#112 4.1-1

chapter4/sections/1/1.tex

@@ -1 +1,47 @@
-\workinprogress % TODO
+Observe, that for $n\times n$ matrices $A$ and $B$,
+\[
+    \PARENS{
+        \begin{matrix}
+            A & 0 \\
+            0 & 0
+        \end{matrix}
+    \,}\PARENS{
+        \begin{matrix}
+            B & 0 \\
+            0 & 0
+        \end{matrix}
+    \,} = \PARENS{
+        \begin{matrix}
+            AB & 0 \\
+            0 & 0
+        \end{matrix}
+    \,},
+\]
+where 0 denotes zero matrices.
+In other words, when multiplying the square matrices $A$ and $B$, we can pad them with zeros to obtain $m\times m$ matrices $A'=\PARENS{\begin{matrix}A&0\\0&0\end{matrix}\,}$ and $B'=\PARENS{\begin{matrix}B&0\\0&0\end{matrix}\,}$, and multiply $A'$ by $B'$ instead.
+The submatrix formed by taking the intersection of the first $n$ rows and the first $n$ columns of $A'B'$ is $AB$.
+
+We implement this trick in the generalized version of the divide-and-conquer algorithm for multiplying $n\times n$ matrices.
+First we find $m$\dash the least power of 2 greater than of equal to $n$.
+Next we create the $m\times m$ matrices $A'$ and $B'$, by copying the corresponding entries of the matrices $A$ and $B$ to the upper left $n\times n$ submatrices of $A'$ and $B'$, while setting the remaining entries to 0.
+For a given $n\times n$ matrix $C$, where the matrix product $AB$ will be added, we similarly pad it with zeros to create the $m\times m$ matrix $C'$.
+Then we can call $\proc{Matrix-Multiply-Recursive}(A',B',C',m)$ and copy the relevant entries of the matrix $C'$ to $C$.
+
+Creating each matrix $A'$, $B'$, and $C'$ takes $\Theta(m^2)$ time, and copying the result to the matrix $C$ takes $\Theta(n^2)$ time.
+Let $T(n)$ and $T'(n)$ be the worst-case time to multiply two $n\times n$ matrices using the standard \proc{Matrix-Multiply-Recursive} procedure, and its generalized version, respectively.
+Then, omitting the base case, we get
+\[
+    T'(n) = T(m)+\Theta(m^2)+\Theta(n^2).
+\]
+Note that $n\le m<2n$, so
+\begin{align*}
+    T'(n) &\ge T(n)+\Theta(n^2)+\Theta(n^2) \\
+    &= \Omega(n^3)
+\end{align*}
+and
+\begin{align*}
+    T'(n) &\le T(2n)+\Theta((2n)^2)+\Theta(n^2) \\
+    &= O((2n)^3) \\
+    &= O(n^3).
+\end{align*}
+Hence, $T'(n)=\Theta(n^3)$.

clrs4e-solutions.tex

@@ -343,31 +343,33 @@
 
     \part{Foundations}
 
-    \MakeChapter{The Role of Algorithms in Computing}{
-        Algorithms,
-        Algorithms as a technology}{}{
-        Comparison of running times}
-
-    \MakeChapter{Getting Started}{
-        Insertion sort,
-        Analyzing algorithms,
-        Designing algorithms}{}{
-        Insertion sort on small arrays in merge sort,
-        Correctness of bubblesort,
-        Correctness of Horner's rule,
-        Inversions}
-
-    \MakeChapter{Characterizing Running Times}{
-        {$O$-notation, $\Omega$-notation, and $\Theta$-notation},
-        Asymptotic notation: formal definitions,
-        Standard notations and common functions}{}{
-        Asymptotic behavior of polynomials,
-        Relative asymptotic growths,
-        Ordering by asymptotic growth rates,
-        Asymptotic notation properties,
-        Manipulating asymptotic notation,
-        Variations on $O$ and $\Omega$,
-        Iterated functions}
+%    \MakeChapter{The Role of Algorithms in Computing}{
+%        Algorithms,
+%        Algorithms as a technology}{}{
+%        Comparison of running times}
+%
+%    \MakeChapter{Getting Started}{
+%        Insertion sort,
+%        Analyzing algorithms,
+%        Designing algorithms}{}{
+%        Insertion sort on small arrays in merge sort,
+%        Correctness of bubblesort,
+%        Correctness of Horner's rule,
+%        Inversions}
+%
+%    \MakeChapter{Characterizing Running Times}{
+%        {$O$-notation, $\Omega$-notation, and $\Theta$-notation},
+%        Asymptotic notation: formal definitions,
+%        Standard notations and common functions}{}{
+%        Asymptotic behavior of polynomials,
+%        Relative asymptotic growths,
+%        Ordering by asymptotic growth rates,
+%        Asymptotic notation properties,
+%        Manipulating asymptotic notation,
+%        Variations on $O$ and $\Omega$,
+%        Iterated functions}
+
+    \setcounter{chapter}{3}
 
     \MakeChapter{Divide-and-Conquer}{
         Multiplying square matrices,
@@ -385,37 +387,37 @@
         Chip testing,
         Monge arrays}
 
-    \appendix
-
-    \MakeChapter{Summations}{
-        Summation formulas and properties,
-        Bounding summations}{}{
-        Bounding summations}
-
-    \MakeChapter{Sets, Etc.}{
-        Sets,
-        Relations,
-        Functions,
-        Graphs,
-        Trees}{}{
-        Graph coloring,
-        Friendly graphs,
-        Bisecting trees}
-
-    \MakeChapter{Counting and Probability}{
-        Counting,
-        Probability,
-        Discrete random variables,
-        The geometric and binomial distributions,
-        The tails of the binomial distribution}{5}{
-        The Monty Hall problem,
-        Balls and bins}
-
-    \MakeChapter{Matrices}{
-        Matrices and matrix operations,
-        Basic matrix properties}{}{
-        Vandermonde matrix,
-        Permutations defined by matrix-vector multiplication over GF(2)}
+%    \appendix
+%
+%    \MakeChapter{Summations}{
+%        Summation formulas and properties,
+%        Bounding summations}{}{
+%        Bounding summations}
+%
+%    \MakeChapter{Sets, Etc.}{
+%        Sets,
+%        Relations,
+%        Functions,
+%        Graphs,
+%        Trees}{}{
+%        Graph coloring,
+%        Friendly graphs,
+%        Bisecting trees}
+%
+%    \MakeChapter{Counting and Probability}{
+%        Counting,
+%        Probability,
+%        Discrete random variables,
+%        The geometric and binomial distributions,
+%        The tails of the binomial distribution}{5}{
+%        The Monty Hall problem,
+%        Balls and bins}
+%
+%    \MakeChapter{Matrices}{
+%        Matrices and matrix operations,
+%        Basic matrix properties}{}{
+%        Vandermonde matrix,
+%        Permutations defined by matrix-vector multiplication over GF(2)}
 
     \backmatter