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%
% Current maintainer: Rogério Brito <rbrito@ime.usp.br>
% Previous maintainer: Peter Williams <pwil3058@bigpond.net.au>
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\documentclass[letterpaper]{article}

\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[pdfstartview={FitH}]{hyperref}
\usepackage{algorithm}
\usepackage{algorithmic}
\usepackage{amsthm}
\usepackage{multicol}

\title{Algorithms}
\author{Rogério Brito\\
  \href{mailto:rbrito@ime.usp.br}{rbrito@ime.usp.br}\thanks{Sincere thanks
    go to the original maintainer of this package, Peter Williams, and for
    being kind enough to allow me to continue with his quite useful work.}}

\newcommand{\keyword}[1]{\texttt{#1}}
\newcommand{\filename}[1]{\texttt{#1}}

\setcounter{tocdepth}{2}

\begin{document}
\addtocontents{toc}{\protect\begin{multicols}{2}}
\maketitle
\tableofcontents
%\listofalgorithms

\section{Introduction}

This package provides two environments, \keyword{algorithmic} and
\keyword{algorithm}, which are designed to be used together but may,
depending on the necessities of the author, 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.

This work may be distributed and/or modified under the conditions of the
GNU Lesser General Public License as published by the Free Software
Foundation (see the file \filename{COPYING} included in this package).
Currently, this package consists of three files:
\filename{algorithm.sty}, \filename{algorithmic.sty} and
\filename{algorithms.tex} (the source of this document). This is likely
to change in the near future.

\section[Environment: \keyword{algorithmic}]%
{The \keyword{algorithmic} Environment}
\label{sec:algorithmic-envir}
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 \emph{if-then-else} Statement}
\subsection{The \emph{if-then-else} Construct}

The \emph{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 \emph{for} Loop}

The \emph{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 \emph{while} Loop}

The \emph{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 \emph{repeat-until} Loop}

The \emph{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{Returning Values}

The \keyword{algorithmic} environment offers a special statement for
explicitly returning values in algorithms. It has the syntax:
\begin{verbatim}
\RETURN <text>
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\RETURN $(x+y)/2$
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
  \RETURN $(x+y)/2$
\end{algorithmic}


\subsubsection{The ``true'' and ``false'' Values}

Since many algorithms have the necessity of returning \emph{true} or
\emph{false} values, \keyword{algorithms}, starting with version
2006-06-02, includes the keywords \verb+\TRUE+ and \verb+\FALSE+,
which are intented to print the values in a standard fashion, like the
following snippet of an algorithm to decide if an integer $n$ is even or
odd:
\begin{verbatim}
\begin{algorithmic}
  \IF{$n$ is odd}
     \RETURN \TRUE
  \ELSE
     \RETURN \FALSE
  \ENDIF
\end{algorithmic}
\end{verbatim}
The code above produces the following output:
\begin{algorithmic}
  \IF{$n$ is odd}
     \RETURN \TRUE
  \ELSE
     \RETURN \FALSE
  \ENDIF
\end{algorithmic}

\subsection{Printing Messages}

Another feature of the \keyword{algorithmic} environment is that it
currently provides a standard way of printing values (which is an
operation used enough to merit its own keyword). It has the syntax:
\begin{verbatim}
\PRINT <text>
\end{verbatim}
For example,
\begin{verbatim}
\begin{algorithmic}
\PRINT \texttt{``Hello, World!''}
\end{algorithmic}
\end{verbatim}
produces
\begin{algorithmic}
  \PRINT \texttt{``Hello, World!''}
\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 \keyword{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/Customization]%
{Options and Customization}

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

\subsubsection{Changing Indentation}
\label{sec:changing-indentation}
In the spirit of saving vertical space (which is especially important
when submitting a paper for a journal, where space is frequently limited
for authors), the \keyword{algorithmic} environment offers, beginning
with the version released in 2005-05-08, a way to control the amount of
indentation that is used by a given algorithm.

