% lua-widow-control
% https://github.com/gucci-on-fleek/lua-widow-control
% SPDX-License-Identifier: MPL-2.0+ OR CC-BY-SA-4.0+
% SPDX-FileCopyrightText: 2022 Max Chernoff
\documentclass[final]{ltugboat}
% This is the LaTeX source for the following article:
% @article{tb133chernoff-widows,
% title={Automatically removing widows and orphans with
% \texttt{lua-widow-control}},
% author={Chernoff, Max},
% journal={TUGboat},
% volume={43},
% number={1},
% pages={28--39},
% year={2022},
% month=may,
% DOI={10.47397/tb/43-1/tb133chernoff-widows},
% }
% Please refer to the PDF on tug.org for the authoritative version.
% Compiling:
% context tb133chernoff-widows-figure.ctx
% lualatex tb133chernoff-widows.ltx
% bibtex tb133chernoff-widows
% lualatex tb133chernoff-widows.ltx
% lualatex tb133chernoff-widows.ltx
% The original article was built with the final/frozen TeX Live 2021.
% Set the publication info
\vol 43, 1.
\issyear 2022.
\issueseqno=133
\setcounter{page}{28}
\PrelimDraftfalse
% Load lwc
\usepackage[balanced]{lua-widow-control}
% Table Stuff
\usepackage{tabularx}
\usepackage{hhline}
\usepackage{booktabs}
\AddToHook{env/tabularx/before}{\smallskip\noindent}
\AddToHook{env/tabularx/after}{\smallskip}
\AddToHook{env/tabular/before}{\smallskip\noindent}
\AddToHook{env/tabular/after}{\smallskip}
\renewcommand{\arraystretch}{1.15}
\usepackage{graphicx}
% Let the macro names in section headings be in boldface
\usepackage{lmodern}
\DeclareRobustCommand{\cs}[1]{\texttt{\textbackslash#1}}
\makeatletter
\DeclareRobustCommand{\eTeX}{%
\ifx\f@series\bfseries@rm%
\ensuremath{\boldsymbol{\varepsilon}}\mbox{-}\kern-.125em\TeX%
\else%
\ensuremath{\varepsilon}\mbox{-}\kern-.125em\TeX%
\fi%
}
\let\goodeTeX=\eTeX
\makeatother
\usepackage{mathtools}
% Abbreviations
% Most of these are just typewriter commands with `\allowbreak`s added.
\def\lwc/{\textsf{lua-\allowbreak widow-\allowbreak control}}
\def\Lwc/{\textsf{lua-\allowbreak widow-\allowbreak control}}
\def\estretch/{%
\texorpdfstring{\cs{emergency}\-\mbox{\ttfamily stretch}}{\textbackslash{}emergencystretch}%
}
\def\openalty/{\cs{output}\-\mbox{\ttfamily penalty}}
\def\waos/{widows and orphans}
\def\wao/{widow and orphan}
\def\woo/{widow or orphan}
\def\woos/{widow or orphans}
\def\latexuse/{%
\cs{use\-package\{lua-\allowbreak widow-\allowbreak control\}}
}
\def\lsness/{\texorpdfstring{%
\cs{loose}\-\mbox{\ttfamily ness}}{\textbackslash{}looseness}%
}
\def\plainop/{Plain~\TeX\slash\OpTeX{}}
\newcommand{\LuaMetaTeX}{Lua\-Meta\-\TeX{}}
\newcommand{\q}[1]{\texorpdfstring{``#1''}{“#1”}}
\def\inlineurl[#1]#2{\href{https://#1}{#2}\footnote{\raggedright\tbsurl{#1}}}
\def\longs/{\char"017F}
\def\endofline#1{\unskip\nobreak\hskip\fontdimen2\font plus 1fill\hbox{#1}}
% Additional macros
\def\dots{\ensuremath{\mathellipsis}}
\def\ttbs{{\tt\char`\\}}
\ifdefined\tubsentencespace\else
\def\tubsentencespace{\spacefactor=3000{}\space\ignorespaces}
\fi
% Figures
\makeatletter
\renewcommand*{\fps@figure}{tb}
\renewcommand*{\fps@table}{tb}
\g@addto@macro\@floatboxreset\centering
\makeatother
% pgfplots
\usepackage{pgfplots}
\usepackage{pgfplotstable}
\usetikzlibrary{patterns}
\pgfkeys{
/pgf/number format/.cd,
sci generic={%
mantissa sep={\times},
exponent={10^{##1}}
},
1000 sep={\,},
}
\pgfplotsset{
compat=1.18,
lua backend=true,
unbounded coords=discard,
filter discard warning=false,
}
\pgfplotstableread{\jobname-plot.dat}{\plotdata}
% Metadata
\title{Automatically removing \waos/ with \lwc/}
\author{Max Chernoff}
\address{Calgary, Alberta\\Canada}
% Load last
\AddToHookNext{shipout/foreground}{%
\put(1in, \dimexpr-2\baselineskip){%
\fbox{\parbox[t]{\textwidth}{%
This document is a part of the \lwc/ documentation. For the
authoritative version of the article, please see
\tbsurl{https://tug.org/TUGboat/tb43-1/tb133chernoff-widows.html}.
}%
}}%
\put(\dimexpr1in + 0.4\textwidth, \dimexpr-\paperheight + 4\baselineskip){%
\fbox{\parbox[t]{0.6\textwidth}{%
Some commands may have changed since publication. Please see the
\lwc/ manual for the current syntax. (Links:
\href{lua-widow-control.pdf}{local},
\href{http://mirrors.ctan.org/macros/luatex/generic/lua-widow-control/lua-widow-control.pdf}{\acro{CTAN}},
\href{https://github.com/gucci-on-fleek/lua-widow-control/releases/latest/download/lua-widow-control.pdf}{GitHub}.)
}%
}}%
}
\usepackage[hidelinks,pdfa]{hyperref}
\usepackage{hyperxmp}
\hypersetup{
pdfdisplaydoctitle=true,
pdftitle={Automatically removing widows and orphans with lua-widow-control},
pdfauthor={Max Chernoff},
pdflang={en},
pdfcontacturl={https://github.com/gucci-on-fleek/lua-widow-control},
pdfcopyright={SPDX-License-Identifier: MPL-2.0+ OR CC-BY-SA-4.0+},
pdflicenseurl={https://creativecommons.org/licenses/by-sa/4.0/},
pdfdoi={10.47397/tb/43-1/tb133chernoff-widows},
pdfpublication={TUGboat},
pdfpubtype={journal},
pdfvolumenum={43},
pdfissuenum={1},
pdfpagerange={28-39},
pdfdate={2022-05}
}
% TODO: Temporary fix
\def\Thanh{H{\`a}n Th\^e\llap{\raise0.5ex\hbox{\'{\relax}}} Th{\`a}nh}
\begin{document}
\maketitle
\begin{abstract}
The \textsf{lua-widow-control} package, for
plain~Lua\TeX\slash{}\LuaLaTeX\slash{}\ConTeXt\slash{}\OpTeX{},
removes widows and orphans without any user intervention.
