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# SeminaireMails
# Seminaire Mails
\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}
\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}
\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}
\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}
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