The amount of indentation to be used is given by the command
\begin{verbatim}
\algsetup{indent=lenght}
\end{verbatim}
where \emph{length} is any valid length used by \TeX. The default value
of the indentation used by the \keyword{algorithmic} environment is $1$
em (for ``backward compatibility reasons''), but a value of $2$ em or
more is recommended, depending on the publication. For example, the
snippet
\begin{verbatim}
\algsetup{indent=2em}
\begin{algorithmic}[1]
  \STATE $a \Leftarrow 1$
  \IF{$a$ is even}
  \PRINT ``$a$ is even''
  \ELSE
  \PRINT ``$a$ is odd''
\end{algorithmic}
\end{verbatim}
produces
\algsetup{indent=2em}
\begin{algorithmic}[1]
  \STATE $a \Leftarrow 1$
  \IF{$a$ is even}
    \PRINT ``$a$ is even''
  \ELSE
    \PRINT ``$a$ is odd''
  \ENDIF
\end{algorithmic}
while
\begin{verbatim}
\algsetup{indent=5em}
\begin{algorithmic}[1]
  \STATE $a \Leftarrow 1$
  \IF{$a$ is even}
  \PRINT ``$a$ is even''
  \ELSE
  \PRINT ``$a$ is odd''
\end{algorithmic}
\end{verbatim}
would produce
\algsetup{indent=5em}
\begin{algorithmic}[1]
  \STATE $a \Leftarrow 1$
  \IF{$a$ is even}
    \PRINT ``$a$ is even''
  \ELSE
    \PRINT ``$a$ is odd''
  \ENDIF
\end{algorithmic}
\algsetup{indent=1em}

The intended use of this option is to allow the author to omit the
\emph{end} (see Section~\ref{kwd:noend} for details) statements without
loosing readability, by increasing the amount of indentation to a
suitable level.

\subsubsection{Changing Line Numbering}

As mentioned in Section~\ref{sec:algorithmic-envir} and illustrated in
Section~\ref{sec:changing-indentation}, \keyword{algorithms} already
provides you with the possibility of numbering lines.

Starting with the version released in 2005-07-05, you can now change two
aspects of line numbering: the size of the line numbers (which, by
default, is \verb+\footnotesize+) and the delimiter used to separate the
line number from the code (which, by default, is \verb+:+, i.e., a
colon).

You can change the size of the line numbers using the command:
\begin{verbatim}
\algsetup{linenosize=size}
\end{verbatim}
where \emph{size} is any of the various commands provided by \LaTeX\ to
change the size of the font to be used. Among others, useful values are
\verb+\tiny+, \verb+\scriptsize+, \verb+\footnotesize+ and
\verb+\small+.  Please see the complete list of sizes in your \LaTeX\
documentation.

As another frequently requested feature, you can change the delimiter
used with the line numbers by issuing the command:
\begin{verbatim}
\algsetup{linenodelimiter=delimiter}
\end{verbatim}
where \emph{delimiter} is any ``well-formed'' string, including the
empty string. With this command, you can change the colon to a period
(\verb+.+) by issuing the command
\begin{verbatim}
\algsetup{linenodelimiter=.}
\end{verbatim}
or even omit the delimiter, by specifying the empty string or a space
(\verb+\ +), whatever seems best for your document.

As an example of such commands, the code produced by
\begin{verbatim}
\algsetup{
   linenosize=\small,
   linenodelimiter=.
}
\begin{algorithmic}[1]
   \STATE $i \leftarrow 10$
   \RETURN $i$ 
\end{algorithmic}
\end{verbatim}
would be something like

\algsetup{
  linenosize=\small,
  linenodelimiter=.
}
\begin{algorithmic}[1]
   \STATE $i \leftarrow 10$
   \RETURN $i$
\end{algorithmic}
\algsetup{linenosize=\footnotesize,
          linenodelimiter=:}

\subsubsection{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}}
\newcommand{\algorithmicprint}{\textbf{print}}
\newcommand{\algorithmicreturn}{\textbf{return}}
\newcommand{\algorithmictrue}{\textbf{true}}
\newcommand{\algorithmicfalse}{\textbf{false}}
\end{verbatim}

If you would like to change the definition of these commands to another
content, then you should use, in your own document, the standard
\LaTeX{} command \keyword{renewcommand}, with an usage like this:
\begin{verbatim}
\renewcommand{\algorithmicrequire}{\textbf{Input:}}
\renewcommand{\algorithmicensure}{\textbf{Output:}}
\end{verbatim}

\paragraph{About the Way Comments Are Formatted}

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}
and another option that may be interesting for users familiar with
C-like languages is to redefine the comments to be
\begin{verbatim}
\renewcommand{\algorithmiccomment}[1]{// #1}
\end{verbatim}
Comments produced this way would be like this:
\renewcommand{\algorithmiccomment}[1]{// #1}
\begin{algorithmic}
  \STATE $i \leftarrow i +1$ \COMMENT{Increments $i$}
\end{algorithmic}
This second way to present comments may become the default in a future
version of this package.

