oops

.gitignore

 build/
 *.tex.mk
+*.pdf

README.md

-# SeminaireMails
+# Seminaire Mails
 

src/1_motivation.tex

-\section{Motivation}
-
-\begin{frame}
-    \frametitle{Static Unit Disk Graphs}
-    \begin{definition}[Unit Disk Graph]
-        \(G = (V, E)\) an undirected graph is a Unit Disk Graph (UDG) in dimension \(n\)
-        when there exists an embedding \(\iota : V \to \RR^n\) such that
-        \(\forall v,v'\in V,\ \set{v,v'} \in E \iff \|\iota(v) - \iota(v')\| \leq 1\)
-    \end{definition}
-    \begin{center}
-        \visible<2->{\includegraphics[width=4.3cm]{udg-examples.pdf}}
-        \visible<2->{\includegraphics[width=2.6cm]{not-pig.pdf}}
-        \visible<3->{\includegraphics[width=3.5cm]{not-udg.pdf}}
-    \end{center}
-\end{frame}
-
-% Left: examples of UDG
-% Right: simple example of not UDG
-% Mention that halving the radius yields intersection model
-
-\begin{frame}
-    \frametitle{Dynamic UDG}
-    \begin{definition}
-        A dynamic UDG is \(\mathcal{G} = (V, E_0, \cdots, E_\tau)\)
-        such that all \(G_i = (V, E_i)\) are UDG and successive embeddings change
-        in limited ways.
-    \end{definition}
-    \(G_i\): ``snapshots''\\
-    \((V, \bigcup_{0 \leq i \leq \tau} E_i)\): ``footprint''\\
-    \begin{itemize}
-        \item To what extent can dynamic UDG be recognized ?
-        \item How to define ``limited ways'' ?
-    \end{itemize}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Plausible Mobility}
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[width=5.3cm]{../figures/ditl-perf.png}
-            \includegraphics[width=5.3cm]{../figures/ditl-perf2.png}
-            \caption{Inferring of positions from contact trace}
-            Screenshots from simulations of reconstructed movements
-        \end{center}
-    \end{figure}
-    Tolerates missing or extra links.\\
-    Reasonable assumption in the case of a low quality trace, but can we do better ?\\
-    {\tiny Whitbeck \& Amorim \& Conan,
-    \textit{Plausible Mobility}, \ttt{https://plausible.lip6.fr} (2011)}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Results}
-    \begin{center}
-        \begin{tabular}{|l|c|c|}
-            \hline
-            setting & static & dynamic (new) \\
-            \hline
-            unrestricted (2D) & NP-hard\(^{(1)}\) & \visible<2->{NP-hard} \\
-            tree (2D) & NP-hard\(^{(2)}\) & \visible<2->{NP-hard} \\
-            caterpillar (2D) & Linear\(^{(2)}\) & \visible<2->{NP-hard\({}^{(*)}\)} \\
-            1D & Linear\(^{(3)}\) & \visible<2->{Linear} \\
-            \hline
-        \end{tabular}
-    \end{center}
-    \visible<2->{
-        \({}^{(*)}\) all snapshots are caterpillars
-    }~\\~\\
-    {\tiny
-        \(^{(1)}\) Breu \& Kirkpatrick, \textit{Unit disk graph recognition is NP-hard}
-        (1998)\\
-        \(^{(2)}\) Bhore \& Nickel \& N\"ollenburg, \textit{Recognition of
-        Unit Disk Graphs for Caterpillars, Embedded Trees, and Outerplanar Graphs} (2021)\\
-        \(^{(3)}\) Booth \& Lueker, \textit{Testing for the consecutive ones property,
-        interval graphs, and graph planarity using PQ-tree algorithms} (1976)
-        (And at least 3 other papers)\\
-    }
-\end{frame}

