### oops

parent d4d5de49
 build/ *.tex.mk *.pdf
 # 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}
Supports Markdown
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!