aboutsummaryrefslogtreecommitdiffstats
path: root/buch/papers/multiplikation/presentation/slides/strassen.tex
diff options
context:
space:
mode:
Diffstat (limited to '')
-rw-r--r--buch/papers/multiplikation/presentation/slides/strassen.tex429
1 files changed, 429 insertions, 0 deletions
diff --git a/buch/papers/multiplikation/presentation/slides/strassen.tex b/buch/papers/multiplikation/presentation/slides/strassen.tex
new file mode 100644
index 0000000..c3398d5
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/strassen.tex
@@ -0,0 +1,429 @@
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \includegraphics[page=1,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+ \includegraphics[page=2,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf} \includegraphics[page=3,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+ \end{frame}
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \centering
+ \large
+\onslide<1->{
+ $
+ \mathbf{A B = C}
+ $
+}
+
+\onslide<2->{
+
+
+\medskip
+ $
+ \begin{bmatrix}
+ A_{11} & A_{12}\\
+ A_{21} & A_{22}
+ \end{bmatrix}
+ \begin{bmatrix}
+ B_{11} & B_{12}\\
+ B_{21} & B_{22}
+ \end{bmatrix}
+ =
+ \begin{bmatrix}
+ C_{11} & C_{12}\\
+ C_{21} & C_{22}
+ \end{bmatrix}
+ $
+ }
+
+
+ \onslide<3->{
+
+\medskip
+$
+C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}
+$
+
+$
+C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}
+$
+
+$
+C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}
+$
+
+$
+C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+$
+}
+\end{frame}
+
+\input{slides/algo.tex}
+
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \onslide<1->{
+ \large
+ \begin{math}
+ \begin{aligned}
+ \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+ \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
+ \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+ \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
+ \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
+ \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+ \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ \end{aligned}
+ \end{math}
+ }
+ \end{column}
+
+ \begin{column}{0.5\textwidth}
+ \onslide<2->{
+ \large
+ \begin{math}
+ \begin{aligned}
+ C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+ C_{21} &= \text{II} + \text{IV} \\
+ C_{12} &= \text{III} + \text{V}\\
+ C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+ \end{aligned}
+ \end{math}
+ }
+ \end{column}
+\end{columns}
+
+\onslide<3->{
+
+\bigskip
+\centering
+\tiny
+\begin{math}
+\begin{aligned}
+ C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
+ C_{11} &= A_{11}B_{11} + A_{12}B_{21}
+\end{aligned}
+\end{math}
+}
+
+\end{frame}
+
+
+\begin{frame}
+\begin{adjustbox}{width=\textwidth}
+\begin{tikzpicture}[ampersand replacement=\&]
+
+ \foreach \i in {1,...,4}
+ {
+ \small{
+ \matrix (X\i)[matrix of math nodes,nodes in empty cells,
+ nodes = {draw, minimum size=10mm,
+ anchor=center,
+ inner sep=0pt, outer sep=0pt},
+ column sep=-\pgflinewidth,
+ row sep=-\pgflinewidth,
+ ] at (0,-\i*5)
+ {
+ A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\
+ A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\
+ A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\
+ A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\
+ };}
+
+ \foreach \j in {1,...,7}
+ {
+ \matrix(M\i\j)[matrix of math nodes,nodes in empty cells,
+ nodes = {draw, minimum size=10mm,
+ anchor=center,
+ inner sep=0pt, outer sep=0pt},
+ column sep=-\pgflinewidth,
+ row sep=-\pgflinewidth,
+ ] at (\j*5,-\i*5)
+ {
+ \& \& \& \\
+ \& \& \& \\
+ \& \& \& \\
+ \& \& \& \\
+ };
+ }
+ }
+
+\huge{
+ \node at (-3,-20) {$C_{22}=$};
+ \node at (-3,-15) {$C_{21}=$} ;
+ \node at (-3,-10) {$C_{12}=$} ;
+ \node at (-3,-5) {$C_{11}=$} ;
+
+ \node at (5,-2) {I};
+ \node at (10,-2) {II};
+ \node at (15,-2) {III};
+ \node at (20,-2) {IV};
+ \node at (25,-2) {V};
+ \node at (30,-2) {VI};
+ \node at (35,-2) {VII};
+ }
+
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+\end{adjustbox}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+ \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
+ \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+ \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
+ \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
+ \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+ \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ \end{aligned}
+ \end{math}
+
+ \end{column}
+