Using the power of Lua\TeX{}, it does so \emph{without} stretching any glue
or shortening any pages or columns. Instead, \textsf{lua-widow-control}
automatically lengthens a paragraph on a page or column where a widow or
orphan would otherwise occur.
To use \textsf{lua-widow-control}, all that most users need do is
place \verb|\usepackage{lua-widow-control}| in their preamble. No further
changes are required.
\end{abstract}
\section{Motivation}
\TeX{} provides top-notch typesetting: even 40 years after its first
release, no other program produces higher quality mathematical
typesetting, and its paragraph-breaking algorithm is still
state-of-the-art. However, its page breaking is not quite as sophisticated
as its paragraph breaking and thus suffers from some minor issues.
Unmodified \TeX{} has only two familiar ways of dealing with \waos/: it can
either shorten a page by one line, or it can stretch vertical
whitespace. \TeX{} was designed for mathematical and scientific typesetting,
where a typical page has multiple section headings, tables, figures, and
equations. For this style of document, \TeX's default behaviour works quite
well, since the slight stretching of whitespace between the various document
elements is nearly imperceptible; however, for prose or other documents
composed almost entirely of paragraphs, there is little vertical whitespace
to stretch.
Since no ready-made and fully-automated solution to remove
\waos/ from all types of documents was available, I decided to
create \lwc/.
\section{What are \waos/?}
\subsection{Widows}
A \q{widow} occurs when the majority of a paragraph is on one page
or column,
but the last line is on the following page or column. It not only looks
quite odd for a lone line to be at the start of the page, but it makes a
paragraph harder to read since the separation of a paragraph and
its last line disconnects the two, causing the reader to lose context for
the widowed line.
\subsection{Orphans}
An \q{orphan} occurs when the first line of a paragraph is at the end
of the page or column preceding the remainder of the paragraph. They are not
as distracting for the reader, but they are still not ideal.
Visually, \waos/ are about equally disruptive; however, orphans tend not to
decrease the legibility of a text as much as widows, so many authors choose
to ignore them.
See figure~\ref{tab:widow} for a visual reference.
\begin{figure}
\def\firstpage#1{%
\parfillskip=0pt%
\spaceskip=0.2em plus 1fill%
\hskip 3em%
#1%
}
\def\lastpage#1{%
\parfillskip=3em%
\spaceskip=0.2em plus 1fill%
#1%
}
\renewcommand{\arraystretch}{1}
\renewcommand{\doublerulesep}{0.5em}
\begin{tabularx}{\linewidth}{|X|@{\hskip\doublerulesep}|X|}
\multicolumn1c{\bfseries Widow} &
\multicolumn1c{\bfseries Orphan}
\\ \hhline{-||-}
\firstpage{A widow occurs when the last line of a paragraph is
placed on a page separate from}
& \vskip2.25\baselineskip\relax\firstpage{An orphan is}
\\ \hhline{-||-}
\lastpage{where it begins.}
& \lastpage{when the first line of a paragraph occurs on the
page before all of the other lines.} \\ \hhline{-||-}
\end{tabularx}
\caption{The difference between \waos/. If we imagine that each box is a
different page, then this roughly simulates how \waos/ appear.}
\label{tab:widow}
\end{figure}
\subsection{Broken hyphens}
\q{Broken} hyphens occur whenever a page break occurs in a
hyphenated word. These are not related to \waos/; however,
breaking a word across two pages is at least as disruptive for the reader
as \waos/. \TeX{} identifies broken hyphens in the same ways as \waos/, so
\lwc/ treats broken hyphens in the same way.
\section{History and etymology}
The concept of \waos/ is nearly as old as printing itself. In \cite{old},
a printers manual from 1683, we have:
\begin{quote}
Nor do good \emph{Compo\longs/iters} account it good Workman\longs/hip
to begin a \emph{Page} with a \emph{Break-line}, unle\longs/s it be a
very \longs/hort \emph{Break}, and cannot be gotten in the foregoing
\emph{Page}\,; but if it be a long \emph{Break}, he will let it be the
\emph{Direction-line} of the fore-going \emph{Page}, and \emph{Set} his
\emph{Direction} at the end of it. \endofline{(p.~226)}
\end{quote}
\subsection{Widows}
However, the terms \q{widow} and \q{orphan} are much newer. The earliest
published source that I could find referencing \q{widows} in typography is
\textsl{Webster's New International Dictionary} from~1934. However, no
one \Dash not even the editors of the dictionary~\cite{widowhistory} \Dash
seems to know how it got there. Even then, the definition is somewhat
different than it is now:
\begin{quote}
widow, n.\ c.\ \emph{Print}\@. A short line or single word carried over from
the foot of one column or page to the head of a succeeding column or
page. \endofline{\cite{widowhistory}}
\end{quote}
Contrast this with the modern definition:
\begin{quote}
\emph{Typography}\@. A short line of text (usually one consisting of one
word or part of a word) which falls undesirably at the end of a
paragraph, esp.\ one set at the top of a page or column.
\endofline{\cite{oed-widow}}
\end{quote}
which includes a single lone line of any length.
\subsection{Orphans}
The term \q{orphan} is even more confusing. Its initial usage seems to have
occurred some time after \q{widow}~\cite{widowhistory}, and it is given many
contradictory definitions. Most sources define an orphan as a first line at
the bottom of the page and a widow as the last line at the
top~\cite{elements, widowhistory, widowhistory2, xIsambert:TB31-1-12,
texbook, widows-and-orphans, oed-line, oed-widow}; however, some sources
define these two terms as \emph{exact opposites} of each other, with a widow
as a first line at the bottom of the page and an orphan as the last
line!~\cite{backwards1, widowhistory, backwards3, oed-line,
backwards2}\tubsentencespace
This usage is plain wrong; nevertheless, it is sufficiently common that you
need to be careful when you see the terms \q{widow} and \q{orphan}.
\subsection{Clubs}
\textsl{The \TeX{}book} never refers to \q{orphans} as such; rather, it
refers to them as \q{clubs}. This term is remarkably rare: I could only find
a \emph{single} source published before \textsl{The \TeX{}book} \Dash a
compilation article about the definition of \q{widow} \Dash that mentions
a \q{club line}:
\begin{quote}
The Dictionary staff informs me that they have no example of the use of
the word widow in the typographical sense.~[\dots]
Mr. Watson of the technical staff says that the Edinburgh printing
houses referred to it as a ``clubline''.
\endofline{\cite[p.~4]{widowhistory}}
\end{quote}\medskip
\begin{quote}
To my knowledge, a `widow', or `widow-line,' is a short line, forming
the end of a paragraph, which is carried over from the foot of a page or
column to the top of the succeeding one.~[\dots]
To my personal knowledge, in typographical parlance in Edinburgh,
Scotland, the `widow' is called a `club-line.'
\endofline{\cite[p.~23]{widowhistory}}
\end{quote}
Both quotes above are from separate authors, and they each define a \q{club}
like we define \q{widow}, not an \q{orphan}. In addition, they both mention
that the term is only used in Scotland. Even the extensive
\acro{OED}\Dash which lists 17~full definitions and
103~subdefinitions for the noun \q{club}\Dash doesn't recognize the
phrase.~\cite{oed-club}
I spent a few hours searching through Google Books and my university library
catalogue, but I could not find a single additional source. If anyone has
any more information on the definition of a \q{club line} or why Knuth chose
to use this archaic Scottish term in \TeX{}, please let me know!
