myprojects-hg-reborn

Fork

% Copyright 2007 by Till Tantau
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.
%
% See the file doc/licenses/LICENSE for more details.



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\newcommand{\jc}{{\it Journal of Colloid and Interface Science}}
\newcommand{\jas}{{\it Journal of Aerosol Science}}
\newcommand{\pra}{{\it Physical Review A}}
\newcommand{\prb}{{\it Physical Review B}}
\newcommand{\pre}{{\it Physical Review E}}
\newcommand{\prl}{{\it Physical Review Letters}}


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%
% DO NOT USE THIS FILE AS A TEMPLATE FOR YOUR OWN TALKS¡!!
%
% Use a file in the directory solutions instead.
% They are much better suited.
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% Standard packages

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% Setup TikZ

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% Author, Title, etc.

\title[Langevin Nanoparticle Dynamics] 
{%
  Langevin Nanoparticle Dynamics%
}

\author[Isella, Drossinos]
{
  Lorenzo~Isella\inst{} and
  Yannis~Drossinos\inst{}
%   Till~Nierhoff\inst{3} \and
%   Roded~Sharan\inst{4} \and
%   \textcolor{green!50!black}{Till~Tantau}\inst{5}
}

\institute[Tübingen and others]
{
  \inst{}%
  Joint Research Centre, Ispra, Italy
%   \and
%   \vskip-2mm
%   \inst{2}%
%   Bar-Ilan University, Ramat-Gan, Israel
%   \and
%   \vskip-2mm
%   \inst{3}%
%   International Computer Science Institute, Berkeley, USA
%   \and
%   \vskip-2mm
%   \inst{4}%
%   Tel-Aviv University, Israel
%   \and
%   \vskip-2mm
%   \inst{5}%
%   Universität zu Lübeck, Germany
}

\date[WABI 2006]
{European Aerosol Conference, Thessaloniki, Greece, 2008}



% The main document

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Outline}
  \tableofcontents
\end{frame}


\section{Introduction}

\subsection{Problem Formulation}

\begin{frame}[t]{Motivation and Goals}
  \begin{itemize}
  \item Ab initio simulation of soot primary particle agglomeration.
\item Evaluate the effects of different monomer-monomer interactions (electrostatic, Van der Waals, short-ranged).
\item Aggregate morphology: radius of gyration, hydrodynamic radius,
  coordination number hence
\item investigation of the fractal dimension of the generated
  clusters.
\item Transport and thermalization properties (diffusion coefficient,
  response time). 
\item Coagulation dynamics and numerical evaluation of the collisional kernel matrix elements. 
  \end{itemize}
\end{frame}

\subsection{Model for Monomer Dynamics}
\begin{frame}[t]{Langevin Equation for Mesoscopic Systems}
\begin{itemize}
\item  3D system of interacting monomers, each obeying
\begin{equation}
  \label{eq:Langevin}
  m_1\ddot{\bf r}_i={\bf F}_i-\beta_1 m_1\dot{\bf r}_i+ 
  {\bf W}_i(t)
\end{equation}
\item force acting on i-th monomer from pairwise potential
\begin{equation}
  \label{eq:potential_pairwise}
  {\bf F}_i=-\nabla_{{\bf r}_i} U_i=-\nabla_{{\bf r}_i}\lro\f{1}2{}\s_{j\neq i}u(r_{ij})\rro
\end{equation}
\item white noise acting on each monomer \begin{equation}
  \label{eq:gaussian_noise}
\langle  W^{j}_i(t) \rangle=0\;\;\;\; {\rm{and}} \;\;\;\; \langle
 W^{j}_i(t) W_{i'}^{j'}(t')\rangle=\Gamma\delta_{ii'}\delta_{j
j'}\delta(t-t'),
\end{equation}
noise~strength~$\Gamma=2\beta_1 m_1k_BT$~fixed~by~fluctuation-dissipation~theorem.

\end{itemize}
\end{frame}


%\section{Physical Scales and Interaction Potential}
\subsection{Dimensionless Formalism}

\begin{frame}[t]{Specification of the Units}
\
\begin{itemize}
\item Natural (but not unique!) choice for time, distance and mass units
\begin{equation}
  \label{eq:dimensionless-scales-chosen}
  r\equiv \sigma\tilde r,\;\;\;\; t\equiv \tau_{\rm mon}\tilde t, \;\;\;\;
  m_1\equiv m_1\tilde m_1.
\end{equation}
\item Temperature unit $T^*$ is a \emph{derived} quantity. For a
$20$nm soot particle ($\rho_p\simeq 1.3$g/cm${^3}$) in air at room temperature
\begin{equation}
  \label{eq:T_star}
  T^*=\f{m_1\sigma^2}{k_B\tau_{\rm mon}^2}=\f{18^2\pi\mu_f^2\sigma}{6k_B\rho_p}\simeq 650{\rm K}
\end{equation}
$\rightarrow \tilde T=0.5$ for exhaust nanoparticles at room
temperature. 
\item Dimensionless quantities used in the following, unless otherwise stated.
\end{itemize}



\end{frame}


\subsection{Interaction Potential}
\begin{frame}[t]{Features of Monomer-Monomer Potential}
%\vspace*{-0.5cm}
%  \begin{block}{General Features}
    \begin{itemize}
    \item Repulsion at short monomer-monomer separations $\sigma$
      (hard-core repulsion) and attraction for separations above
      $\sigma$ 
      (sticking upon collision).
\item Simulations with  Van Der Waals potentials are being run; results for a
potential well 
%\begin{itemize}
%\item Potential used in the simulations:
\end{itemize}
\vspace*{-.5cm}
\begin{figure}
\includegraphics[height=5.2cm, width=8cm]{presentation_potential.pdf} 
%\caption{show an example picture}
\end{figure}


\end{frame}


\section{Simulations  and Data Post-Processing}
\subsection{Numerical Implementation}
\begin{frame}[t]{Setting up of the System}
  \begin{itemize}
\item 5000 monomers displaced randomly in a box with density
  $\rho=0.01$.
\item Bulk simulations: periodic boundary conditions assumed.
\item Mitigate the role of initial conditions $\to$ results averaged
  on $10$ simulations. 
\item At the end of the simulation, the aggregate concentration is
  almost two orders of magnitude below the initial one.
  \item MD EspreSSo package to solve the 3D Langevin
    equations ($4$th-order Verlet)
\item Unlike early studies (Meakin, Mountain), \emph{not} necessary to
  look for aggregation events while evolving the system
\item Only monomers positions and velocities are returned $\to$ how to
  identify the clusters?
  \end{itemize}
\end{frame}

\subsection{Post-Processing}
\begin{frame}[t]{Clusters as Networks} 
\begin{figure}
\includegraphics[height=2cm, width=6cm]{clusters_fig.pdf} 
%\caption{show an example picture}
\end{figure}
\begin{itemize}
\item Two monomers are bound together whenever their distance falls
  below a threshold $d_{\rm thr}$ close to
  the position of the minimum of $u(r)$.
\item I can ``step'' onto any monomer in a cluster by making strides of
  length  $d_{\rm thr}$.
\item Cluster determination from the monomers positions
  $\leftrightarrow$ determination of the connected components in a non-directed graph.
\item Information on the number of clusters, their composition, their
  geometrical properties.
\end{itemize}
\end{frame}

\begin{frame}[t]{Collision Statistic}
\vspace{-0.5cm}
 \begin{figure}
% \setlength{\textwidth}{1cm}
% \setlength{\textheight}{3cm}
% \includegraphics[height=3cm, width=6cm]{clusters_coll.pdf} 
\input{clusters_coll.pdf_t}
% %\caption{show an example picture}
\end{figure}
\begin{itemize}
\item Two independent clusters at time $t$ become a proper subset of
  the same cluster at time $t+\delta t$ $\to$ collision.
\item Determination of the kernel element from $i$ and $j$-mer
  concentrations and from number of collisions per unit volume in
  $\delta t$ 
  \begin{equation}
    \label{eq:1}
    \f{n_{ij}}{\delta t}=\mathcal{K}_{ij}n_in_j.
  \end{equation}
\end{itemize}
\end{frame}

