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papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.tex Property changes : Deleted: svn:eol-style ## -1 +0,0 ## -native \ No newline at end of property Deleted: svn:keywords ## -1 +0,0 ## -Author Date Id Rev URL \ No newline at end of property Index: papers/PHR/uEA2014/slide/beamer/template.tex =================================================================== --- papers/PHR/uEA2014/slide/beamer/template.tex (revision 286) +++ papers/PHR/uEA2014/slide/beamer/template.tex (nonexistent) @@ -1,744 +0,0 @@ -% 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. - - - -\documentclass{beamer} - -% -% 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. -% - - -% Setup appearance: - -\usetheme{Darmstadt} -\usefonttheme[onlylarge]{structurebold} -\setbeamerfont*{frametitle}{size=\normalsize,series=\bfseries} -\setbeamertemplate{navigation symbols}{} - - -% Standard packages - -\usepackage[english]{babel} -\usepackage[latin1]{inputenc} -\usepackage{times} -\usepackage[T1]{fontenc} - - -% Setup TikZ - -\usepackage{tikz} -\usetikzlibrary{arrows} -\tikzstyle{block}=[draw opacity=0.7,line width=1.4cm] - - -% Author, Title, etc. - -\title[Block Partitioning and Perfect Phylogenies] -{% - On the Complexity of SNP Block Partitioning Under the Perfect - Phylogeny Model% -} - -\author[Gramm, Hartman, Nierhoff, Sharan, Tantau] -{ - Jens~Gramm\inst{1} \and - Tzvika~Hartman\inst{2} \and - Till~Nierhoff\inst{3} \and - Roded~Sharan\inst{4} \and - \textcolor{green!50!black}{Till~Tantau}\inst{5} -} - -\institute[Tübingen and others] -{ - \inst{1}% - Universität Tübingen, Germany - \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] -{Workshop on Algorithms in Bioinformatics, 2006} - - - -% The main document - -\begin{document} - -\begin{frame} - \titlepage -\end{frame} - -\begin{frame}{Outline} - \tableofcontents -\end{frame} - - -\section{Introduction} - -\subsection{The Model and the Problem} - -\begin{frame}{What is haplotyping and why is it important?} - You hopefully know this after the previous three talks\dots -\end{frame} - -\begin{frame}[t]{General formalization of haplotyping.} - \begin{block}{Inputs} - \begin{itemize} - \item A \alert{genotype matrix} $G$. - \item The \alert{rows} of the matrix are \alert{taxa / individuals}. - \item The \alert{columns} of the matrix are \alert{SNP sites / - characters}. - \end{itemize} - \end{block} - \begin{block}{Outputs} - \begin{itemize} - \item A \alert{haplotype matrix} $H$. - \item Pairs of rows in $H$ \alert{explain} the rows of $G$. - \item The haplotypes in $H$ are \alert{biologically plausible}. - \end{itemize} - \end{block} -\end{frame} - - -\begin{frame}[t]{Our formalization of haplotyping.} - \begin{block}{Inputs} - \begin{itemize} - \item A genotype matrix $G$. - \item The rows of the matrix are individuals / taxa. - \item The columns of the matrix are SNP sites / characters. - \item - The problem is directed: one haplotype is known. - \item - The input is biallelic: there are only two homozygous - states (0 and 1) and one heterozygous state (2). - \end{itemize} - \end{block} - \begin{block}{Outputs} - \begin{itemize} - \item A haplotype matrix $H$. - \item Pairs of rows in $H$ explain the rows of $G$. - \item The haplotypes in $H$ form a perfect phylogeny. - \end{itemize} - \end{block} -\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 - 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 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 $\chi_{{\operatorname{PPP}}}$: - \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{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} - -

papers/PHR/uEA2014/slide/beamer/template.tex Property changes : Deleted: svn:eol-style ## -1 +0,0 ## -native \ No newline at end of property Deleted: svn:keywords ## -1 +0,0 ## -Author Date Id Rev URL \ No newline at end of property Index: papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.pdf =================================================================== Cannot display: file marked as a binary type. svn:mime-type = application/octet-stream Index: papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.pdf =================================================================== --- papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.pdf (revision 286) +++ papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.pdf (nonexistent)

papers/PHR/uEA2014/slide/beamer/beamer-Warsaw.pdf Property changes : Deleted: svn:mime-type ## -1 +0,0 ## -application/octet-stream \ No newline at end of property