% language=us runpath=texruns:manuals/lowlevel

\environment lowlevel-style

\startdocument
  [title=balancing,
   color=middlecyan]

\startsectionlevel[title=Introduction]

{\em This is work in progress as per end 2024 these mechanisms are still in flux.
We expect them to be stable around the \CONTEXT\ meeting in 2025. The text is not
corrected, so feel free to comment.}

This manual is about a new (sort of fundamental) feature that got added to
\LUAMETATEX\ when we started upgrading column sets. In \TEX\ we have a par
builder that does a multi|-|pass optimization where it considers various
solutions based on tolerance, penalties, demerits etc. The page builder on the
other hand is forward looking and backtracks to a previous break when there is an
overflow. The balancing mechanism discussed here is basically a page builder
operating like the par builder: it looks at the whole picture.

In order to make this a useful mechanism the engine also permits intercepting the
main vertical list, so we start by introducing this.

\stopsectionlevel

\startsectionlevel[title=Intercepting the MVL]

When content gets processed it's added to a list. We can be in horizontal mode or
vertical mode (let's forget about math mode). In vertical mode we can be in a box
context (say \type {\vbox}) or in what is called the main vertical list: the one
that makes the page. But what is page? When \TEX\ has collected enough to match
the criteria set by \type {\pagegoal} which starts out as \type {\vsize}, it will
call the so called output routine which basically is expanding the \type
{\output} token list. That routine had do so something with the box that has the
collected material. It can become a page, likely with the content wrapped in a
page body with headers and footers and such, but it can also be stored for later
assembly, for instance in multiple columns, or after some analysis fed back into
the main vertical list.

For various mechanisms it matters if they are used inside a contained boxed
environment or in the more liberal main vertical list (from now on called mvl).
That's why we can intercept the mvl and use it later. Intercepting works as
follows:

\starttyping
\beginmvl 1
various content
\endmvl

\beginmvl 2
various content
\endmvl
\stoptyping

When at some point you want this content, you can do this:

\starttyping
\setbox\scratchboxone\flushmvl 2
\setbox\scratchboxtwo\flushmvl 1
\stoptyping

and then do whatever is needed. You can see what goes on with:

\starttyping
\tracingmvl 1
\stoptyping

There is not much more to say other than that this is the way to operate on
content as if it were added to the page which can be different from collecting
something in a vertical box. Think of various callbacks that can differ for the
mvl and a box.

The \type {\beginmvl} primitive takes a number or a set of keywords, as in:

\starttyping
\beginmvl
    index   1
    options \numexpr "01 + "04\relax
\relax
\stoptyping

There is of course some possible interference with mechanism that check the page
properties like \type {\pagegoal}. If needed one can check this:

\starttyping
\ifcase\mvlcurrentlyactive
  % main mvl
\or
  % first one
\else
  % other ones
\fi
\stoptyping

Possible applications of this mechanism are the mentioned columns and parallel,
independent, streams. However for that we need to be able to manipulate the
collected content. Actually, the next manipulator preceded the capturing, because
we first wanted to make sure that what we had in mind made sense.

The \type {beginmvl} also accepts keywords. You can specify an \type {index} (an
integer), a \type {prevdepth} (dimensions) and \type {options} (an integer
bitset). Possible option bit related values are:

\starttabulate[|Tr|||]
\NC 0x\tohexadecimal\ignoreprevdepthmvloptioncode \NC ignore prevdepth \NC \type {\ignoreprevdepthmvloptioncode} \NC \NR
\NC 0x\tohexadecimal\noprevdepthmvloptioncode     \NC no prevdepth     \NC \type {\noprevdepthmvloptioncode    } \NC \NR
\NC 0x\tohexadecimal\discardtopmvloptioncode      \NC discard top      \NC \type {\discardtopmvloptioncode     } \NC \NR
\NC 0x\tohexadecimal\discardbottommvloptioncode   \NC discard bottom   \NC \type {\discardbottommvloptioncode  } \NC \NR
\stoptabulate

Here the last column is a numeric alias available in \CONTEXT. More options are
likely to show up. When we eventually will balance these lists the routine will
deal with the discardables (like glue) but one can also remove them via the
options.