\section{Results and Discussion}
\subsection{Determination of the Fractal Dimension}
\begin{frame}[t]{Distribution of Aggregate Morphologies} 
\vspace*{-0.5cm}
\begin{figure}
\includegraphics[height=4cm, width=0.45\columnwidth]{my_output.png}
\vspace*{0.1cm} 
\includegraphics[height=4cm, width=0.45\columnwidth]{50_monomers.png} 
%\caption{show an example picture}
\end{figure} 

\vspace{-0.5cm}

\begin{figure}
\includegraphics[height=4cm, width=0.45\columnwidth]{50_monomers-2.png} 
\vspace*{0.1cm} 
\includegraphics[height=4cm, width=0.45\columnwidth]{50_monomers-3.png} 
%\caption{show an example picture}
\end{figure} 



\end{frame}


\begin{frame}[t]{Time-averaged Fractal Dimension}
  \begin{itemize}
  \item time-independent $d_f$ obtained from fitting mean radius of gyration $R_g\sim k^{1/d_f}$ from $k=5$.
\item results better fitted using \emph{two} slopes $\to$ effect of
  kinetic on the aggregate morphology.
  \end{itemize}
\vspace*{-0.5cm}
\begin{figure}
\includegraphics[height=5.5cm, width=8cm]{figure_two_slopes_extended_complete.pdf} 
%\caption{show an example picture}
\end{figure}
\end{frame}


\begin{frame}[t]{Time-dependent Fractal Dimension}
  \begin{itemize}
  \item For each saved system configuration, fit $R_g\sim
    k^{1/d_f^t}$.
\item $d_f^t$ determined by the interplay of the large and small
  cluster populations.
\item Fractal dimension decreases and tends to $d_f^{\rm large}$. 
  \end{itemize}
\vspace*{-0.5cm}
\begin{figure}
\includegraphics[height=5.5cm, width=8cm]{evolution_df.pdf} 
%\caption{show an example picture}
\end{figure}
\end{frame}

\subsection{Transport Properties and Response Time}
\begin{frame}[t]{Numerical determination of the diffusion coefficient}
\vspace*{-0.8cm}
\begin{figure}
\includegraphics[height=4.8cm, width=8cm]{inset_plot.pdf} 
%\caption{show an example picture}
\end{figure}
\vspace*{-0.5cm}
\begin{itemize}
\item Ensemble average on $800$ trajectories
\vspace*{-0.2cm}
\begin{equation}
  \label{eq:cluster-diffusion-coefficient}
  D_k \xrightarrow{t\to\infty} \f{\langle \delta{ r}^2_{CM}(t) \rangle}
  {6t}=\f{k_BT}{km_1\beta_k}C_s(Kn).
  \end{equation}
\item $D_k\sim t^\gamma$, $\gamma\simeq 3$ at early times $\to$
  similar to Brownian monomers. 
\item Simulations consistent with  $C_s(Kn)=1$ and $\beta_k=\beta_1$ 
  $\to$ continuum regime and ``transparent'' clusters. 
\end{itemize}

\end{frame}
 



% \begin{frame}{We can do perfect phylogeny haplotyping efficiently, but
%     \dots}
%   \begin{enumerate}
%   \item \alert{Data may be missing.}
%     \begin{itemize}
%     \item This makes the problem NP-complete \dots
%     \item \dots even for very restricted cases.
%     \end{itemize}
%     \textcolor{green!50!black}{Solutions:}
%     \begin{itemize}
%     \item Additional assumption like the rich data hypothesis. 
%     \end{itemize}
%   \item \alert{No perfect phylogeny is possible.}
%     \begin{itemize}
%     \item This can be caused by chromosomal crossing-over effects.
%     \item This can be caused by incorrect data.
%     \item This can be caused by multiple mutations at the same sites.
%     \end{itemize}
%     \textcolor{green!50!black}{Solutions:}
%     \begin{itemize}
%     \item Look for phylogenetic networks.
%     \item Correct data.
%     \item<alert@1->
%        Find blocks where a perfect phylogeny is possible.
%     \end{itemize}
%   \end{enumerate}
% \end{frame}


