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Small fixes

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Antti H S Laaksonen 2017-02-26 11:21:04 +02:00
parent bc02261210
commit 9f5bef4b7e
4 changed files with 18 additions and 15 deletions
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4
chapter01.tex
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@ -778,7 +778,9 @@ n! & = & n \cdot (n-1)! \\
\index{Fibonacci number} \index{Fibonacci number}
The \key{Fibonacci numbers}\footnote{Fibonacci (c. 1175--1250) was an Italian mathematician.} arise in many situations. The \key{Fibonacci numbers}
%\footnote{Fibonacci (c. 1175--1250) was an Italian mathematician.}
arise in many situations.
They can be defined recursively as follows: They can be defined recursively as follows:
\[ \[
\begin{array}{lcl} \begin{array}{lcl}

6
chapter02.tex
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@ -282,7 +282,7 @@ no polynomial algorithm is known, i.e.,
nobody knows how to solve them efficiently. nobody knows how to solve them efficiently.
\key{NP-hard} problems are an important set \key{NP-hard} problems are an important set
of problems for which no polynomial algorithm of problems for which no polynomial algorithm
is known\footnote{A classic book on this topic is is known\footnote{A classic book on the topic is
M. R. Garey's and D. S. Johnson's M. R. Garey's and D. S. Johnson's
\emph{Computers and Intractability: A Guide to the Theory \emph{Computers and Intractability: A Guide to the Theory
of NP-Completeness} \cite{gar79}.}. of NP-Completeness} \cite{gar79}.}.
@ -357,7 +357,7 @@ time and even in $O(n)$ time.
Given an array of $n$ integers $x_1,x_2,\ldots,x_n$, Given an array of $n$ integers $x_1,x_2,\ldots,x_n$,
our task is to find the our task is to find the
\key{maximum subarray sum}\footnote{J. Bentley's \key{maximum subarray sum}\footnote{J. Bentley's
book \emph{Programming Pearls} \cite{ben86} made this problem popular.}, i.e., book \emph{Programming Pearls} \cite{ben86} made the problem popular.}, i.e.,
the largest possible sum of numbers the largest possible sum of numbers
in a contiguous region in the array. in a contiguous region in the array.
The problem is interesting when there may be The problem is interesting when there may be
@ -474,7 +474,7 @@ After this change, the time complexity is $O(n^2)$.
\subsubsection{Algorithm 3} \subsubsection{Algorithm 3}
Surprisingly, it is possible to solve the problem Surprisingly, it is possible to solve the problem
in $O(n)$ time\footnote{In \cite{ben86}, this linear algorithm in $O(n)$ time\footnote{In \cite{ben86}, this linear-time algorithm
is attributed to J. B. Kadene, and the algorithm is sometimes is attributed to J. B. Kadene, and the algorithm is sometimes
called \index{Kadene's algorithm} \key{Kadene's algorithm}.}, which means that we can remove called \index{Kadene's algorithm} \key{Kadene's algorithm}.}, which means that we can remove
one more loop. one more loop.

7
chapter07.tex
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@ -984,9 +984,10 @@ $2^m$ distinct rows and the time complexity is
$O(n 2^{2m})$. $O(n 2^{2m})$.
As a final note, there is also a surprising direct formula As a final note, there is also a surprising direct formula
for calculating the number of tilings\footnote{Surprisingly, for calculating the number of tilings:
this formula was discovered independently % \footnote{Surprisingly,
by \cite{kas61} and \cite{tem61} in 1961.}: % this formula was discovered independently
% by \cite{kas61} and \cite{tem61} in 1961.}:
\[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\] \[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\]
This formula is very efficient, because it calculates This formula is very efficient, because it calculates
the number of tilings in $O(nm)$ time, the number of tilings in $O(nm)$ time,

16
list.tex
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@ -198,10 +198,10 @@
Efficient randomized pattern-matching algorithms. Efficient randomized pattern-matching algorithms.
\emph{IBM Journal of Research and Development}, 31(2):249--260, 1987. \emph{IBM Journal of Research and Development}, 31(2):249--260, 1987.
\bibitem{kas61} % \bibitem{kas61}
P. W. Kasteleyn. % P. W. Kasteleyn.
The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. % The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice.
\emph{Physica}, 27(12):1209--1225, 1961. % \emph{Physica}, 27(12):1209--1225, 1961.
\bibitem{knu982} \bibitem{knu982}
D. E. Knuth. D. E. Knuth.
@ -296,10 +296,10 @@
Finding biconnected componemts and computing tree functions in logarithmic parallel time. Finding biconnected componemts and computing tree functions in logarithmic parallel time.
\emph{25th Annual Symposium on Foundations of Computer Science}, 12--20, 1984. \emph{25th Annual Symposium on Foundations of Computer Science}, 12--20, 1984.
\bibitem{tem61} % \bibitem{tem61}
H. N. V. Temperley and M. E. Fisher. % H. N. V. Temperley and M. E. Fisher.
Dimer problem in statistical mechanics -- an exact result. % Dimer problem in statistical mechanics -- an exact result.
\emph{Philosophical Magazine}, 6(68):1061--1063, 1961. % \emph{Philosophical Magazine}, 6(68):1061--1063, 1961.
\bibitem{war23} \bibitem{war23}
H. C. von Warnsdorf. H. C. von Warnsdorf.