\section{Pagination in \TeX}
Let's move on to looking at how \TeX{} treats these \waos/.
\subsection{Algorithm}
It is tricky to understand how \lwc/ works if you aren't familiar with how
\TeX{} breaks pages and columns. For a full description, you should
consult Chapter~15 of \textit{\TB}~\cite{texbook} (\q{How \TeX{} Makes Lines
into Pages}); however, this goes into much more detail than most users
require, so here is a \emph{very} simplified summary of \TeX{}'s page
breaking algorithm:
\TeX{} fills the page with lines and other objects until the next object
will no longer fit. Once no more objects will fit, \TeX{} will align the
bottom of the last line with the bottom of the page by stretching any
available vertical spaces if (in \LaTeX) \cs{flushbottom} is set;
otherwise, it will
break the page and leave the bottom empty.
However, some objects have penalties attached. Penalties encourage or
discourage page breaks from occurring at specific places. For example,
\LaTeX{} sets a negative penalty before section headings to encourage a
page break there; conversely, it sets a positive penalty after section
headings to discourage breaking.
To reduce \waos/, \TeX{} sets weakly\hyph positive penalties between the
first and second lines of a paragraph to prevent orphans, and between the
penultimate and final lines to prevent widows.
One important note: once \TeX{} begins breaking a page, it never goes
back to modify any content on the page. Page breaking is a localized
algorithm, without any backtracking.
\subsection{Behaviour}
Merely describing the algorithm doesn't allow us to intuitively
understand how
\TeX{} deals with \waos/.
Due to the penalties attached to \waos/, \TeX{} tries to avoid
them. Widows and orphans with small penalties attached \Dash like
\LaTeX's default values of 150 \Dash are only lightly coupled to the rest
of the paragraph, while \waos/ with large penalties \Dash values of
10\,000 or more \Dash are treated as infinitely bad and are thus
unbreakable. Intermediate values behave just as you would expect,
discouraging page breaks proportional to their value.
When \TeX{} goes to break a page, it tries to avoid breaking at a
location with a high penalty. How it does so depends on a few settings:
\subsubsection{\cs{flushbottom} and \cs{normalbottom}}
With the settings \cs{normalbottom} (Plain \TeX{}) or
\cs{flushbottom} (\LaTeX{}), \TeX{} is willing to stretch any glue on the
page by an amount roughly commensurate to the magnitude of the
penalty: for small \cs{clubpenalty} and \cs{widowpenalty} values, \TeX{} will
only slightly stretch the glue on the page before creating a \woo/;
for very large penalties, \TeX{} will stretch the glue by a
near-infinite amount.
This corresponds to the \q{Stretch} column in
Figure~\ref{fig:demo}. It is
the default behaviour of Plain~\TeX{}, and of the standard \LaTeX{} classes
when the \verb|twocolumn| option is given.
\subsubsection{\cs{raggedbottom}}
When \cs{raggedbottom} is set, \TeX{} won't stretch any glue. Instead,
for sufficiently-high \cs{clubpenalty} and \cs{widowpenalty} values, \TeX{} will
shorten the page or column by one~line in order to prevent the \woo/ from
being created.
This corresponds to the \q{Shorten} column in Figure~\ref{fig:demo} and is
the default behaviour of the \LaTeX{} classes when the
\verb|twocolumn| option is not given.
\section{\lsness/}\label{sec:looseness}
Before we can continue further, we need to discuss one more \TeX{}
command: \lsness/. The following is excerpted from Chapter~14 of
\cite{texbook} (\q{How \TeX{} Breaks Paragraphs into Lines}):
\begin{quote}\parskip=0pt
If you set \lsness/\verb|=1|, \TeX{} will try to make the current
paragraph one line longer than its optimum length, provided that
there is a way to choose such breakpoints without exceeding the
tolerance you have specified for the badnesses of individual lines.
Similarly, if you set \lsness/\verb|=2|, \TeX{} will try to make the
paragraph two lines longer; and \lsness/\verb|=-1| causes an attempt
to make it shorter.~[\dots]
For example, you can set \lsness/\verb|=1| if you want to avoid a
lonely \q{club line} or \q{widow line} on some page that does not
have sufficiently flexible glue, or if you want the total number of
lines in some two-column document to come out to be an even number.
It's usually best to choose a paragraph that is already pretty
\q{full}, i.e., one whose last line doesn't have much white space,
since such paragraphs can generally be loosened without much harm.
You might also want to insert a tie between the last two words of
that paragraph, so that the loosened version will not end with only
one \q{widow word} on the orphans line; this tie will cover your
tracks, so that people will find it hard to detect the fact that you
have tampered with the spacing. On the other hand, \TeX{} can take
almost any sufficiently long paragraph and stretch it a bit, without
substantial harm.
\end{quote}
The \wao/ removal strategy suggested in the second paragraph works quite
well; however, it requires manual editing each and every time a page
or paragraph is rewritten or repositioned.
\begin{figure*}[p]
% Note: this figure may appear incorrect on some systems.
\divide\abovecaptionskip by 2
\includegraphics{\jobname-figure}
\caption{A visual comparison of various automated widow-handling
techniques.
}\label{fig:demo}
\end{figure*}
\section{Alternate removal strategies}
\looseness=1 There have been a few previous attempts to improve upon \TeX's
previously-discussed \wao/-handling abilities; however, none of these
have been able to automatically remove \waos/ without stretching any glue
or shortening any pages.
The articles \q{Strategies against
widows} by Paul Isambert~\cite{xIsambert:TB31-1-12} and
\q{Managing forlorn paragraph lines} by Frank
Mittelbach~\cite{Mittelbach:2018:MFP} both
begin with comprehensive discussions of the methods of preventing \waos/.
They agree that \waos/ are bad and ought to be avoided; however, they
differ in their solutions. \textsl{Strategies}~proposes an output routine
that reduces the length of facing pages by one line when necessary to
remove \waos/, while \textsl{Managing}~proposes that the author manually
rewrites or adjusts \lsness/ when needed.
\looseness=1 The post \q{Paragraph callback \dots} by
jeremie~\cite{widow-assist} contains a file
\verb|widow-assist.lua| that
automatically detects which paragraphs can be safely shortened or
lengthened by one line. Mittelbach's \textsf{widows-and-orphans}
package~\cite{widows-and-orphans} alerts the author to
the pages that contain widows or orphans. Combined, these packages make
it simple for the author to quickly remove \waos/ by adjusting the
values of \lsness/; however, it still requires the author to make manual
source changes after each revision.
Another article by Mittelbach~\cite{global} suggests an fully-automated
solution to remove \waos/. This would seem to offer a complete solution;
however, it requires multiple passes, an external tool, and has not yet been
publicly released.