\startbuffer
\beginmvl
    index     1
    prevdepth 0pt
    options  \discardtopmvloptioncode
\relax
\scratchdimen\prevdepth
\dontleavehmode
\quad\the\mvlcurrentlyactive\quad\the\scratchdimen
\quad\blackrule[height=\strutht,depth=\strutdp,color=darkred]
\endmvl

\ruledhbox {\llap{1\quad}\flushmvl 1}
\stopbuffer

\typebuffer \start \showmakeup[line] \getbuffer \stop

\startbuffer
\beginmvl
    index  2
    options \numexpr
                \ignoreprevdepthmvloptioncode
              + \discardtopmvloptioncode
            \relax
\relax
\scratchdimen\prevdepth
\dontleavehmode
\quad\the\mvlcurrentlyactive\quad\the\scratchdimen
\quad\blackrule[height=\strutht,depth=\strutdp,color=darkred]
\endmvl

\ruledhbox {\llap{2\quad}\flushmvl 2}
\stopbuffer

\typebuffer \start \showmakeup[line] \getbuffer \stop

\startbuffer
\beginmvl 3 % when no keywords are used we expect a number
\scratchdimen\prevdepth
\dontleavehmode
\quad\the\mvlcurrentlyactive\quad\the\scratchdimen
\quad\blackrule[height=\strutht,depth=\strutdp,color=darkred]
\endmvl

\ruledhbox {\llap{3\quad}\flushmvl 3}
\stopbuffer

\typebuffer \start \showmakeup[line] \getbuffer \stop

\startbuffer
\beginmvl index 4 options 1
\scratchdimen\prevdepth
\dontleavehmode
\quad\the\mvlcurrentlyactive\quad\the\scratchdimen
\quad\blackrule[height=\strutht,depth=\strutdp,color=darkred]
\endmvl

\ruledhbox {\llap{4\quad}\flushmvl 4}
\stopbuffer

\typebuffer \start \showmakeup[line] \getbuffer \stop

\stopsectionlevel

\startsectionlevel[title=Balancing]

Balancing is not referring to balancing columns but to \quote {a result that
looks well balanced}. Just like we want lines in a paragraph to look consistent
with each other, something that is reflected in the (adjacent) demerits, we want
the same with vertical split of pieces. For this purpose we took elements of the
par builders to construct a (page) snippet builder. Here are some highlights:

\startitemize

\startitem
    Instead of a pretolerance, tolerance and emergency pass we only enable the
    last two. In the par builder the pretolerance pass is the one without
    hyphenation.
\stopitem

\startitem
    We seriously considered vertical discretionaries but eventually rejected the
    idea: we just don't expect users to go through the trouble of adding lots of
    split related pre, post and replace content. It's not hard to support it but
    in the end it also interfered with other demands that we had. We kept the
    code around for a while but then removed it. To mention one complication: if
    we add some new node we also need to intercept it in various callbacks that
    we already have in place in \CONTEXT. As with horizontal discretionaries, we
    then need to go into the components and sometimes even need to make decisions
    what can not yet be made.
\stopitem

\startitem
    As with the par builder, \TEX\ will happily produce an overfull box when no
    solution is possible that fits the constraints. In a paragraph there are
    plenty spaces (with stretch) and discretionaries (with components that vary
    in width) which enlarges the solution space. In vertical material there is
    less possible so there an emergency pass really makes sense: better be
    underful than overful.
\stopitem

\startitem
    In many cases there is no stretch available. There are also widow, club,
    shape and orphan penalties that can limit the solution space.
\stopitem

\startitem
    When we look at splitting pages (and boxes) we see (split) top skip kick in.
    This is something that we need to provide one way ot the other. And as we
    have to do that, we can as well provide support for bottom skip. A horizontal
    analogue is protrusion, something that also has to be taken into account in a
    rather dynamic way, at the beginning or end of the currently analyzed line.
\stopitem