% \subsection{The Integrated Approach}

% \begin{frame}{How blocks help in perfect phylogeny haplotyping.}
%   \begin{enumerate}
%   \item Partition the site set into overlapping contiguous blocks.
%   \item Compute a perfect phylogeny for each block and combine them.
%   \item Use dynamic programming for finding the partition.
%   \end{enumerate}

%   \begin{tikzpicture}
%     \useasboundingbox (0,-1) rectangle (10,2);
    
%     \draw[line width=2mm,dash pattern=on 1mm off 1mm]
%       (0,1) -- (9.99,1) node[midway,above] {Genotype matrix}
%       (0,0.6666) -- (9.99,0.6666)
%       (0,0.3333) -- (9.99,0.3333)
%       (0,0) -- (9.99,0) node[midway,below] {\only<1>{no perfect phylogeny}};

%     \begin{scope}[xshift=-.5mm]
%       \only<2->
%       {
%         \draw[red,block]            (0,.5)   -- (3,.5)
%           node[midway,below] {perfect phylogeny};
%       }
        
%       \only<3->
%       {
%         \draw[green!50!black,block] (2.5,.5)   -- (7,.5)
%           node[pos=0.6,below] {perfect phylogeny};
%       }

%       \only<4->
%       {
%         \draw[blue,block]           (6.5,.5) -- (10,.5)
%           node[pos=0.6,below] {perfect phylogeny};
%       }
%     \end{scope}
%   \end{tikzpicture}
% \end{frame}

% \begin{frame}{Objective of the integrated approach.}
%   \begin{enumerate}
%   \item Partition the site set into \alert{noncontiguous} blocks. 
%   \item Compute a perfect phylogeny for each block and combine them. 
%   \item<alert@1-> Compute partition while computing perfect
%     phylogenies. 
%   \end{enumerate}

%   \begin{tikzpicture}
%     \useasboundingbox (0,-1) rectangle (10,2);

%     \draw[line width=2mm,dash pattern=on 1mm off 1mm]
%       (0,1) -- (9.99,1) node[midway,above] {Genotype matrix}
%       (0,0.6666) -- (9.99,0.6666)
%       (0,0.3333) -- (9.99,0.3333)
%       (0,0) -- (9.99,0) node[midway,below] {\only<1>{no perfect phylogeny}};

%     \only<2->
%     {
%       \begin{scope}[xshift=-0.5mm]
%         \draw[red,block] (0,.5)   -- (3,.5) 
%           node[midway,below] {perfect phylogeny}
%                          (8,.5) -- (9,.5);

%         \draw[green!50!black,block]
%           (3,.5)   -- (6,.5)
%             node[pos=0.6,below] {perfect phylogeny}
%           (6.4,.5)   -- (8,.5)
%           (9,.5) -- (10,.5);

%         \draw[blue,block] (6,.5) -- (6.4,.5)
%           node[midway,below=5mm] {perfect phylogeny};
%       \end{scope}
%     }
%   \end{tikzpicture}
% \end{frame}


% \begin{frame}{The formal computational problem.}
%   We are interested in the computational complexity of \\
%   \alert{the function \alert{$\chi_{\operatorname{PP}}$}}:
%   \begin{itemize}
%   \item It gets genotype matrices as input.
%   \item It maps them to a number $k$.
%   \item This number is minimal such that the sites can be
%     covered by $k$ sets, each admitting a perfect phylogeny.
%     \\
%     (We call this a \alert{pp-partition}.)
%   \end{itemize}
% \end{frame}