\pagebreak
\Lwc/ is essentially a combination of
\verb|widow-assist.lua|~\cite{widow-assist} and
\textsf{widows-and-orphans}~\cite{widows-and-orphans} (although its
implementation is independent of both): when the \openalty/
value indicates
that a \woo/ has occurred, Lua is used to find a stretchable paragraph. What
\lwc/ mainly adds on top of these packages is automation: it eliminates the
requirement for any manual adjustments or changes to your document's
source.
\section{Visual comparison}
Although \TeX{}'s page breaking algorithm is reasonably
straightforward, it can lead to
complex behaviour when \waos/ are involved. The usual
choices, when rewriting is not possible, are to ignore them,
stretch some glue, or shorten the
page. Figure~\ref{fig:demo} has a visual comparison of these
options, which we'll discuss in the following:
\subsection{\q{Ignore}}
As you can see, the last line of the page is on a separate page from the
rest of its paragraph, creating a widow. This is usually highly
distracting for the reader, so it is best avoided for the reasons previously
discussed.
\subsection{\q{Shorten}}
This page did not leave any widows, but it did shorten the previous page
by one line. Sometimes this is acceptable, but usually it looks bad because
pages will then have different text-block heights. This can make the pages
look quite uneven, especially when typesetting with columns or in a book
with facing pages.
\subsection{\q{Stretch}}
This page also has no widows and it has a flush bottom margin. However,
the space between each pair of paragraphs had to be stretched.
If this page had many equations, headings, and other elements with
natural space between them, the stretched out space would be much less
noticeable. \TeX{} was designed for mathematical typesetting, so it makes
sense that this is its default behaviour. However, in a page with mostly
text, these paragraph gaps look unsightly.
Also, this method is incompatible with grid typesetting, where
all glue stretching must be quantised to the height of a line.
\subsection{\q{\lwc/}}
\Lwc/ has none of these issues: it eliminates the widows in a document
while keeping a flush bottom margin and constant paragraph spacing.
To do so, \lwc/ lengthened the second paragraph by one line. If you look
closely, you can see that this stretched the interword spaces. This
stretching is noticeable when typesetting in a narrow text block, but
is mostly imperceptible with larger widths.
\Lwc/ automatically finds the \q{best} paragraph to stretch, so the
increase in interword spaces should almost always be minimal.
\section{Installation and standard usage}
The \lwc/ package was first released in
October~2021. It is available in the default installations of both
MiK\TeX{} and \TeX{}~Live, although you will need recent versions
of either.
You may also download \lwc/ manually from either
\inlineurl[ctan.org/pkg/lua-widow-control]{\acro{CTAN},}
the \inlineurl%
[modules.contextgarden.net/cgi-bin/module.cgi/action=view/id=127]%
{\ConTeXt{} Garden,} or \inlineurl%
[github.com/gucci-on-fleek/lua-widow-control/releases/latest/]%
{GitHub,} although it is best if you can install it through your
\TeX~distribution.
As its name may suggest, \lwc/ \textit{requires} \LuaTeX{}\footnote{Or
\LuaMetaTeX{} in the case of \ConTeXt{}.} regardless of the format used.
With that in mind, using \lwc/ is quite simple:
\begin{tabularx}{\linewidth}{@{}l@{}>{\raggedleft\arraybackslash}X@{}}
Plain \TeX{} &
\cs{input lua-widow-control}\phantom{\tt]} \\
\OpTeX {} &
\cs{load[lua-widow-control]} \\
\LaTeX{} &
\latexuse/ \\
\ConTeXt{} &
\cs{usemodule[lua-widow-control]} \\
\end{tabularx}
And that's usually enough. Most users won't need to do anything else since
\lwc/ comes preconfigured and ready-to-go.
\section{Options}
Nevertheless, \lwc/ does have a few options.
\Lwc/ tries very hard to have a \q{natural} user interface with each
format, so how you set an option heavily depends on how you are running
\lwc/. Also note that not every option is available in every format.
Some general guidelines:
\begin{description}
\item[\rm \plainop/\ ] Some options are set
by modifying a register, while others must be set manually using
\cs{directlua}.
\item[\rm \LaTeX{}\ ] Options can be set either as package options
or at any point in the document with \cs{lwcsetup}.
\item[\rm \ConTeXt{}\ ] Always use \cs{setuplwc}.
\end{description}
\subsection{Disabling}
You may want to disable \lwc/ for certain portions of your
document. You can do so with the following commands:
\begin{tabularx}{\linewidth}{@{}Xl@{}}
\plainop/ &
\cs{lwcdisable} \\
\LaTeX{} &
\cs{lwcsetup\{disable\}} \\
\ConTeXt{} &
\cs{setuplwc[state=stop]}\hphantom{\texttt{a}} \\
\end{tabularx}
This prevents \lwc/ from stretching any paragraphs that follow. If a page
has earlier paragraphs where \lwc/ was still enabled and a \woo/ is
detected, \lwc/ will still attempt to remove the \woo/.
\subsection{Enabling}
\Lwc/ is enabled as soon as the package is loaded. If you
have previously disabled it, you will need to re-enable it to save new paragraphs.
\begin{tabularx}{\linewidth}{@{}Xl@{}}
\plainop/ &
\cs{lwcenable} \\
\LaTeX{} &
\cs{lwcsetup\{enable\}} \\
\ConTeXt{} &
\cs{setuplwc[state=start]} \\
\end{tabularx}
\subsection{Automatically disabling}
You may want to disable \lwc/ for certain commands where
stretching is undesirable such as section headings. Of course, manually
disabling and
then enabling \lwc/ multiple times
throughout a document would quickly become tedious, so \lwc/ provides
some options to do this automatically for you.
\Lwc/ automatically patches the default \LaTeX{}, \ConTeXt{},
Plain~\TeX{}, \OpTeX{}, \textsf{\mbox{memoir}}, \KOMAScript, and
\textsf{titlesec} section commands, so you don't need to patch these.
Any others, though, you'll need to patch yourself.
\begin{tabularx}{\linewidth}{@{}Xl@{}}
\leavevmode\rlap{\plainop/} & \texttt{ }%
\cs{lwcdisablecmd\{\meta{\ttbs macro}\}} \\
\LaTeX{} & \texttt{ }%
\cs{lwcsetup\{disablecmds=\{} \\
& \hfill\texttt{\meta{csnameone},
\meta{csnametwo}\}\}} \\
\ConTeXt{} &
\cs{prependtoks\textbackslash{}lwc@patch@pre} \\
& \hfill\cs{to\textbackslash{}everybefore\meta{hook}} \\
& \cs{prependtoks\textbackslash{}lwc@patch@post} \\
& \hfill\cs{to\textbackslash{}everyafter\meta{hook}} \\
\end{tabularx}
\subsection{\estretch/}
\Lwc/ defaults to an \estretch/ value of 3~em for stretched paragraphs,
but you can configure this.
\Lwc/ will only use the \estretch/ when it cannot lengthen a paragraph
in any other way, so it is fairly safe to set this to a large value.
