Small fixes

chapter23.tex

@@ -245,8 +245,9 @@ two $n \times n$ matrices
 in $O(n^3)$ time.
 There are also more efficient algorithms
 for matrix multiplication\footnote{The first such
-algorithm, with time complexity $O(n^{2.80735})$,
-was published in 1969 \cite{str69}, and
+algorithm was Strassen's algorithm,
+published in 1969 \cite{str69},
+whose time complexity is $O(n^{2.80735})$;
 the best current algorithm
 works in $O(n^{2.37286})$ time \cite{gal14}.},
 but they are mostly of theoretical interest
@@ -749,8 +750,9 @@ $2 \rightarrow 1 \rightarrow 4 \rightarrow 2 \rightarrow 5$.
 \index{Kirchhoff's theorem}
 \index{spanning tree}
 
-\key{Kirchhoff's theorem}\footnote{G. R. Kirchhoff (1824--1887) was
-a German physicist.} provides a way
+\key{Kirchhoff's theorem}
+%\footnote{G. R. Kirchhoff (1824--1887) was a German physicist.}
+provides a way
 to calculate the number of spanning trees
 of a graph as a determinant of a special matrix.
 For example, the graph

chapter24.tex

@@ -359,8 +359,10 @@ The expected value for $X$ in a geometric distribution is
 
 \index{Markov chain}
 
-A \key{Markov chain}\footnote{A. A. Markov (1856--1922)
-was a Russian mathematician.} is a random process
+A \key{Markov chain}
+% \footnote{A. A. Markov (1856--1922)
+% was a Russian mathematician.}
+is a random process
 that consists of states and transitions between them.
 For each state, we know the probabilities
 for moving to other states.
@@ -515,11 +517,13 @@ just to find one element?
 
 It turns out that we can find order statistics
 using a randomized algorithm without sorting the array.
-The algorithm is a Las Vegas algorithm:
+The algorithm, called \key{quickselect}\footnote{In 1961,
+C. A. R. Hoare published two algorithms that
+are efficient on average: \index{quicksort} \index{quickselect}
+\key{quicksort} \cite{hoa61a} for sorting arrays and
+\key{quickselect} \cite{hoa61b} for finding order statistics.}, is a Las Vegas algorithm:
 its running time is usually $O(n)$
-but $O(n^2)$ in the worst case\footnote{C. A. R. Hoare
-discovered both this algorithm, known as \key{quickselect} \cite{hoa61b},
-and a similar sorting algorithm, known as \key{quicksort} \cite{hoa61a}.}.
+but $O(n^2)$ in the worst case.
 
 The algorithm chooses a random element $x$
 in the array, and moves elements smaller than $x$
@@ -563,9 +567,10 @@ but one could hope that verifying the
 answer would by easier than to calculate it from scratch.
 
 It turns out that we can solve the problem
-using a Monte Carlo algorithm whose
-time complexity is only $O(n^2)$\footnote{This algorithm is sometimes
-called \index{Freivalds' algoritm} \key{Freivalds' algorithm} \cite{fre77}.}.
+using a Monte Carlo algorithm\footnote{R. M. Freivalds published
+this algorithm in 1977 \cite{fre77}, and it is sometimes
+called \index{Freivalds' algoritm} \key{Freivalds' algorithm}.} whose
+time complexity is only $O(n^2)$.
 The idea is simple: we choose a random vector
 $X$ of $n$ elements, and calculate the matrices
 $ABX$ and $CX$. If $ABX=CX$, we report that $AB=C$,

chapter25.tex

@@ -249,7 +249,7 @@ It turns out that we can easily classify
 any nim state by calculating
 the \key{nim sum} $x_1 \oplus x_2 \oplus \cdots \oplus x_n$,
 where $\oplus$ is the xor operation\footnote{The optimal strategy
-for nim was published in 1901 by C. L. Bouton \cite{bou01}}.
+for nim was published in 1901 by C. L. Bouton \cite{bou01}.}.
 The states whose nim sum is 0 are losing states,
 and all other states are winning states.
 For example, the nim sum for
@@ -369,7 +369,7 @@ so the nim sum is not 0.
 \index{Sprague–Grundy theorem}
 
 The \key{Sprague–Grundy theorem}\footnote{The theorem was discovered
-independently by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}} generalizes the
+independently by R. Sprague \cite{spr35} and P. M. Grundy \cite{gru39}.} generalizes the
 strategy used in nim to all games that fulfil
 the following requirements: