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Add references to the tiling formula

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Antti H S Laaksonen 2017-04-18 20:31:08 +03:00
parent a8e456ce15
commit 180fe097e8
2 changed files with 17 additions and 13 deletions
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7
chapter07.tex
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@ -987,10 +987,9 @@ $2^m$ 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: for calculating the number of tilings\footnote{Surprisingly,
% \footnote{Surprisingly, this formula was discovered in 1961 by two research teams \cite{kas61,tem61}
% this formula was discovered independently that worked 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,

23
list.tex
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@ -40,10 +40,15 @@
\emph{Programming Pearls}. \emph{Programming Pearls}.
Addison-Wesley, 1999 (2nd edition). Addison-Wesley, 1999 (2nd edition).
\bibitem{ben80}
J. Bentley and D. Wood.
An optimal worst case algorithm for reporting intersections of rectangles.
\emph{IEEE Transactions on Computers}, C-29(7):571--577, 1980.
\bibitem{bou01} \bibitem{bou01}
C. L. Bouton. C. L. Bouton.
Nim, a game with a complete mathematical theory. Nim, a game with a complete mathematical theory.
pro \emph{Annals of Mathematics}, 3(1/4):35--39, 1901. \emph{Annals of Mathematics}, 3(1/4):35--39, 1901.
% \bibitem{bur97} % \bibitem{bur97}
% W. Burnside. % W. Burnside.
@ -218,10 +223,10 @@ pro \emph{Annals of Mathematics}, 3(1/4):35--39, 1901.
J. Kleinberg and É. Tardos. J. Kleinberg and É. Tardos.
\emph{Algorithm Design}, Pearson, 2005. \emph{Algorithm Design}, Pearson, 2005.
% \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.
@ -335,10 +340,10 @@ pro \emph{Annals of Mathematics}, 3(1/4):35--39, 1901.
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.