\startitem
    There is no equivalent of hanging indentation but a shape makes sense. Here
    the shape defines heights, top and bottom skips and maybe more in the future.
    For that reason we use a keyword driven shape.
\stopitem

\startitem
    Because we have so called par passes, it made sense to have something similar
    for balancing. This gives is the opportunity to experiment with various
    variables that drive the process.
\stopitem

\startitem
    For those who read what we wrote about the par builder, it will not come as
    surprise that we also added extensive tracing and a callback for intercepting
    the results. This makes it possible to show the same detailed output as we
    can do for par passes.
\stopitem

\stopitemize

It's about time for some examples but before we come to that it is good to
roughly explain how the page builder works. When the page builder is triggered it
will take elements from the contributions list and add them to the page. When
doing that it keeps track of the height and depth as contributed by boxes and
rules. Because it will discard glue and kerns it does some checking there. An
important feature is that the depth is added in a next iteration. The routine
also needs to look at inserts. The variables \type {\pagegoal} (original \type
{\vsize} minus accumulated insert heights) and \type {\pagetotal} are compared
and when we run over the target height the accumulated stretch and shrink in glue
(when present) will be used to determine how bad this break is. If it is too bad,
the previous best break will be taken. Penalties can make a possible break more
or less attractive. When the output routine gets a split of page, the total is
not reliable because we can have backtracked to the previous break. In
\LUAMETATEX\ we have some more variables, like \type {\pagelastheight}, that give
a better estimate of what we got.

In order to make the first lines align properly relative to the top of the page
there is a variable \type {\topskip}. The height of the first line is at least
that amount. The correction is calculated when the first contribution happens: a
box or rule.

When we look at the balancer it is good to keep in mind that where the page
builder stepwise adds and checks, the balancer looks at the whole picture. The
page builder does a decent job but is less sophisticated than the par builder.
There is a badness calculation, penalties are looked at, glue is taken into
account but there are no demerits.

We want the balancer to work well with column sets that are very much grid based.
But in getting there we had some hurdles to take. Because the algorithm (like the
par builder) happily results in overfull boxes unless emergency stretch is set,
pages can overflow. When there is no stretch and|/|or shrink using emergency
stretch can give an underfull page.

The way out of this is to have non destructive trial passes and decrease the
number of lines. Of course we can get short pages but when for instance it
concerns a section title that gets moved this is no big deal. In a similar
fashion splitting a multi|-|line formula is also okay.

\startitemize
\startitem
    Collect the content in an mvl list and after that's done put the result in a
    box.
\stopitem
\startitem
    Set up a balance shape that specifies the slots in in columns (normally a
    column is just a blob of text).
\stopitem
\startitem
    Perform a trial balance run. As soon as an overfull page is seen, adapt the
    balance shape and do a new trial run.
\stopitem
\startitem
    When we're fine, either because we reached the end without overfull column or
    by passing the set deadcycles value, quit the trial process and balance the
    original list using the most recent balance shape.
\stopitem
\startitem
    Flush the result by fetching the topmost from the result split collection and
    feed it into the page flow. The boxed pseudo page will happily trigger the
    output routine that in turn construct the final page.
\stopitem
\stopitemize

At some point we decided to support multiple mvl streams and therefore changed
the last mentioned step. Because we store the whole column set we can as well
also store the assembled page bodies. This way we can flush different streams into
the same result.

\startitemize
\startitem
    Flush the result by fetching the topmost from the result split collection and
    feed it into the page flow. Do this for every saved (mvl) stream.
\stopitem
\startitem
    When we're done, the boxed pseudo pages will be flushed as pages. In the
    process, for every page we identify marks.
\stopitem
\stopitemize

We are now ready to look at some examples. Here we also show what balance shapes
do. These basically describe a sequence of slots to be filled. The last
specification is used when we exceed the number of defined slots. These are just
examples of simple situations, for real applications more code is needed.