% \section{Bad News: Hardness Results}

% \subsection{Hardness of PP-Partitioning of Haplotype Matrices}

% \begin{frame}{Finding pp-partitions of haplotype matrices.}
%   We start with a special case:
%   \begin{itemize}
%   \item The inputs $M$ are \alert{already haplotype matrices}.
%   \item The inputs $M$ \alert{do not allow a perfect phylogeny}.
%   \item What is $\chi_{\operatorname{PP}}(M)$?
%   \end{itemize}
%   \begin{example}
%     \begin{columns}
%       \column{.3\textwidth}
%       $M\colon$
%       \footnotesize
%       \begin{tabular}{cccc}
%         0 & 0 & 0 & 1 \\
%         0 & 1 & 0 & 0 \\
%         1 & 0 & 0 & 0 \\
%         0 & 1 & 0 & 0 \\
%         1 & 0 & 0 & 0 \\
%         0 & 1 & 0 & 1 \\
%         1 & 1 & 0 & 0 \\
%         0 & 0 & 1 & 0 \\
%         1 & 0 & 1 & 0
%       \end{tabular}%
%       \only<2>
%       {%
%         \begin{tikzpicture}
%           \useasboundingbox (2.9,0);

%           \draw [red, opacity=0.7,line width=1cm] (1.7 ,1.9) -- (1.7 ,-1.7);
%           \draw [blue,opacity=0.7,line width=5mm] (0.85,1.9) -- (0.85,-1.7)
%                                                   (2.55,1.9) -- (2.55,-1.7);
%         \end{tikzpicture}
%       }
%       \column{.6\textwidth}
%       \begin{overprint}
%         \onslide<1>
%         No perfect phylogeny is possible.
        
%         \onslide<2>
%         \textcolor{blue!70!bg}{Perfect phylogeny}
        
%         \textcolor{red!70!bg}{Perfect phylogeny}
        
%         $\chi_{\operatorname{PP}}(M) = 2$.
        
%       \end{overprint}
%     \end{columns}
%   \end{example}
% \end{frame}

% \begin{frame}{Bad news about pp-partitions of haplotype matrices.}
%   \begin{theorem}
%     Finding \alert{optimal pp-partition of haplotype matrices}\\
%     is equivalent to finding \alert{optimal graph colorings}.
%   \end{theorem}

%   \begin{proof}[Proof sketch for first direction]
%     \begin{enumerate}
%     \item Let $G$ be a graph.
%     \item Build a matrix with a column for each vertex of $G$.
%     \item For each edge of $G$ add four rows inducing\\the
%       submatrix $\left(
%         \begin{smallmatrix}
%           0 & 0 \\
%           0 & 1 \\
%           1 & 0 \\
%           1 & 1
%         \end{smallmatrix}\right)$.
%     \item The submatrix enforces that the columns lie in different
%       perfect phylogenies. \qedhere  
%     \end{enumerate}
%   \end{proof}
% \end{frame}

% \begin{frame}{Implications for pp-partitions of haplotype matrices.}
%   \begin{corollary}
%     If $\chi_{\operatorname{PP}}(M) = 2$ for a haplotype matrix $M$,
%     we can find an optimal pp-partition in polynomial time. 
%   \end{corollary}

%   \begin{corollary}
%     Computing $\chi_{\operatorname{PP}}$ for haplotype matrices is
%     \begin{itemize}
%     \item $\operatorname{NP}$-hard,
%     \item not fixed-parameter tractable, unless
%       $\operatorname{P}=\operatorname{NP}$, 
%     \item very hard to approximate.
%     \end{itemize}
%   \end{corollary}
% \end{frame}


% \subsection{Hardness of PP-Partitioning of Genotype Matrices}


% \begin{frame}{Finding pp-partitions of genotype matrices.}
%   Now comes the general case:
%   \begin{itemize}
%   \item The inputs $M$ are \alert{genotype matrices}.
%   \item The inputs $M$ \alert{do not allow a perfect phylogeny}.
%   \item What is $\chi_{\operatorname{PP}}(M)$?
%   \end{itemize}
%   \begin{example}
%     \begin{columns}
%       \column{.3\textwidth}
%       $M\colon$
%       \footnotesize
%       \begin{tabular}{cccc}
%         2 & 2 & 2 & 2 \\
%         1 & 0 & 0 & 0 \\
%         0 & 0 & 0 & 1 \\
%         0 & 0 & 1 & 0 \\
%         0 & 2 & 2 & 0 \\
%         1 & 1 & 0 & 0 
%       \end{tabular}%
%       \only<2>
%       {%
%         \begin{tikzpicture}
%           \useasboundingbox (2.9,0);
          