\TeX{} accumulates badness when \estretch/ is
used~\cite{Knuth:TB10-3-325}, so it's pretty rare that a paragraph that
requires any \estretch/ will actually be used on the page.
\begin{tabularx}{\linewidth}{@{}Xl@{}}
\leavevmode\rlap{\plainop/} &
\hskip2em\cs{lwcemergencystretch=} \\
& \hfill\texttt{\meta{dimension}} \\
\LaTeX{} &
\cs{lwcsetup\{emergencystretch=\hphantom{\}}}\\
& \hfill\texttt{\meta{dimension}\}} \\
\ConTeXt{} &
\cs{setuplwc[emergencystretch=\hphantom{]}} \\
& \hfill\texttt{\meta{dimension}]} \\
\end{tabularx}
\subsection{Penalties}
You can also manually adjust the penalties that \TeX{} assigns to \waos/.
Usually, the defaults are fine, but there are a few circumstances where you
may want to change them.
\begin{tabular}{@{}p{.175\linewidth}@{}r@{\texttt{=\meta{integer}}}l@{}}
\leavevmode\rlap{\plainop/} &
\hskip6em\cs{widowpenalty} \\
& \hskip6em\cs{clubpenalty} \\
& \hskip6em\cs{brokenpenalty} \\
\LaTeX{} &
\cs{lwcsetup\{ widowpenalty} & \texttt{\}} \\
& \cs{lwcsetup\{orphanpenalty} & \texttt{\}} \\
& \cs{lwcsetup\{brokenpenalty} & \texttt{\}} \\
\leavevmode\hbox{\ConTeXt{}} &
\cs{setuplwc[ widowpenalty} & \texttt{]} \\
& \cs{setuplwc[orphanpenalty} & \texttt{]} \\
& \cs{setuplwc[brokenpenalty} & \texttt{]} \\
\end{tabular}
The value of these penalties determines how much \TeX{} should attempt
to stretch glue before passing the \woo/ to \lwc/. If you set the values to~1
(default), \TeX{} will stretch nothing and immediately trigger \lwc/;
if you set the values to 10\,000, \TeX{} will stretch infinitely and
\lwc/ will never be triggered. If you set the value to some intermediate
number, \TeX{} will first attempt to stretch some glue to remove the \woo/;
only if it fails will \lwc/ come in and lengthen a paragraph. As a special
case, if you set the values to~0, both \TeX{} and \lwc/ will completely
ignore the \woo/.
\subsection{\cs{nobreak} behaviour}
When \lwc/ encounters an orphan, it removes it by moving the orphaned
line to the next page. The majority of the time, this is an appropriate
solution. However, if the orphan is immediately preceded by a section
heading (or \cs{nobreak}\slash\cs{penalty 10000}), \lwc/ would na\"ively separate a section heading from
the paragraph that follows. This is almost always undesirable, so \lwc/
provides some options to configure this.
\begin{tabularx}{\linewidth}{@{}Xr@{}}
\leavevmode\rlap{\plainop/} &
\cs{directlua\{lwc.}\hskip4em\null \\
& \hfill\texttt{nobreak\_behaviour="\meta{value}"\}} \\
\LaTeX{} &
\cs{lwcsetup\{nobreak=\meta{value}\}\hphantom{"}} \\
\ConTeXt{} &
\cs{setuplwc[nobreak=\meta{value}]\hphantom{"}} \\
\end{tabularx}
The default value, \texttt{keep}, \emph{keep}s the section heading with
the orphan by moving both to the next page. The advantage to this option
is that it removes the orphan and retains any \cs{nobreak}s; the
disadvantage is that moving the section heading can create a large blank
space at the end of the page.
The value \texttt{split} \emph{split}s up the section heading and the
orphan by moving the orphan to the next page while leaving the heading
behind. This is usually a bad idea, but exists for the sake of
flexibility.
The value \texttt{warn} causes \lwc/ to give up on the page and do nothing,
leaving an orphaned line. \Lwc/ \emph{warn}s the user so that they can
manually remove the orphan.
See figure~\ref{tab:nobreak} for a visual reference.
\begin{figure}
\renewcommand{\arraystretch}{1}
\renewcommand{\doublerulesep}{0.5em}
\begin{tabularx}{\linewidth}{%
|X|@{\hskip\doublerulesep}|X|@{\hskip\doublerulesep}|X|%
}
\multicolumn1c{\ttfamily keep} &
\multicolumn1c{\ttfamily split} &
\multicolumn1c{\ttfamily warn}
\\ \hhline{-||-||-}
&
&
\textbf{Heading} \\
&
\textbf{Heading} &
The\hfill first\hfill line
\\ \hhline{-||-||-}
\textbf{Heading} &
The\hfill first\hfill line &
text\hfill text\hfill text \\
The\hfill first\hfill line &
text\hfill text\hfill text &
last line. \\
text\hfill text\hfill text &
last line. &
% Nothing
\\ \hhline{-||-||-}
\end{tabularx}
\caption{A visual comparison of the \texttt{nobreak} option values.}
\label{tab:nobreak}
\end{figure}
\subsection{Maximum cost}
\Lwc/ ranks each paragraph on the page by how much it would \q{cost} to
lengthen that paragraph. By default, \lwc/ selects the paragraph on
the page with the lowest cost; however, you can configure it to only
select paragraphs below a selected cost.
If there aren't any paragraphs below the set threshold, then \lwc/ won't
remove the \woo/ and will instead issue a warning.
\begin{tabularx}{\linewidth}{@{}Xr@{\texttt{=\meta{integer}}}l@{}}
\leavevmode\rlap{\plainop/} &
\cs{lwcmaxcost} \\
\LaTeX{} &
\cs{lwcsetup\{max-cost} & \texttt{\}} \\
\leavevmode\hbox{\ConTeXt{}} &
\cs{setuplwc[maxcost} & \texttt{]} \\
\end{tabularx}
Based on my testing, \texttt{max-cost} values less than 1\,000
cause completely imperceptible changes in interword spacing; values less
than 5\,000 are only noticeable if you are specifically trying to pick out the
expanded paragraph on the page; values less than 15\,000 are typically
acceptable; and larger values may become distracting. \Lwc/ defaults to an
infinite \texttt{max-cost}, although the \q{strict} and \q{balanced} modes
sets the values to~5\,000 and 10\,000 respectively.
\section{Presets}
As you can see, \lwc/ provides quite a few options. Luckily, there are a few
presets that you can use to set multiple options at once. These presets are
a good starting point for most documents, and you can always manually
override individual options.
Currently, these presets are \LaTeX{}-only.
\begin{tabular}{@{}rl@{}}
\LaTeX{} &
\cs{lwcsetup\{\meta{preset}\}} \\
\end{tabular}
\subsection{\texttt{default}}
If you use \lwc/ without any options, it defaults to this preset. In default
mode, \lwc/ takes all possible measures to remove \waos/ and will not
attempt to stretch any vertical glue. This usually
removes~$\mathord{>}\,95\%$ of all
possible \waos/. The catch here is that this mode is quite aggressive, so
it often leaves behind some fairly \q{spacey} paragraphs.
This mode is good if you want to remove (nearly) all \waos/ from your
document, without fine-tuning the results.
\subsection{\texttt{strict}}
\Lwc/ also offers a strict mode. This greatly restricts \lwc/'s tolerance
and makes it so that it will only lengthen paragraphs where the change will
be imperceptible.
The caveat with strict mode is that\Dash depending on the document\Dash
\lwc/ will be able to remove less than a third of the \waos/.