src/2_two-dim.tex

-\section{2-dimensional}
-
-\begin{frame}
-    \frametitle{Overview and intuition}
-    \begin{itemize}
-        \item reduction from 3\tsf{-SAT}: \(F = C_1 \wedge \cdots \wedge C_k\);
-            \(C_i = l_i \vee l'_i \vee l''_i\)
-        \item one group of disks for each variable
-        \item each variable can take two states, interpreted as true or false
-        \item clauses are handled sequentially over a sequence of consecutive snapshots\\~\\
-
-            \includegraphics[width=10cm]{timeline.pdf}
-    \end{itemize}
-    \visible<2->{
-        Hypothesis: ``slow enough''. Speed is bounded by a constant fraction
-        of the radius.\\
-        This makes variables unable to change state in the middle of the process.
-    }
-\end{frame}
-
-\begin{frame}
-    \frametitle{Two configurations of variables}
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[width=11cm]{variable.pdf}
-            Left: \tbf{true}, Right: \tbf{false}
-        \end{center}
-    \end{figure}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Clause assembling}
-    \begin{figure}[H]
-        \begin{minipage}[t]{0.45\linewidth}
-            \begin{center}
-                \includegraphics[height=7cm]{clause.pdf}
-            \end{center}
-        \end{minipage}
-        \hfill
-        \begin{minipage}[t]{0.45\linewidth}
-            The clause \(C = \neg x_1 \vee x_2 \vee \bot\).\\
-            With \(x_1 = x_2 = \tbf{true}\).\\
-            Satisfied thanks to \(x_2\).\\
-
-            The central 12-cycle can fit \(4\) disks but not \(6\).
-        \end{minipage}
-    \end{figure}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Extension of the result}
-    This shows NP-hardness in the general case.\\~\\
-
-    Simpler proof than in the static case\\
-    + linear number of disks instead of quadratic\\
-    + fewer restrictions on initial \tsc{3-SAT} instance\\
-    ~\\
-
-    \visible<2->{
-        Still NP-hard under the modified constraints (separately):
-        \begin{itemize}
-            \item integer coordinates \visible<3->{{\color{red} (static: unknown)}}
-            \item footprint is a tree \visible<3->{{\color{red} (static: NP-hard)}}
-            \item snapshots are caterpillars \visible<3->{{\color{red} (static: linear)}}
-            \item snapshots have CCs of size at most 2
-                \visible<3->{{\color{red} (static: \(O(1)\))}}
-            \item one event at a time \visible<3->{{\color{red} (static: irrelevant)}}
-        \end{itemize}
-    }
-    \visible<2->{\small (caterpillar: tree with all vertices within distance 1 of a
-    central path)}
-\end{frame}