%           \draw [red, opacity=0.7,line width=1cm] (1.7 ,1.3) -- (1.7 ,-1.1);
%           \draw [blue,opacity=0.7,line width=5mm] (0.85,1.3) -- (0.85,-1.1)
%                                                   (2.55,1.3) -- (2.55,-1.1);
%         \end{tikzpicture}
%       }
%       \column{.6\textwidth}
%       \begin{overprint}
%         \onslide<1>
%         No perfect phylogeny is possible.
        
%         \onslide<2>
%         \textcolor{blue!70!bg}{Perfect phylogeny}
        
%         \textcolor{red!70!bg}{Perfect phylogeny}
        
%         $\chi_{\operatorname{PP}}(M) = 2$.
        
%       \end{overprint}
%     \end{columns}
%   \end{example}
% \end{frame}


% \begin{frame}{Bad news about pp-partitions of haplotype matrices.}
%   \begin{theorem}
%     Finding \alert{optimal pp-partition of genotype matrices}
%     is at least as hard as finding \alert{optimal colorings of
%       3-uniform hypergraphs}. 
%   \end{theorem}

%   \begin{proof}[Proof sketch]
%     \begin{enumerate}
%     \item Let $G$ be a 3-uniform hypergraph.
%     \item Build a matrix with a column for each vertex of $G$.
%     \item For each hyperedge of $G$ add four rows inducing\\ the submatrix
%       $\left(
%         \begin{smallmatrix}
%           2 & 2 & 2 \\
%           1 & 0 & 0 \\
%           0 & 1 & 0 \\
%           0 & 0 & 1
%         \end{smallmatrix}\right)
%       $.
%     \item The submatrix enforces that the three columns do not all lie
%       in the same perfect phylogeny. \qedhere
%     \end{enumerate}
%   \end{proof}
% \end{frame}

% \begin{frame}{Implications for pp-partitions of genotype matrices.}
%   \begin{corollary}
%     Even if we know $\chi_{\operatorname{PP}}(M) = 2$ for a genotype matrix $M$,\\
%     finding a pp-partition of any fixed size is still
%     \begin{itemize}
%     \item $\operatorname{NP}$-hard,
%     \item not fixed-parameter tractable, unless
%       $\operatorname{P}=\operatorname{NP}$, 
%     \item very hard to approximate.
%     \end{itemize}
%   \end{corollary}
% \end{frame}


% \section{Good News: Tractability Results}

% \subsection{Perfect Path Phylogenies}

% \begin{frame}{Automatic optimal pp-partitioning is hopeless, but\dots}
%   \begin{itemize}
%   \item The hardness results are \alert{worst-case} results for\\
%     \alert{highly artificial inputs}.
%   \item \alert{Real biological data} might have special properties
%     that make the problem \alert{tractable}.
%   \item One such property is that perfect phylogenies are often
%     perfect \alert{path} phylogenies:

%     In HapMap data, in 70\% of the blocks where a perfect phylogeny
%     is possible a perfect path phylogeny is also possible.
%   \end{itemize}  
% \end{frame}


% \begin{frame}{Example of a perfect path phylogeny.}
%   \begin{columns}[t]
%     \column{.3\textwidth}
%     \begin{exampleblock}{Genotype matrix}
%       $G\colon$
%       \begin{tabular}{ccc}
%         A & B & C \\\hline
%         2 & 2 & 2 \\
%         0 & 2 & 0 \\
%         2 & 0 & 0 \\
%         0 & 2 & 2 
%       \end{tabular}
%     \end{exampleblock}

