From 227eadd5404a2005fbffc021b198471afa5f5a62 Mon Sep 17 00:00:00 2001 From: Antti H S Laaksonen Date: Tue, 21 Feb 2017 21:59:03 +0200 Subject: [PATCH] References --- luku07.tex | 4 +++- luku20.tex | 4 +--- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/luku07.tex b/luku07.tex index ac14546..13de7b0 100644 --- a/luku07.tex +++ b/luku07.tex @@ -983,7 +983,9 @@ $2^m$ 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 \cite{kas61,tem61}: +for calculating the number of tilings\footnote{Surprisingly, +this formula was discovered independently +by \cite{kas61} and \cite{tem61} in 1961.}: \[ \prod_{a=1}^{\lceil n/2 \rceil} \prod_{b=1}^{\lceil m/2 \rceil} 4 \cdot (\cos^2 \frac{\pi a}{n + 1} + \cos^2 \frac{\pi b}{m+1})\] This formula is very efficient, because it calculates the number of tilings in $O(nm)$ time, diff --git a/luku20.tex b/luku20.tex index 28c8fbb..1e7a8d0 100644 --- a/luku20.tex +++ b/luku20.tex @@ -431,8 +431,6 @@ by using one of the following techniques: \index{Edmonds–Karp algorithm} The \key{Edmonds–Karp algorithm} \cite{edm72} -is a variant of the -Ford–Fulkerson algorithm that chooses each path so that the number of edges on the path is as small as possible. This can be done by using breadth-first search @@ -443,7 +441,7 @@ of the algorithm is $O(m^2 n)$. \index{scaling algorithm} -The \key{scaling algorithm} uses depth-first +The \key{scaling algorithm} \cite{ahu91} uses depth-first search to find paths where each edge weight is at least a threshold value. Initially, the threshold value is