src/3_one-dim.tex

-\section{1-dimensional}
-
-\begin{frame}
-    \frametitle{Takeaway and 1D restriction}
-    Main source of problems: structures can be forced to ``choose'' one of
-    several embeddings, which they are then unable to escape from.\\
-    ~\\
-
-    In one dimension, an efficient representation
-    of all possible configurations\\
-    \(\longrightarrow\) extension of \(PQ\)-trees\\
-\end{frame}
-
-\begin{frame}
-    \frametitle{Physical 1D model}
-    \begin{itemize}
-        \item one event at a time \(\tsc{LinkUp}\) or \(\tsc{LinkDown}\)
-            \((|E_i \Delta E_{i+1}| = 1)\)\\
-            \(\longrightarrow\) perfect trace\\
-        \item continuous transition from one embedding to the next
-    \end{itemize}
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[width=7cm]{movements.pdf}
-        \end{center}
-    \end{figure}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Equivalent permutations}
-    \begin{theorem}
-        For \(\pi\in\mathfrak{S}(V)\), there exists an injective embedding
-        \(\iota\) of \(G\) with the same ordering of vertices
-        iff all neighborhoods \(N[v]\) are contiguous subsequences of \(\pi\)
-    \end{theorem}
-    \(\longrightarrow\) The set of all valid embeddings can be represented by a set of
-    permutations.\\
-
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[height=3.8cm]{thm-1.pdf}
-        \end{center}
-    \end{figure}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Equivalent transitions}
-    \begin{theorem}
-        There exists a continuous transition without event from \(\iota\) to \(\iota'\)
-        iff \(\iota\) and \(\iota'\) differ only in the order of vertices that have the same
-        neighborhood
-    \end{theorem}
-    \(\longrightarrow\) From now on, only manipulations on sets of permutations
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[height=4cm]{thm-2.pdf}
-        \end{center}
-    \end{figure}
-\end{frame}
-
-\begin{frame}
-    \frametitle{\(PQ\)-tree}
-    \begin{minipage}{0.62\textwidth}
-        \begin{figure}[H]
-            \begin{center}
-                \includegraphics[width=6cm]{pqtree.pdf}
-            \end{center}
-        \end{figure}
-    \end{minipage}
-    \hfill
-    \vline
-    \hfill
-    \begin{minipage}{0.35\textwidth}
-        Example:
-        \begin{figure}[H]
-            \begin{center}
-                \includegraphics[width=3cm]{pqtree-example.pdf}
-            \end{center}
-        \end{figure}
-        \begin{tiny}
-            A tree for the set \\
-            1234567, 1324567, 2134567,\\
-            2314567, 3124567, 3214567,\\
-            7654321, 7654231, 7654312,\\
-            7654132, 7654213, 7654123,\\
-        \end{tiny}
-    \end{minipage}
-\end{frame}
-
-
-\begin{frame}
-    \frametitle{\(PQ\)-forest}
-    \begin{itemize}
-        \item set of \(PQ\)-trees
-        \item \(P\)-nodes as leaves contain disks with the same neighborhood
-        \item toplevel trees can be arbitrarily permuted
-    \end{itemize}
-    \begin{figure}[H]
-        \begin{center}
-            \includegraphics[width=10cm]{pqforest.pdf}
-        \end{center}
-    \end{figure}
-\end{frame}
-
-%\begin{frame}
-%    \frametitle{Initialization and input}
-%    Input: sequence of events \(\tsc{LinkUp}(v,v')\) or \(\tsc{LinkDown}(v,v')\)\\
-%    starting from an empty contact trace\\
-%    ~\\
-%
-%    Initial forest:
-%    \begin{figure}[H]
-%        \begin{center}
-%            \includegraphics[width=5cm]{initforest.pdf}
-%        \end{center}
-%    \end{figure}
-%    \textit{i.e.} no information available
-%\end{frame}
-
-
-\begin{frame}
-    \frametitle{\(\tsc{LinkUp}(v,v')\)}
-    \begin{minipage}{0.48\linewidth}
-        \visible<1->{
-            \begin{figure}[H]
-                \includegraphics[width=5cm]{up_start.pdf}
-                Initial
-            \end{figure}
-        }
-    \end{minipage}
-    \hfill
-    \begin{minipage}{0.48\linewidth}
-        \visible<2->{
-            \begin{figure}[H]
-                \includegraphics[width=5cm]{up_rotate.pdf}
-                Rotate
-            \end{figure}
-        }
-    \end{minipage}
-    \vspace{1cm}\\
-    \begin{minipage}{0.48\linewidth}
-        \visible<3->{
-            \begin{figure}[H]
-                \includegraphics[width=5cm]{up_extract.pdf}
-                Extract
-            \end{figure}
-        }
-    \end{minipage}
-    \hfill
-    \begin{minipage}{0.48\linewidth}
-        \visible<4->{
-            \begin{figure}[H]
-                \includegraphics[width=5cm]{up_flatten.pdf}
-                \vspace{1.02cm}\\
-                Flatten
-            \end{figure}
-        }
-    \end{minipage}
-\end{frame}
-
-\begin{frame}
-    \frametitle{\(\tsc{LinkDown}(v,v')\)}
-    \begin{minipage}{0.48\linewidth}
-        \visible<1->{
-            \begin{figure}[H]
-                \includegraphics[height=2cm]{down_start.pdf}\\
-                Initial
-            \end{figure}
-        }
-    \end{minipage}
-    \hfill
-    \begin{minipage}{0.48\linewidth}
-        \visible<2->{
-            \begin{figure}[H]
-                \includegraphics[height=2cm]{down_extract.pdf}\\
-                Extract
-            \end{figure}
-        }
-    \end{minipage}
-    \vspace{1cm}\\
-    \begin{minipage}{0.48\linewidth}
-        \visible<3->{
-            \begin{figure}[H]
-                \includegraphics[height=2.5cm]{down_flip.pdf}\\
-                Allow flip
-            \end{figure}
-        }
-    \end{minipage}
-\end{frame}
-
-\begin{frame}
-    \frametitle{Final result}
-    \begin{itemize}
-        \item each new event requires amortized \(O(\log n)\)\\
-            (\(n\): number of vertices)
-        \item linear overall: \(O(\tau \cdot \log n)\)\\
-            (\(\tau\): number of events)
-        \item online algorithm: updates the \(PQ\)-forest in real time
-    \end{itemize}
-\end{frame}

src/4_conclusion.tex

-\section{Conclusion}
-
-\begin{frame}
-    \frametitle{Open questions \& future works}
-    \begin{itemize}
-        \item characterization of forbidden 1D patterns
-        \item exact algorithm for 2D (even if exponential) ?
-        \item 2D when the \textit{footprint} is a caterpillar\\
-            (despite it being too restrictive for practical purposes)
-        \item 1D algorithm implementation
-    \end{itemize}
-\end{frame}