\documentclass{article}
\usepackage{algorithm}
\usepackage{algorithmic}
%\usepackage[dcu]{harvard}
\title{Algorithms}
\author{Peter Williams \\ Peter.Williams@dsto.defence.gov.au}
\date{7 April 1996}
\newcommand{\keyword}[1]{\texttt{#1}}
\newcommand{\latcom}[1]{\texttt{$\backslash$#1}}
\begin{document}
\maketitle
\tableofcontents
\listofalgorithms

\section{Introduction}

This package provides two environments, \keyword{algorithmic} and
\keyword{algorithm}, which are designed to be used together but may be used
separately.
The \keyword{algorithmic} environment provides an environment for describing
algorithms and the \keyword{algorithm} environment provides a ``float''
wrapper for algorithms (implemented using \keyword{algorithmic} or some
other method at the author's option).
The reason that two environments are provided is to allow the author
maximum flexibility.

\section{The \keyword{algorithmic} Environment}

Within an \keyword{algorithmic} a number of commands for typesetting
popular algorithmic constructs are available.
In general, the commands provided can be arbitrarily nested to
describe quite complex algorithms.
An optional argument to the \verb+\begin{algorithmic}+ statement can be
used to turn on line numbering by giving a positive integer indicating the
required frequency of line numbering.
For example, \verb+\begin{algorithmic}[5]+ would cause every fifth line to
be numbered.

\subsection{The Simple Statement}

The simple statement takes the form
\begin{verbatim}
\STATE <text>
\end{verbatim}
and is used for simple statements, e.g.
\begin{verbatim}
\begin{algorithmic}
\STATE $S \leftarrow 0$
\end{algorithmic}
\end{verbatim}
would produce
\begin{algorithmic}
\STATE $S \leftarrow 0$
\end{algorithmic}
and with line numbering selected for every line using
\begin{verbatim}
\begin{algorithmic}[1]
\STATE $S \leftarrow 0$
\end{algorithmic}
\end{verbatim}
would produce
\begin{algorithmic}[1]
\STATE $S \leftarrow 0$
\end{algorithmic}
For users of earlier versions of \keyword{algorithmic} this construct is
a cause of an incompatibility.
In the earlier version, instead of starting simple statements with the
\verb+\STATE+
command, simple statements were entered as free text and terminated with
\verb+\\+ command.
Unfortunately, this simpler method failed to survive the modifications
necessary for statement numbering.
However, the \verb+\\+ command can still be used to force a line break within
a simple statement.

\subsection{The {\em if-then-else} Construct}

The {\em if-then-else} construct takes the forms.
\begin{verbatim}
\IF{<condition>} <text> \ENDIF
\IF{<condition>} <text1> \ELSE <text2> \ENDIF
\IF{<condition1>} <text1> \ELSIF{<condition2>} <text2> \ELSE <text3> \ENDIF
\end{verbatim}
In the third of these forms there is no limit placed on the number
of \verb+\ELSIF{<C>}+ that may be used.
For example,
\begin{verbatim}
\begin{algorithmic}
\IF{some condition is true}
\STATE do some processing
\ELSIF{some other condition is true}
\STATE do some different processing
\ELSIF{some even more bizarre condition is met}
\STATE do something else
\ELSE
\STATE do the default actions
\ENDIF
\end{algorithmic}
\end{verbatim}
would produce
\begin{algorithmic}
\IF{some condition is true}
\STATE do some processing
\ELSIF{some other condition is true}
\STATE do some different processing
\ELSIF{some even more bizarre condition is met}
\STATE do something else
\ELSE
\STATE do the default actions
\ENDIF
\end{algorithmic}
with appropriate indentations.

\subsection{The {\em for} Loop}

The {\em for} loop takes the forms.
\begin{verbatim}
\FOR{<condition>} <text> \ENDFOR
\FORALL{<condition>} <text> \ENDFOR
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\FOR{$i=0$ to $10$}
\STATE carry out some processing 
\ENDFOR
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\FOR{$i=0$ to $10$}
\STATE carry out some processing 
\ENDFOR
\end{algorithmic}
and
\begin{verbatim}
\begin{algorithmic}[1]
\FORALL{$i$ such that $0\leq i\leq 10$}
\STATE carry out some processing 
\ENDFOR
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}[1]
\FORALL{$i$ such that $0\leq i\leq 10$}
\STATE carry out some processing 
\ENDFOR
\end{algorithmic}

\subsection{The {\em while} Loop}

The {\em while} loop takes the form.
\begin{verbatim}
\WHILE{<condition>} <text> \ENDWHILE
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\WHILE{some condition holds}
\STATE carry out some processing 
\ENDWHILE
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\WHILE{some condition holds}
\STATE carry out some processing 
\ENDWHILE
\end{algorithmic}