+ \begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+ C_{21} &= \text{II} + \text{IV} \\
+ C_{12} &= \text{III} + \text{V}\\
+ C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+ \end{aligned}
+ \end{math}
+
+ \end{column}
+\end{columns}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+
+\begin{columns}
+ \begin{column}{0.5\textwidth}
+\large
+\begin{math}
+\begin{aligned}
+\text{\textbf{I}} &= (\mathbf{A_{11}} + \mathbf{A_{22}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{22}}) \\
+\text{\textbf{II}} &= (\mathbf{A_{21}} + \mathbf{A_{22}}) \cdot \mathbf{B_{11}} \\
+\text{\textbf{III}} &= \mathbf{A_{11}} \cdot (\mathbf{B_{12}}-\mathbf{B_{22}}) \\
+\text{\textbf{IV}} &= \mathbf{A_{22}} \cdot (-\mathbf{B_{11}}+\mathbf{B_{21}}) \\
+\text{\textbf{V}} &= (\mathbf{A_{11}} + \mathbf{A_{12}}) \cdot \mathbf{B_{22}} \\
+\text{\textbf{VI}} &= (-\mathbf{A_{11}} + \mathbf{A_{21}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{12}}) \\
+\text{\textbf{VII}} &= (\mathbf{A_{12}} - \mathbf{A_{22}}) \cdot (\mathbf{B_{21}} + \mathbf{B_{22}}) \\
+\end{aligned}
+\end{math}
+
+\end{column}
+
+\begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ \mathbf{C_{11}} &= \text{\textbf{I}} + \text{\textbf{IV}} - \text{\textbf{V}} + \text{\textbf{VII}} \\
+ \mathbf{C_{21}} &= \text{\textbf{II}} + \text{\textbf{IV}} \\
+ \mathbf{C_{12}} &= \text{\textbf{III}} + \text{\textbf{V}}\\
+ \mathbf{C_{22}} &= \text{\textbf{I}} + \text{\textbf{III}} - \text{\textbf{II}} + \text{\textbf{VI}} \\
+ \end{aligned}
+ \end{math}
+
+\end{column}
+\end{columns}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Algorithm}
+ \onslide<1->{
+
+ \scalebox{0.45}{\parbox{\linewidth}{
+ \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{strassen}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros((n, n))$
+ \State $P \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$
+ \State $Q \gets (A[1][0]+A[1][1])\cdot B[0][0]$
+ \State $R \gets A[0][0]\cdot (B[0][1]-B[1][1])$
+ \State $S \gets A[1][1]\cdot (B[1][0]-B[0][0])$
+ \State $T \gets (A[0][0]+A[0][1])\cdot B[1][1]$
+ \State $U \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$
+ \State $V \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$
+ \State $C[0][0] \gets P+S-T+V$
+ \State $C[0][1] \gets R+T$
+ \State $C[1][0] \gets Q+S$
+ \State $C[1][1] \gets P+R-Q+U$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$
+ \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$
+ \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}- \mathbf{B22}),m)$
+ \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}- \mathbf{B11}),m)$
+ \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$
+ \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$
+ \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$
+
+
+
+ \State $\mathbf{C11} \gets \mathbf{P+S-T+V}$
+ \State $\mathbf{C12} \gets \mathbf{R+T}$
+ \State $\mathbf{C21} \gets \mathbf{Q+S}$
+ \State $\mathbf{C22} \gets \mathbf{P+R-Q+U}$
+ \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+ }}}
+% \[
+% \mathcal{T}(n) = \left\{\begin{array}{lr}
+% 1, & \text{if} n \leq 2\\
+% 7 \mathcal{T}(\frac{n}{2}) + n^2, & \text{if} n > 2\\
+% \end{array}\right\}
+% \]
+\only<2>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{\log_2 7})$
+
+}
+\only<3>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{2.81})$
+
+}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Algorithm}
+ \onslide<1->{
+
+ \scalebox{0.45}{\parbox{\linewidth}{
+ \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros((n, n))$
+ \State $C[0, 0] \gets A[0][0]*B[0][0]+A[0][1]*B[1][0]$
+ \State $C[0, 1] \gets A[0][0]*B[0][1]+A[0][1]*B[1][1]$
+ \State $C[1, 0] \gets A[1][0]*B[0][0]+A[1][1]*B[1][0]$
+ \State $C[1, 1] \gets A[1][0]*B[0][1]+A[1][1]*B[1][1]$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11}) + \text{MM}(\mathbf{A12}, \mathbf{B21})$
+ \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12}) + \text{MM}(\mathbf{A12},\mathbf{B22})$
+ \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11}) + \text{MM}(\mathbf{A22}, \mathbf{B21})$
+ \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12}) + \text{MM}(\mathbf{A22}, \mathbf{B22})$
+ \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+ \bigskip
+ \bigskip
+ \bigskip
+ \bigskip
+ \bigskip
+ }}}
+
+\only<2>{
+
+
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{\log_2 8})$
+
+}
+\only<3>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{3})$
+
+}
+
+\end{frame}
generated by cgit v1.2.1 (git 2.18.0) at 2024-07-02 20:19:41 +0200