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Antti H S Laaksonen 2017-02-21 21:59:03 +02:00
parent cdfae6862e
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4
luku07.tex
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@ -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,

4
luku20.tex
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@ -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