diff --git a/chapter07.tex b/chapter07.tex index e5b16e9..b21943c 100644 --- a/chapter07.tex +++ b/chapter07.tex @@ -987,10 +987,9 @@ $2^m$ distinct rows and the time complexity is $O(n 2^{2m})$. As a final note, there is also a surprising direct formula -for calculating the number of tilings: -% \footnote{Surprisingly, -% this formula was discovered independently -% by \cite{kas61} and \cite{tem61} in 1961.}: +for calculating the number of tilings\footnote{Surprisingly, +this formula was discovered in 1961 by two research teams \cite{kas61,tem61} +that worked independently.}: \[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\] This formula is very efficient, because it calculates the number of tilings in $O(nm)$ time, diff --git a/list.tex b/list.tex index 8ae33c1..c53ea82 100644 --- a/list.tex +++ b/list.tex @@ -40,10 +40,15 @@ \emph{Programming Pearls}. 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} C. L. Bouton. 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} % W. Burnside. @@ -218,10 +223,10 @@ pro \emph{Annals of Mathematics}, 3(1/4):35--39, 1901. J. Kleinberg and É. Tardos. \emph{Algorithm Design}, Pearson, 2005. -% \bibitem{kas61} -% P. W. Kasteleyn. -% The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. -% \emph{Physica}, 27(12):1209--1225, 1961. +\bibitem{kas61} + P. W. Kasteleyn. + The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. + \emph{Physica}, 27(12):1209--1225, 1961. \bibitem{knu982} 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. \emph{25th Annual Symposium on Foundations of Computer Science}, 12--20, 1984. -% \bibitem{tem61} -% H. N. V. Temperley and M. E. Fisher. -% Dimer problem in statistical mechanics -- an exact result. -% \emph{Philosophical Magazine}, 6(68):1061--1063, 1961. +\bibitem{tem61} + H. N. V. Temperley and M. E. Fisher. + Dimer problem in statistical mechanics -- an exact result. + \emph{Philosophical Magazine}, 6(68):1061--1063, 1961. \bibitem{war23} H. C. von Warnsdorf.