From 180fe097e835fb8527a5b9b80884cfa63be1b10d Mon Sep 17 00:00:00 2001
From: Antti H S Laaksonen <ahslaaks@cs.helsinki.fi>
Date: Tue, 18 Apr 2017 20:31:08 +0300
Subject: [PATCH] Add references to the tiling formula

---
 chapter07.tex |  7 +++----
 list.tex      | 23 ++++++++++++++---------
 2 files changed, 17 insertions(+), 13 deletions(-)

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.