u)$ of the connected component.

Then, by ignoring the tree edges, shrink the graph $G$ such that $E' = {(v, w)~|~v \in F', w \in F''}$, leaving only cross links behind. This step takes another $\mathcal{O}(E+V)$ time.

Now run the topological sort algorithm presented in \emph{page 84} of the \nth{3} lecture notes this operation is yet again $\mathcal{O}(E+V)$.

Finally, reverse the direction of the edges on the graph that have been output by the topological sort and starting from the new root node, traverse the graph downwards and update the $min(u)$ of every SCC as follows;

{\centering
    \begin{minipage}{.7\linewidth}
        \begin{algorithm}[H]
            \DontPrintSemicolon{}
            \SetAlgoLongEnd{}
            \KwData{$G'$ = topological sorted G with reversed edges}
            \KwResult{$\min(u)$ for all vertices $u \in V$}
            $\text{mostmin} \longleftarrow \min(\text{root})$\;
            \While{traversing $G'$ downwards with current node $v$}{
                \uIf{$\min(v) < mostmin$}{
                    $\text{mostmin} \longleftarrow \min(v)$\;
                }
                \Else{
                    label $v$ as $mostmin$
                }
            }
            \caption{Updating $\min(u)$ of the SCCs}%
            \label{alg:question_2}
        \end{algorithm}
    \end{minipage}
    \par
}

\printbibliography

\end{document}