\startbuffer[one]
\setbox\scratchboxone\vbox\bgroup
    \hsize.30\hsize
    \samplefile{tufte}
\egroup
\stopbuffer

\startbuffer[two]
\balanceshape 3
    vsize      12\lineheight
    topskip    \strutht
    bottomskip \strutdp
next
    vsize       5\lineheight
    topskip    \strutht
    bottomskip \strutdp
next
    vsize      8\lineheight
    topskip    \strutht
    bottomskip \strutdp
\relax
\stopbuffer

\startbuffer[three]
\setbox\scratchboxtwo\vbalance\scratchboxone
\stopbuffer

\startbuffer[four]
\hbox \bgroup
    \localcontrolledendless {%
        \ifvoid\scratchboxtwo
            \expandafter\quitloop
        \else
            \setbox\scratchbox\ruledhbox\bgroup
                \vbalancedbox\scratchboxtwo
            \egroup
            \vbox to 12\lineheight \bgroup
                \box\scratchbox
                \vfill
            \egroup
            \hskip1em
        \fi
    }\unskip
\egroup
\stopbuffer

We start with some content in a box. This can of course be a flushed
mvl but here we just set it directly:

\typebuffer[one]

We will split this box in columns. If you are familiar with \TEX\ you might know
that a paragraph of text can follow a shape defined by \type {\parshape}. In a
similar way as lines are split by width, we can split a vertical list by height.
For that we define a balance shape:

\typebuffer[two]

\typebuffer[three]

Contrary to a \type {\parshape}, a \type {\balanceshape} is not wiped after the
work is done. It also expects keys and values. As with \type {\parpasses} each
step is separated by \type {next}. This makes it an extensible mechanism. Finally
we will split the box according to this shape:

\typebuffer[four]

The result is shown here:

\startlinecorrection
    \small
    \setuptolerance[tolerant,stretch]
    \getbuffer[one,two,three,four]
\stoplinecorrection

Like the par builder we can end up with overfull boxes but we can deal with that
by using trial runs.

\starttyping
\setbox\scratchboxtwo\vbalance\scratchboxone trial
\stoptyping

\startbuffer[one]
\setbox\scratchboxone\vbox\bgroup
    \hsize.30\hsize
    \samplefile{knuthmath} \blank
    \framed[height=4\lineheight]{test}
    \samplefile{knuthmath} \blank
\egroup
\stopbuffer

In that case the result is made from empty boxes so the original is not
disturbed. Here we show an overflow, so in the first resulting box you
can compare the height with the requested one and when it's larger you
can decide to decrease the first height in the shape and try again.

\startlinecorrection
    \small
    \setuptolerance[tolerant,stretch]
    \getbuffer[one,two,three,four]
\stoplinecorrection

Of course that involves some juggling of the shape but after all we have \LUA\ at
our disposal so in the end it's all quite doable.

\startbuffer[three]
\setbox\scratchboxtwo\vbalance\scratchboxone trial
\stopbuffer

\startbuffer[four]
\global\globalscratchtoks\emptytoks
\localcontrolledendless {%
    \ifvoid\scratchboxtwo
        \expandafter\quitloop
    \else
        \setbox\scratchbox\vbalancedbox\scratchboxtwo
        \xtoksapp\globalscratchtoks {
            \NC \the\currentloopiterator
            \NC \the\ht\scratchbox
            \NC \the\balanceshapevsize\currentloopiterator
            \NC \NR
        }
    \fi
}
\stopbuffer

\start
    \small
    \setuptolerance[tolerant,stretch]
    \getbuffer[one,two,three,four]
\stop

\starttabulate[||||]
\BC \BC real \BC target \NC \NR
\the\globalscratchtoks
\stoptabulate

Because the balancer can produce what otherwise the page builder produces, we
need to handle the equivalent of top skip which is what the already shown \type
{top} keyword takes care of. This means that the current slice (think current
line in the par builder) has to take that into account. This can be compared to the
left- and right protrusion in the par builder. When we typeset on a grid we have an
additional demand.

When we surround (for instance a formula) with halfline spacing, we eventually
have to return on the grid. One complication is that when we are in grid mode and
use half line vertical spacing, we can end up in a situation where the initial
half line space is on a previous page. That means that we need to use a larger
top skip. This is not something that we want to burden the balancer with but we
have ways to trick it into taking that compensation into account.