For the \waos/ that can't be automatically removed, a warning will be
printed to your terminal and log file so that a human can manually fix the
situation.
This mode is good if you want the best possible typesetting and are willing
to do some manual editing.
\subsection{\texttt{balanced}}
Balanced mode sits somewhere between default mode and strict mode. This mode
first lets \TeX{} stretch a little glue to remove the \woo/; only if that
fails will it then trigger \lwc/. Even then, the maximum paragraph cost is
capped. Here, \lwc/ can usually remove 90\% of a document's
potential \waos/, and it does so while making a minimal visual impact.
This mode is recommended for most users who care about their document's
typography. This mode is not the default since it doesn't remove all
\waos/: it
still requires a little manual intervention.
\begin{table}
\caption{\Lwc/ options set by each mode.}\label{tab:modes}
\ttfamily\setlength{\tabcolsep}{4pt}
\begin{tabular}{l*3r}\toprule
\textrm{Option} & default & balanced & strict \\ \midrule
max-cost & $\infty$ & 10000 & 5000 \\
\rlap{emergencystretch} & 3em & 1em & 0pt \\
nobreak & keep & keep & warn \\
widowpenalty & 1 & 500 & 1 \\
orphanpenalty & 1 & 500 & 1 \\
brokenpenalty & 1 & 500 & 1 \\
\bottomrule\end{tabular}
\end{table}
\section{Compatibility}
The \lwc/ implementation is almost entirely in Lua, with only a minimal
\TeX{} footprint. It
doesn't modify the output routine, inserts\slash floats, \cs{everypar}, and
it doesn't insert any whatsits. This means that it should be compatible with
nearly any \TeX{} package, class, and format. Most changes that \lwc/ makes
are not observable on the \TeX{} side.
However, on the Lua side, \lwc/ modifies much of a page's internal
structure.
This should not affect any \TeX{} code; however, it may surprise
Lua code that modifies or depends on the page's low-level structure. This
does not matter for Plain~\TeX{} or \LaTeX{}, where even most Lua-based
packages don't depend on the node list structure; nevertheless, there are
a few issues with \ConTeXt{}.
Simple \ConTeXt{} documents tend to be fine, but many advanced
\ConTeXt{} features rely heavily on Lua and can thus be disturbed by
\lwc/. This is not a huge issue\Dash the \lwc/ manual is
written in \ConTeXt{}\Dash but \lwc/ is inevitably more reliable
with Plain \TeX{} and \LaTeX{} than with \ConTeXt{}.
Finally, keep in mind that adding \lwc/ to a document will almost certainly
change its page break locations.
\subsection{Formats}
\Lwc/ runs on all known \LuaTeX{}-based formats: Plain~\LuaTeX{},
\LuaLaTeX{}, \CMkIV{}, \ConTeXt{} Mk\acro{XL}\slash\acro{LMTX},
and~\OpTeX{}. Unless otherwise documented, all features should work
equally well in all formats.
\subsection{Columns}
Since \TeX{} and the formats implement column breaking and page
breaking through the
same internal mechanisms, \lwc/ removes \waos/ between columns just
as it does with \waos/ between pages.
\Lwc/ is known to work with the \LaTeX{} class option \verb|twocolumn|
and the two-column output routine from Chapter~23 of \cite{texbook}.
\subsection{Performance}
\Lwc/ runs entirely in a single pass, without depending on any
\verb|.aux| files or the like. Thus, it shouldn't meaningfully
increase compile times. Although \lwc/ internally breaks each paragraph
twice, modern computers break paragraphs near-instantaneously, so you
are not likely to notice any slowdown.
\subsection{\eTeX{} penalties}
Knuth's original \TeX{} has three basic line penalties:
\cs{interlinepenalty}, which
is inserted between all lines; \cs{club\-penalty}, which is inserted after
the first line; and \cs{widow\-penalty}, which is inserted before the last
line. The \eTeX{} extensions~\cite{etex} generalize these commands with a
syntax similar to \cs{parshape}: with \cs{widow\-penalties} you can set the
penalty between the last, second last, and $n$th last lines of a paragraph;
\cs{inter\-line\-penalties} and \cs{club\-penalties} behave similarly.
\Lwc/ makes no explicit attempts to support these new -\texttt{penalties}
commands. Specifically, if you give a line a penalty that matches either
\cs{widowpenalty} or \cs{clubpenalty}, \lwc/ will treat the lines
exactly as it would a \woo/. So while these commands won't break \lwc/, they
are likely to lead to some unexpected behaviour.
\section{Short last lines}
\looseness=1
When lengthening a paragraph with \lsness/, it is common advice to insert
ties (\verb|~|) between the last few words of the paragraph to avoid
overly-short last lines \cite{texbook}. \Lwc/ does this automatically,
but instead of using ties or \cs{hbox}es, it uses the
\cs{par\allowbreak fill\allowbreak skip}
parameter~\cite{texbook, Wermuth:2018:ECP}. When lengthening a paragraph
(and only when lengthening a paragraph\Dash remember, \lwc/ doesn't
interfere with \TeX{}'s output unless it detects a \woo/), \lwc/ sets
\cs{parfillskip} to \verb|0pt plus 0.8\hsize|.
This normally makes the last line of a paragraph be at least
20\% of the overall paragraph's width, thus preventing
ultra-short~lines.
\section{How it works}
\Lwc/ uses a fairly simple algorithm to eliminate \waos/, but there
are a few subtleties.
\subsection{Setup}
\Lwc/ sets the parameters \cs{clubpenalty}, \cs{widowpenalty}, and
\cs{brokenpenalty} to sentinel values of~1. This will signal to \lwc/ when
a \woo/ occurs, yet it is small enough that it won't stretch any glue.
\Lwc/ also enables \LuaTeX{}'s micro\-typographic
extensions~\cite{xThanh:2000:MTE}. This isn't strictly necessary;
however, it significantly increases the number of paragraphs that can
be acceptably \q{loosened}.
That is all that happens on the \TeX{} end. The rest of \lwc/ is pure Lua.
\subsection{Paragraph breaking}
First, \lwc/ hooks into the paragraph breaking process, before any output
routines or page breaking.
Before a paragraph is broken by \TeX{}, \lwc/ grabs the unbroken
paragraph. Then \lwc/ breaks the paragraph one line longer than its natural
length and stores it for later. It does this in the background,
\emph{without} interfering with how \TeX{} breaks paragraphs into their
natural length.
After \TeX{} has broken its paragraph into its natural length, \lwc/
appears again. Before the broken paragraph is added to the main
vertical list, \lwc/ \q{tags} the first and last nodes of the paragraph
using a \LuaTeX{} attribute. These attributes associate the
previously-saved lengthened paragraph with the naturally-typeset
paragraph on the page.
\subsection{Page breaking}
\Lwc/ intercepts \cs{box255} (the \cs{vbox} output by \TeX) immediately
before the output routine runs,
after all the paragraphs have been typeset.
First, \lwc/ looks at the \openalty/ of the page or column. If the page
was broken at a \woo/, the \openalty/ will be equal to either
the \cs{widowpenalty} or the \cs{clubpenalty}. If the \openalty/ does not
indicate a \woo/, \lwc/ will stop and return \cs{box255} unmodified to
the output~routine, and \TeX{} continues as normal.