\subsection{The {\em repeat-until} Loop}

The {\em repeat-until} loop takes the form.
\begin{verbatim}
\REPEAT <text> \UNTIL{<condition>}
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\REPEAT
\STATE carry out some processing 
\UNTIL{some condition is met}
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\REPEAT
\STATE carry out some processing 
\UNTIL{some condition is met}
\end{algorithmic}

\subsection{The Infinite Loop}

The infinite loop takes the form.
\begin{verbatim}
\LOOP <text> \ENDLOOP
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\LOOP
\STATE this processing will be repeated forever
\ENDLOOP
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\LOOP
\STATE this processing will be repeated forever
\ENDLOOP
\end{algorithmic}

\subsection{The Precondition}

The precondition (that must be met if an algorithm is to correctly
execute) takes the form.
\begin{verbatim}
\REQUIRE <text>
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\REQUIRE $x \neq 0$ and $n \geq 0$
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\REQUIRE $x \neq 0$ and $n \geq 0$
\end{algorithmic}

\subsection{The Postcondition}

The postcondition (that must be met after an algorithm has correctly
executed) takes the form.
\begin{verbatim}
\ENSURE <text>
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\ENSURE $x \neq 0$ and $n \geq 0$
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\ENSURE $x \neq 0$ and $n \geq 0$
\end{algorithmic}

\subsection{Comments}

Comments may be inserted at most points in an algorithm using the
form.
\begin{verbatim}
\COMMENT{<text>}
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\STATE do something \COMMENT{this is a comment}
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
\STATE do something \COMMENT{this is a comment}
\end{algorithmic}
Because the mechanisms used to build the various algorithmic structures
make it difficult to use the above mechanism for placing comments at the
end of the first line of a construct, the commands \verb+\IF+,
\verb+\ELSIF+, \verb+\ELSE+, \verb+\WHILE+, \verb+\FOR+,
\verb+\FORALL+, \verb+\REPEAT+
and \verb+\LOOP+ all take an optional argument which will be treated
as a comment to be placed at the end of the line on which they appear.
For example,
\begin{algorithmic}
\REPEAT[this is comment number one]
\IF[this is comment number two]{condition one is met}
\STATE do something
\ELSIF[this is comment number three]{condition two is met}
\STATE do something else
\ELSE[this is comment number four]
\STATE do nothing
\ENDIF
\UNTIL{hell freezes over}
\end{algorithmic}

\subsection{An Example}

The following example demonstrates the use of the \texttt{algorithmic}
environment to describe a complete algorithm.
The following input
\begin{verbatim}
\begin{algorithmic}
\REQUIRE $n \geq 0$
\ENSURE $y = x^n$
\STATE $y \Leftarrow 1$
\STATE $X \Leftarrow x$
\STATE $N \Leftarrow n$
\WHILE{$N \neq 0$}
\IF{$N$ is even}
\STATE $X \Leftarrow X \times X$
\STATE $N \Leftarrow N / 2$
\ELSE[$N$ is odd]
\STATE $y \Leftarrow y \times X$
\STATE $N \Leftarrow N - 1$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{verbatim}
will produce
\begin{algorithmic}
\REQUIRE $n \geq 0$
\ENSURE $y = x^n$
\STATE $y \Leftarrow 1$
\STATE $X \Leftarrow x$
\STATE $N \Leftarrow n$
\WHILE{$N \neq 0$}
\IF{$N$ is even}
\STATE $X \Leftarrow X \times X$
\STATE $N \Leftarrow N / 2$
\ELSE[$N$ is odd]
\STATE $y \Leftarrow y \times X$
\STATE $N \Leftarrow N - 1$
\ENDIF
\ENDWHILE
\end{algorithmic}
which is an algorithm for finding the value of a number taken to a
non-negative power.

\subsection{Options}

There is a single option, \texttt{noend} that may be invoked when
the \texttt{algorithmic} package is loaded.
With this option invoked the {\em end} statements are omitted in the
output.
This allows space to be saved in the output document when this is an issue.

\subsection{Customization}

In order to facilitate the use of this package with foreign languages, all
of the words in the output are produced via redefinable macro commands.
The default definitions of these macros are:
\begin{verbatim}
\newcommand{\algorithmicrequire}{\textbf{Require:}}
\newcommand{\algorithmicensure}{\textbf{Ensure:}}
\newcommand{\algorithmicend}{\textbf{end}}
\newcommand{\algorithmicif}{\textbf{if}}
\newcommand{\algorithmicthen}{\textbf{then}}
\newcommand{\algorithmicelse}{\textbf{else}}
\newcommand{\algorithmicelsif}{\algorithmicelse\ \algorithmicif}
\newcommand{\algorithmicendif}{\algorithmicend\ \algorithmicif}
\newcommand{\algorithmicfor}{\textbf{for}}
\newcommand{\algorithmicforall}{\textbf{for all}}
\newcommand{\algorithmicdo}{\textbf{do}}
\newcommand{\algorithmicendfor}{\algorithmicend\ \algorithmicfor}
\newcommand{\algorithmicwhile}{\textbf{while}}
\newcommand{\algorithmicendwhile}{\algorithmicend\ \algorithmicwhile}
\newcommand{\algorithmicloop}{\textbf{loop}}
\newcommand{\algorithmicendloop}{\algorithmicend\ \algorithmicloop}
\newcommand{\algorithmicrepeat}{\textbf{repeat}}
\newcommand{\algorithmicuntil}{\textbf{until}}
\end{verbatim}