\section[Environment: \keyword{algorithm}]%
{The \keyword{algorithm} Environment}

\subsection{General}

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.\footnote{This is the
  expected behaviour for floats in \LaTeX. If you don't care about
  having your algorithm split between pages, then one option that you
  have is to ignore the \texttt{algorithm} environment.} 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.

\subsection{An Example}
To illustrate the use of the \texttt{algorithm} environment, the
following 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.

\begin{algorithm}[H]
  \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}

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}

\subsubsection{Placement of Algorithms}

One important fact that many users may not have noticed is that the
\texttt{algorithm} environment is actually built with the \texttt{float}
package and \texttt{float}, in turn, uses David Carlisle's \textsf{here}
style option. This means that the floats generated by the
\texttt{algorithm} environment accept a special option, namely,
\textbf{[H]}, with a capital `H', instead of the usual `h' offered by
plain \LaTeX.

This option works as a stronger request of ``please put the float
here'': instead of just a suggestion for \LaTeX, it actually means ``put
this float HERE'', which is something desired by many. The two
algorithms typeset in this document use this option.

\theoremstyle{definition}
\newtheorem*{warning*}{Warning}

\medskip
\begin{warning*}
  You \emph{can't} use the `H' positioning option together with the
  usual `h' (for ``here''), `b' (for ``bottom'') etc. This is a
  limitation (as far as I know) of the \texttt{float.sty} package.
\end{warning*}

\section[References in Algorithms]%
{Labels and References in Algorithms}

With the release of 2005-07-05, now \keyword{algorithmic} accepts labels
and references to specific lines of a given algorithm, so you don't have
to hardcode the line numbers yourself when trying to explain what the
code does in your texts.  Thanks to Arnaud Legrand for the suggestion
and patch for this highly missed feature.

An example of its use is shown in Algorithm~\ref{alg2}.
\begin{algorithm}[H]
  \caption{Calculate $y = x^n$}
  \label{alg2}
  \begin{algorithmic}[1]
    \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}\label{alg:n-is-even}
    \STATE $X \Leftarrow X \times X$
    \STATE $N \Leftarrow N / 2$
    \ELSE\label{alg:n-is-odd}
    \STATE $y \Leftarrow y \times X$
    \STATE $N \Leftarrow N - 1$
    \ENDIF
    \ENDWHILE
  \end{algorithmic}
\end{algorithm}
See that, in line~\ref{alg:n-is-even}, we deal with the case of $N$
being even, while, in line~\ref{alg:n-is-odd}, we give treatment to the
case of $N$ being odd. The numbers you see on this document were
generated automatically from the source document.


\section[Known Issues]{Issues Between \texttt{algorithms} and
  \href{http://www.ctan.org/tex-archive/help/Catalogue/entries/tocbibind.html}{\texttt{tocbibind}}
  or \href{http://www.ctan.org/tex-archive/help/Catalogue/entries/memoir.html}{\texttt{memoir}}}

It
\href{http://groups.google.com/group/comp.text.tex/browse_thread/thread/4094e0c4f4fbd83e/a80a3f4666c794f0?fwc=1}{has
  been discussed} in late 2005 that \texttt{algorithms} may have bad
interactions with the
\href{http://www.ctan.org/tex-archive/help/Catalogue/entries/tocbibind.html}{\texttt{tocbibind}}
or the
\href{http://www.ctan.org/tex-archive/help/Catalogue/entries/memoir.html}{\texttt{memoir}}
package (which includes \texttt{tocbibind}).

A workaround has been suggested for the problem. After including
something like
\begin{verbatim}
\usepackage[nottoc]{tocbibind}
\end{verbatim}
in the preamble of your document, you can put, after
\verb+\begin{document}+, the following snippet of code:
\begin{verbatim}
\renewcommand{\listofalgorithms}{\begingroup
  \tocfile{List of Algorithms}{loa}
\endgroup}

\makeatletter
\let\l@algorithm\l@figure
\makeatother
\end{verbatim}
which should make the command \verb+\listofalgorithms+ work as expected.

\addtocontents{toc}{\protect\end{multicols}}
\end{document}