Otherwise, we assume that we have a \woo/ on the page,
meaning that we should lengthen the page by 1~line. We iterate through
the list of saved paragraphs to find the lengthened paragraph with the
least cost. After we've selected a good paragraph, we traverse
through the page to find the original version of this paragraph\Dash the
one that unmodified \TeX{} originally typeset. Having found the original
paragraph, we splice in the lengthened paragraph in place of the original.
Since the page is now 1~line longer than it was before, we pull the last
line off the page to bring it back to its original length, and place
that line onto the top of \TeX's \q{recent contributions} list. When
the next page begins, this line will be inserted before all other
paragraphs, right at the top. Now, we can return the new, widow-free page
(updated \cs{box255}) to the output routine, which proceeds
as normal.
\section{Choosing the \q{best} paragraph}
As we discussed previously, \lwc/ lengthens the paragraph with the lowest
cost. However, assigning a cost to each paragraph is not quite as simple as
it sounds. Before we look at how \lwc/ assigns costs, let's look at how
\TeX{} scores paragraphs when breaking them naturally.
\subsection{How \TeX{} scores paragraphs}
All glue in \TeX{} has a certain natural size: the size that it would be
in an ideal scenario. However, most glue also has stretch and shrink
components so that the glue can change in size to adapt to its
surroundings. For each line, \TeX{} individually sums the total
stretch/shrink for the line and the stretch/shrink that was actually used.
We define the stretch/shrink ratio~$r$ as the quotient of the
stretch/shrink used and the stretch/shrink available. Then the badness~$b$
of a line is approximately defined as
\begin{equation*}
b = 100r^3.
\end{equation*}
This is the badness referenced in the commonly-seen
\texttt{Underfull \cs{hbox}
(badness 1234)} warnings that \TeX{} produces.
\TeX{} calculates the badness for each line individually; however, we also
need to assess the paragraph as a whole. To do so, \TeX{} defines the
demerits for a whole paragraph~$d$ as approximately\footnotemark{} the sum of
the squared badnesses for each line. The natural paragraph that \TeX{}
breaks is the one that minimizes~$d$.
\footnotetext{We ignore any additional demerits or penalties that
\TeX{} may add.}
One important thing to realize is that demerits grow incredibly fast:
demerits are proportional to the \emph{sixth} power of glue stretch. This
means that you can expect to see extremely large demerit values, even for
a relatively \q{good} paragraph.
\subsection{Possible cost functions}
Now, let's return to how \lwc/ assigns costs to each paragraph. This is
surprisingly more complicated than it sounds, so we'll go through a few
possible cost functions first.
Here, we use $c$~for the cost of a paragraph, $d$~for the total demerits,
and $l$~for the number of lines (\cs{prevgraf}).
\subsubsection{The original implementation}
The original implementation of \lwc/ used the very simple cost function
\begin{equation*}
c = d.
\end{equation*}
This cost function works reasonably well, but has one major issue: it doesn't
take into account the number of lines in the paragraph. The demerits for a
paragraph is the sum of the demerits for each line. This means this cost
function will prefer using shorter paragraphs since they tend to have fewer
demerits. However, long paragraphs tend to have much more available glue
stretch, so this strategy can lead to suboptimal solutions.
\subsubsection{Scaling by the number of lines}
Once I realized this issue, I tried correcting it by dividing by the number
of lines in the paragraph to get the average demerits instead of the total
demerits:
\begin{equation*}
c = \frac{d}{l}
\end{equation*}
This works better than the previous function, but still has an issue.
If we have a fairly bad ten-line paragraph with total demerits $10d$ and an
almost-equally bad two-line paragraph with total demerits $2d + 1$, then by
this cost function, the ten-line paragraph will have a lower cost and will
be chosen. This means that our page now has ten bad lines instead of two bad
lines, which is not ideal.
\subsubsection{Current implementation}
Our first cost function, $c=dl^0$, doesn't consider the number of lines at
all, while our second cost function, $c=dl^{-1}$, considers the number of
lines too much. Splitting the difference between the two functions, we get
the current implementation:
\begin{equation*}
c = \frac{d}{\sqrt{l}}
\end{equation*}
I didn't arrive at this function through any sort of scientific testing;
rather, I picked the simplest function that I could think of that satisfies
the following two properties:
\begin{itemize}
\item Given a long paragraph and a short paragraph with different
average badnesses per line, prefer the one with the least average
badness.
\item Given two paragraphs with equal average badnesses per line,
prefer the shorter one.
\end{itemize}
\section{Quantitative analysis}
\begin{figure}
\begin{tikzpicture}\begin{axis}[
ybar interval,
xticklabels={1, ..., 15,
{$\,\ge\! 16$}},
x tick label style={font=\small},
y tick label style={font=\small},
enlarge y limits=upper,
enlarge x limits={abs=1},
grid=none,
scaled y ticks=base 10:-3,
ytick scale label code/.code={},
xlabel={Paragraph length (lines)},
ylabel={Count (thousands)}
]
\addplot+ [
draw=black,
fill=black!10,
semithick,
] table {
Length Count
1 4429
2 3704
3 2045
4 1320
5 894
6 717
7 498
8 406
9 379
10 251
11 175
12 152
13 111
14 95
15 79
16 437
18 0
};
\filldraw [fill=black!25, draw=black] (16, 0) rectangle (18, 437);
\end{axis}\end{tikzpicture}
\caption{Histogram of natural paragraph lengths in the sample text.}
\label{fig:hist}
\end{figure}
Let's look at some statistics for \lwc/. For testing, I
downloaded the top~ten books on \textsl{Project Gutenberg},\footnotemark{}
converted them to \LaTeX{} using \textsf{pandoc}, concatenated them into a
single \textsf{article} file, and compiled twice. This gives us a \acro{PDF}
with 1\,381~pages, 15\,692~paragraphs, 61\,865~lines, and 399~\waos/
(if they aren't removed).
\footnotetext{\textsl{Frankenstein},
\textsl{Pride and Prejudice},
\textsl{Alice's Adventures in Wonderland},
\textsl{The Great Gatsby},
\textsl{The Adventures of Sherlock Holmes},
\textsl{Simple Sabotage Field Manual},
\textsl{A Tale of Two Cities},
\textsl{The Picture of Dorian Gray},
\textsl{Moby Dick},
and \textsl{A Doll's House}.
}
This is a fairly challenging test: almost every third page has a \woo/, over
half of the paragraphs have two lines or fewer, and the text block is set to
the fairly wide \textsf{article} defaults. An average document is
much less challenging for \lwc/, so we can consider this to be a
worst-case scenario.