In addition, the formatting of comments is implemented via a single
argument command macro which may also be redefined.
The default definition is
\begin{verbatim}
\newcommand{\algorithmiccomment}[1]{\{#1\}}
\end{verbatim}

\section{The \keyword{algorithm} Environment}

\subsection{General}
\begin{algorithm}
\caption{Calculate $y = x^n$}
\label{alg1}
\begin{algorithmic}
\REQUIRE $n \geq 0 \vee x \neq 0$
\ENSURE $y = x^n$
\STATE $y \Leftarrow 1$
\IF{$n < 0$}
\STATE $X \Leftarrow 1 / x$
\STATE $N \Leftarrow -n$
\ELSE
\STATE $X \Leftarrow x$
\STATE $N \Leftarrow n$
\ENDIF
\WHILE{$N \neq 0$}
\IF{$N$ is even}
\STATE $X \Leftarrow X \times X$
\STATE $N \Leftarrow N / 2$
\ELSE[$N$ is odd]
\STATE $y \Leftarrow y \times X$
\STATE $N \Leftarrow N - 1$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}

When placed within the text without being encapsulated in a floating
environment \texttt{algorithmic} environments may be split over a page
boundary greatly detracting from their appearance.
In addition, it is useful to have algorithms numbered for reference
and for lists of algorithms to be appended to the list of contents.
The \texttt{algorithm} environment is meant to address these concerns
by providing a floating environment for algorithms.
For example, the input text
\begin{verbatim}
\begin{algorithm}
\caption{Calculate $y = x^n$}
\label{alg1}
\begin{algorithmic}
\REQUIRE $n \geq 0 \vee x \neq 0$
\ENSURE $y = x^n$
\STATE $y \Leftarrow 1$
\IF{$n < 0$}
\STATE $X \Leftarrow 1 / x$
\STATE $N \Leftarrow -n$
\ELSE
\STATE $X \Leftarrow x$
\STATE $N \Leftarrow n$
\ENDIF
\WHILE{$N \neq 0$}
\IF{$N$ is even}
\STATE $X \Leftarrow X \times X$
\STATE $N \Leftarrow N / 2$
\ELSE[$N$ is odd]
\STATE $y \Leftarrow y \times X$
\STATE $N \Leftarrow N - 1$
\ENDIF
\ENDWHILE
\end{algorithmic}
\end{algorithm}
\end{verbatim}
produces Algorithm~\ref{alg1} which is a slightly modified version of
the earlier algorithm for determining the value of a number taken to an
integer power.
In this case, provided the power may be negative provided the number is
not zero.

The command \verb+\listofalgorithms+ may be used to produce a list
of algorithms as part of the table contents as shown at the beginning of
this document.
An auxiliary file with a suffix of \texttt{.loa} is produced when this
feature is used.

\subsection{Options}

The appearance of the typeset algorithm may be changed by use of the
options: \texttt{plain}, \texttt{boxed} or \texttt{ruled} during the
loading of the \texttt{algorithm} package.
The default option is \texttt{ruled}.

The numbering of algorithms can be influenced by providing the name of
the document component within which numbering should be recommenced.
The legal values for this option are: \texttt{part}, \texttt{chapter},
\texttt{section}, \texttt{subsection}, \texttt{subsubsection}
or \texttt{nothing}.
The default value is \texttt{nothing} which causes algorithms to be
numbered sequentially throughout the document.

\subsection{Customization}

In order to facilitate the use of this package with foreign languages,
methods have been provided to facilitate the necessary modifications.

The title used in the caption within \texttt{algorithm} environment
can be set by use of the standard \verb+\floatname+ command which is
provided as part of the \texttt{float} package which was used to
implement this package.
For example,
\begin{verbatim}
\floatname{algorithm}{Procedure}
\end{verbatim}
would cause \textbf{Procedure} to be used instead of \textbf{Algorithm}
within the caption of algorithms.

In a manner analogous to that available for the built in floating
environments, the heading used for the list of algorithms may be changed
by redefining the command \verb+listalgorithmname+.
The default definition for this command is
\begin{verbatim}
\newcommand{\listalgorithmname}{List of Algorithms}
\end{verbatim}

\end{document}