%     \column{.3\textwidth}
%     \begin{exampleblock}{Haplotype matrix}
%       $H\colon$
%       \begin{tabular}{ccc}
%         A & B & C \\\hline
%         1 & 0 & 0 \\
%         0 & 1 & 1 \\
%         0 & 0 & 0 \\
%         0 & 1 & 0 \\
%         0 & 0 & 0 \\
%         1 & 0 & 0 \\
%         0 & 0 & 0 \\
%         0 & 1 & 1 
%       \end{tabular}
%     \end{exampleblock}

%     \column{.4\textwidth}
%     \begin{exampleblock}{Perfect path phylogeny}
%       \begin{center}
%         \begin{tikzpicture}[auto,thick]
%           \tikzstyle{node}=%
%           [%
%             minimum size=10pt,%
%             inner sep=0pt,%
%             outer sep=0pt,%
%             ball color=example text.fg,%
%             circle%
%           ]
        
%           \node [node] {} [->]
%             child {node [node] {} edge from parent node[swap]{A}}
%             child {node [node] {}
%               child {node [node] {} edge from parent node{C}}
%               edge from parent node{B}
%             };
%         \end{tikzpicture}
%       \end{center}
%     \end{exampleblock}
%   \end{columns}
% \end{frame}


% \begin{frame}{The modified formal computational problem.}
%   We are interested in the computational complexity of \\
%   the function $\chi_{\alert{\operatorname{PPP}}}$:
%   \begin{itemize}
%   \item It gets genotype matrices as input.
%   \item It maps them to a number $k$.
%   \item This number is minimal such that the sites can be
%     covered by $k$ sets, each admitting a perfect \alert{path} phylogeny.
%     \\
%     (We call this a ppp-partition.)
%   \end{itemize}
% \end{frame}



% \subsection{Tractability of PPP-Partitioning of Genotype Matrices}

% \begin{frame}{Good news about ppp-partitions of genotype matrices.}
%   \begin{theorem}
%     \alert{Optimal ppp-partitions of genotype matrices} can be
%     computed in \alert{polynomial time}. 
%   \end{theorem}
%   \begin{block}{Algorithm}
%     \begin{enumerate}
%     \item Build the following partial order:
%       \begin{itemize}
%       \item Can one column be above the other in a phylogeny?
%       \item Can the columns be the two children of the root of a
%         perfect path phylogeny?
%       \end{itemize}
%     \item Cover the partial order with as few compatible chain pairs 
%       as possible. 

%       For this, a maximal matching in a special graph needs to be
%       computed.
%     \end{enumerate}
%   \end{block}
%   \hyperlink{algorithm<1>}{\beamergotobutton{The algorithm in action}}
%   \hypertarget{return}{}
% \end{frame}

% \section*{Summary}

% \begin{frame}
%   \frametitle<presentation>{Summary}

%   \begin{itemize}
%   \item
%     Finding optimal pp-partitions is \alert{intractable}. 
%   \item
%     It is even intractable to find a pp-partition when \alert{just two 
%       noncontiguous  blocks are known to suffice}.
%   \item
%     For perfect \alert{path} phylogenies, optimal partitions can be
%     computed \alert{in polynomial time}.
%   \end{itemize}
% \end{frame}


% \appendix

% \section*{Appendix}

% \begin{frame}[label=algorithm]{The algorithm in action.}{Computation of
%     the partial order.}
%   \begin{columns}[t]
%     \column{.4\textwidth}
%     \begin{exampleblock}{Genotype matrix}
%       $G\colon$
%       \begin{tabular}{ccccc}
%         A & B & C & D & E \\\hline
%         2 & 2 & 2 & 2 & 2 \\
%         0 & 1 & 2 & 1 & 0 \\
%         1 & 0 & 0 & 1 & 2 \\
%         0 & 2 & 2 & 0 & 0
%       \end{tabular}
%     \end{exampleblock}
%     \column{.6\textwidth}
%     \begin{exampleblock}{Partial order}
%       \begin{tikzpicture}[node distance=15mm]
%         \tikzstyle{every node}=
%         [%
%           fill=green!50!black!20,%
%           draw=green!50!black,%
%           minimum size=7mm,%
%           circle,%
%           thick%
%         ]