\subsection{Widows and orphans removed}
\begin{figure}
\begin{tikzpicture}\begin{axis}[
ybar=0pt,
bar width=0.8,
xtick=data,
ylabel={Widows and orphans removed},
width=\linewidth,
height=0.8\linewidth,
xticklabels={
\shortstack[c]{\hfill Maximum\\\hfill possible},
\textsf{lwc} \texttt{default},
\texttt{balanced},
\LaTeX{},
\texttt{strict},
},
x tick label style={
font=\small,
rotate=45,
anchor=east,
},
enlarge x limits=0.2,
]
\addplot+ [
draw=black,
fill=black!10,
semithick,
] table [x expr=\coordindex, y index=0] {
399
392
348
179
52
};
\filldraw [fill=black!25, draw=black]
(-0.4, 0) rectangle (0.4, 399);
\end{axis}\end{tikzpicture}
\divide\abovecaptionskip by 2
\caption{The number of \waos/ removed by each method.}\label{fig:modes}
\end{figure}
When we run \LaTeX{} with its default settings on the file, 179~(47\%) of
the \waos/ are removed. When we add \lwc/ with default settings, we remove
392~(98\%). Switching to strict mode, we can only remove 52~(13\%) of the
\waos/. In balanced mode, we remove 348~(87\%). See figure~\ref{fig:modes}
for a visual comparison.
\subsection{Paragraph costs}
\begin{figure}
\begin{tikzpicture}\begin{axis}[
width=\linewidth,
height=0.8\linewidth,
xlabel={Percentile},
ylabel={Cost},
ymode=log,
legend entries={Long, Natural},
legend pos=north west,
cycle list={
{black, thick},
{black!35, thick},
},
]
\addplot+ table [x=Percentile, y=Long] {\plotdata};
\addplot+ table [x=Percentile, y=Natural] {\plotdata};
\end{axis}\end{tikzpicture}
\divide\abovecaptionskip by 2
\caption{Paragraph costs by percentile rank for naturally-broken and
one-line lengthened paragraphs.}\label{fig:costs}
\end{figure}
The last section showed us that \lwc/ is quite effective at removing \waos/,
so now let's look at the paragraphs that \lwc/ expands. As \TeX{}
processes a document, \lwc/ is recording the costs for the naturally-broken
and expanded versions of each paragraph in the document. Costs don't
mean that much on their own, but a lower cost is always better.
As you can see in figure~\ref{fig:costs}, the lengthened paragraphs tend to
have \emph{much} higher costs than the naturally-broken paragraphs. This
is not surprising, since (as we've seen) a paragraph's demerits
scale with the sixth
power of glue stretch, so even a small amount of glue stretch can cause a
huge increase in demerits.
The empty space on the left of the \q{long} line is from the paragraphs
that \lwc/ was unable to lengthen at any cost. \LuaTeX{} assigns these
paragraphs zero~demerits, so they disappear on a logarithmic plot.
\subsection{Lengthening vs.\ shortening paragraphs}
\begin{figure}
\begin{tikzpicture}\begin{axis}[
xbar stacked,
height=0.2\linewidth,
width=\dimexpr\linewidth-1em,
scale only axis,
bar width=1,
enlargelimits=false,
xmin=0,
ymin=-0.5,
ymax=1,
ymajorticks=false,
xtick style={draw=none},
xlabel={Paragraphs (thousands)},
scaled x ticks=base 10:-3,
xtick scale label code/.code={},
legend style={at={(0.5,1)}, anchor=north},
legend columns=5,
legend cell align=left,
legend style={
/tikz/every even column/.append style={column sep=1em},
draw=none,
fill=none,
},
legend entries={
{$n=1$},
{$n$},
{$n+1$},
{$n\pm1$},
{$n-1$}
},
]
\addplot [fill=black!10 ] coordinates {(4429, 0)}; % One
\addplot [fill=white ] coordinates {(4474, 0)}; % None
\addplot [pattern=north east lines] coordinates {(5457, 0)}; % Long
\addplot [pattern=crosshatch ] coordinates {( 482, 0)}; % Both
\addplot [pattern=north west lines] coordinates {( 850, 0)}; % Short
\end{axis}\end{tikzpicture}
\divide\abovecaptionskip by 2
\caption{The number of paragraphs in the test sample that
(respectively) have exactly
one line, cannot be stretched or shrunk, can be only stretched
by one~line, can be either stretched or shrunk, and can be
only shrunk.}
\label{fig:stretchshrink}
\end{figure}
Figure~\ref{fig:stretchshrink} shows the number of paragraphs that \lwc/
could potentially stretch or shrink. The one-line paragraphs are broken out
separately since this test sample has an anomalous number of them.
Otherwise, we can see that \lwc/ is capable of stretching the majority of
paragraphs.
We can also see that of non-single-line paragraphs, only about 8\%
of paragraphs can only be shrunk (the last segment of
figure~\ref{fig:stretchshrink}), and this is in a document where 13\%
of paragraphs have at least eight~lines. Most documents rarely have
such long paragraphs, and it is these long paragraphs that are the
easiest to shrink.
Because of this, \lwc/ doesn't even attempt to shrink paragraphs; it
only stretches them.
\section{Known issues}
\Lwc/ is quite stable these days, a few issues remain:
\begin{itemize}
\item When a three-line paragraph is at the end of a page forming a
widow, \lwc/ will remove the widow; however, it will leave an orphan.
This issue is inherent to any process that removes widows through
paragraph expansion and is thus unavoidable. Orphans are considered
to be better than widows~\cite{elements}, so this is still an
improvement.
\item Sometimes a \woo/ cannot be eliminated because no paragraph has
enough stretch. Sometimes this can be remediated by
increasing \lwc/'s \estretch/; however, some pages just don't have
any suitable paragraph.
Long paragraphs with short words tend to be stretchier than short
paragraphs with long words since these long paragraphs have more
interword glue. Narrow columns also stretch more easily than wide columns
since you need to expand a paragraph by less to make a new line.
\item When running under \LuaMetaTeX{} (\ConTeXt{}), the log may
contain many lines like \q{\texttt{%
\spaceskip=\fontdimen2\font plus1.25pt minus1.25pt
luatex warning > tex: left parfill skip is gone}}. These messages
are completely harmless (although admittedly quite annoying).
\item \TeX{} may warn about overfull \cs{vbox}es on pages where
\lwc/ removed a \woo/. This happens due to the way that \lwc/
corrects for the \cs{prevdepth} when replacing paragraphs. It
does not actually produce an overfull \texttt{vbox}, but there
is a warning nevertheless. You can set \cs{vfuzz=2.5pt} to
hide the
warning.
\item \Lwc/ only attempts to expand paragraphs on a page with
a \woo/. A global system like in~\cite{global} would solve this;
however, this is both \acro{NP}-complete~\cite{plass} and
impossible to solve in a single pass. Very rarely would such a
system remove \woos/ that \lwc/ cannot.
\end{itemize}
\section{Conclusion}
All this probably makes \lwc/ look quite complicated, and this is true to
some extent. However, this complexity is hidden from the end~user:
as stated at the outset, most
users merely need to place \latexuse/ in their \LaTeX{} document
preamble, and \lwc/ will remove all the troublesome \waos/, without needing
any manual intervention.
Should you have any issues, questions, or suggestions for \lwc/, please
visit the project's GitHub page:
\href{https://github.com/gucci-on-fleek/lua-widow-control}
{\ttfamily github.com/gucci-on-fleek/lua-widow-control}.
Any feedback is greatly appreciated!
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