References etc.

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Antti H S Laaksonen 2017-02-25 18:21:27 +02:00
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@ -326,7 +326,7 @@ of the algorithm is at least $O(n^2)$.
It is possible to sort an array efficiently It is possible to sort an array efficiently
in $O(n \log n)$ time using algorithms in $O(n \log n)$ time using algorithms
that are not limited to swapping consecutive elements. that are not limited to swapping consecutive elements.
One such algorithm is \key{mergesort}\footnote{According to \cite{knu98}, One such algorithm is \key{mergesort}\footnote{According to \cite{knu983},
mergesort was invented by J. von Neumann in 1945.} mergesort was invented by J. von Neumann in 1945.}
that is based on recursion. that is based on recursion.

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@ -657,7 +657,9 @@ achieves these properties.
\index{Floyd's algorithm} \index{Floyd's algorithm}
\key{Floyd's algorithm} walks forward \key{Floyd's algorithm}\footnote{The idea of the algorithm is mentioned in \cite{knu982}
and attributed to R. W. Floyd; however, it is not known if Floyd was the first
who discovered the algorithm.} walks forward
in the graph using two pointers $a$ and $b$. in the graph using two pointers $a$ and $b$.
Both pointers begin at a node $x$ that Both pointers begin at a node $x$ that
is the starting node of the graph. is the starting node of the graph.

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@ -368,7 +368,10 @@ performs two depth-first searches.
\index{2SAT problem} \index{2SAT problem}
Strongly connectivity is also linked with the Strongly connectivity is also linked with the
\key{2SAT problem} \cite{asp79}. \key{2SAT problem}\footnote{The algorithm presented here was
introduced in \cite{asp79}.
There is also another well-known linear-time algorithm \cite{eve75}
that is based on backtracking.}.
In this problem, we are given a logical formula In this problem, we are given a logical formula
\[ \[
(a_1 \lor b_1) \land (a_2 \lor b_2) \land \cdots \land (a_m \lor b_m), (a_1 \lor b_1) \land (a_2 \lor b_2) \land \cdots \land (a_m \lor b_m),

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@ -676,7 +676,9 @@ nodes in $O(\log n)$ time using this technique.
\subsubsection{Method 2} \subsubsection{Method 2}
Another way to solve the problem is based on Another way to solve the problem is based on
a tree traversal array \cite{ben00}. a tree traversal array\footnote{This lowest common ancestor algorithm is based on \cite{ben00}.
This technique is sometimes called the \index{Euler tour technique}
\key{Euler tour technique} \cite{tar84}.}.
Once again, the idea is to traverse the nodes Once again, the idea is to traverse the nodes
using a depth-first search: using a depth-first search:
@ -719,8 +721,7 @@ However, we use a bit different tree
traversal array than before: traversal array than before:
we add each node to the array \emph{always} we add each node to the array \emph{always}
when the depth-first search walks through the node, when the depth-first search walks through the node,
and not only at the first visit\footnote{A similar technique is sometimes called the and not only at the first visit.
\key{Euler tour technique} \cite{tar84}.}.
Hence, a node that has $k$ children appears $k+1$ times Hence, a node that has $k$ children appears $k+1$ times
in the array and there are a total of $2n-1$ in the array and there are a total of $2n-1$
nodes in the array. nodes in the array.

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@ -54,6 +54,11 @@
Theoretical improvements in algorithmic efficiency for network flow problems. Theoretical improvements in algorithmic efficiency for network flow problems.
\emph{Journal of the ACM}, 19(2):248--264, 1972. \emph{Journal of the ACM}, 19(2):248--264, 1972.
\bibitem{eve75}
S. Even, A. Itai and A. Shamir.
On the complexity of time table and multi-commodity flow problems.
\emph{16th Annual Symposium on Foundations of Computer Science}, 184--193, 1975.
\bibitem{fan94} \bibitem{fan94}
D. Fanding. D. Fanding.
A faster algorithm for shortest-path -- SPFA. A faster algorithm for shortest-path -- SPFA.
@ -141,7 +146,11 @@
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{knu98} \bibitem{knu982}
D. E. Knuth.
\emph{The Art of Computer Programming. Volume 2: Seminumerical Algorithms}, AddisonWesley, 1998 (3rd edition).
\bibitem{knu983}
D. E. Knuth. D. E. Knuth.
\emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, AddisonWesley, 1998 (2nd edition). \emph{The Art of Computer Programming. Volume 3: Sorting and Searching}, AddisonWesley, 1998 (2nd edition).