%         \node (A) {A};
%         \node (B) [right of=A] {B};
%         \node (C) [below of=B] {C};
%         \node (D) [above of=A] {D};
%         \node (E) [below of=A] {E};

%         \path [thick,shorten >=1pt,-stealth'] (A) edge (E)
%                          (B) edge (C)
%                          (D) edge (A)
%                              edge[bend right] (E);

%         \uncover<2>{
%         \path [-,blue,thick](A) edge (B)
%                                 edge (C)  
%                             (B) edge (E)
%                             (C) edge (E);}
%       \end{tikzpicture}

%       Partial order: \tikz[baseline] \draw[thick,-stealth'] (0pt,.5ex)
%       -- (5mm,.5ex); 

%       \uncover<2>{\textcolor{blue}{Compatible as children of root:
%           \tikz[baseline] \draw[thick] (0pt,.5ex) -- (5mm,.5ex);}} 
%     \end{exampleblock}
%   \end{columns}  
% \end{frame}

% \begin{frame}{The algorithm in action.}{The matching in the special graph.}
%   \begin{columns}[t]
%     \column{.3\textwidth}
%     \begin{exampleblock}{Partial order}
%       \begin{tikzpicture}[node distance=15mm]
%         \tikzstyle{every node}=%
%         [%
%           fill=green!50!black!20,%
%           draw=green!50!black,%
%           minimum size=8mm,%
%           circle,%
%           thick%
%         ]

%         \node (A)              {$A$};
%         \node (B) [right of=A] {$B$};
%         \node (C) [below of=B] {$C$};
%         \node (D) [above of=A] {$D$};
%         \node (E) [below of=A] {$E$};

%         \path [thick,shorten >=1pt,-stealth'] (A) edge (E)
%                          (B) edge (C)
%                          (D) edge (A)
%                              edge[bend right] (E);

%         \path [-,blue,thick](A) edge (B)
%                                 edge (C)  
%                             (B) edge (E)
%                             (C) edge (E);

%         \only<3->
%         {
%           \path[very thick,shorten >=1pt,-stealth',red] (D) edge (A) (B) edge (C);
%           \path [-,red,very thick](E) edge (B);
%         }
%       \end{tikzpicture}
%     \end{exampleblock}
%     \column{.7\textwidth}
%     \begin{exampleblock}{Matching graph}
%       \begin{tikzpicture}[node distance=15mm]
%         \tikzstyle{every node}=%
%         [%
%           fill=green!50!black!20,%
%           draw=green!50!black,%
%           minimum size=8mm,%
%           circle,%
%           thick,%
%           inner sep=0pt%
%         ]

%         \node (A)              {$A$};
%         \node (B) [right of=A] {$B$};
%         \node (C) [below of=B] {$C$};
%         \node (D) [above of=A] {$D$};
%         \node (E) [below of=A] {$E$};

%         \begin{scope}[xshift=4.75cm]
%           \node (A')               {$A'$};
%           \node (B') [right of=A'] {$B'$};
%           \node (C') [below of=B'] {$C'$};
%           \node (D') [above of=A'] {$D'$};
%           \node (E') [below of=A'] {$E'$};
%         \end{scope}
        
%         \path [thick]    (A) edge (E')
%                          (B) edge (C')
%                          (D) edge (A')
%                              edge (E');

%         \path [blue,thick](A') edge (B')
%                                edge (C')  
%                           (B') edge (E')
%                           (C') edge (E');

%         \only<2->
%         {
%           \path[very thick,red] (D) edge (A')
%                            (B) edge (C')
%                            (B') edge (E');
%         }
%       \end{tikzpicture}
%     \end{exampleblock}
%   \end{columns}

%   \medskip
%   \uncover<2->{A \alert{maximal matching} in the matching graph
%     \uncover<3>{induces\\ \alert{perfect path phylogenies}.}}

%   \hfill\hyperlink{return}{\beamerreturnbutton{Return}}
% \end{frame}

\end{document}