\startlinecorrection
\hpack \bgroup
    \ruledvpack to 8\lineheight \bgroup \forgetall \raggedcenter \offinterlineskip \hsize 3cm
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=\strutht,  depth=\strutdp,  color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=middlegray]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=\strutht,  depth=\strutdp,  color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=darkgray]
        \vfill
    \egroup
    \quad
    \ruledvpack to 8\lineheight \bgroup \forgetall \raggedcenter \offinterlineskip \hsize 3cm
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=.5\strutht,depth=.5\strutdp,color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=middlegray]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=.5\strutht,depth=.5\strutdp,color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=darkgray]
        \vfill
    \egroup
    \quad
    \ruledvpack to 8\lineheight \bgroup \forgetall \raggedcenter \offinterlineskip \hsize 3cm
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=middlegray]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=.5\strutht,depth=.5\strutdp,color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=darkgray]
        \vfill
    \egroup
    \quad
    \ruledvpack to 8\lineheight \bgroup \forgetall \raggedcenter \offinterlineskip \hsize 3cm
        \blackrule[width=.2\hsize,height=.5\strutht,depth=.5\strutdp,color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=middlegray]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.2\hsize,height=.5\strutht,depth=.5\strutdp,color=darkred]\par
        \blackrule[width=\hsize,  height=\strutht,  depth=\strutdp,  color=darkgray]\par
        \blackrule[width=.6\hsize,height=\strutht,  depth=\strutdp,  color=darkgray]
        \vfill
    \egroup
\egroup
\stoplinecorrection

However, when we split in the middle of that segment, we can end up with a half
line skip in a next slot because \TEX\ will remove glue at the edge. So we end up
with what we see in the third sequence above. We deal with that in a somewhat
special way: a box as a discardable field which value will be taken into account
as additional top value. That field is set and reset by glue options {\tt
0x\tohexadecimal \cldcontext {tex . glueoptioncodes . setdiscardable}} and {\tt
0x\tohexadecimal \cldcontext {tex . glueoptioncodes . resetdiscardable}} that can
be manipulated in \LUA\ as part of some spacing model. Here we suffice by
mentioning that it makes sure that (as in the fourth blob above) at the top we
have a half line spacing.

\stopsectionlevel

\startsectionlevel[title=Forcing breaks]

Because the initial application of balancing was in column sets, we also need the
ability to goto a next slot (step in a shape), column (possibly more steps), page
(depending on the page state), and spread (for instance if we are doubles ided).
For this we use \type {\balanceboundary}. It takes two values and when the
boundary node triggers a callback in the builder these are passed along with a
shape identifier and current shape slot. That callback can then signal back that
we need to try a break here with a given penalty. Assuming that at the \LUA\ end
we know at which slot we have a slot, column, page or spread break. Multiple
slots can be skipped by multiple boundaries. There is one pitfall: we need
something in a slot in order to break at all, so one ends up with for instance:

\starttyping
\balanceboundary 3 1\relax
\vskip\zeropoint
\balanceboundary 3 0\relax
\vskip\zeropoint
\balanceboundary 3 0\relax
\stoptyping

Here the \type {3} is just some value that the callback can use to determine its
action (like goto a next page) and the second value provides a detail. Of course
all depends on the intended usage. By using a callback we can force breaks while
not burdening the engine with some hard coded solution. For example, in \CONTEXT\
we used these (the values are these from experiments and might change:

\starttabulate[|c|c|||]
\BC first \BC second \BC action                                    \BC user interface          \NC \NR
\NC 1     \NC 1 or 0 \NC goto next spread (1 initial, 0 follow up) \NC \type {\page[spread]}   \NC \NR
\NC 2     \NC 1 or 0 \NC goto next page (idem)                     \NC \type {\page}           \NC \NR
\NC 3     \NC 1 or 0 \NC goto next column (idem)                   \NC \type {\column}         \NC \NR
\NC 4     \NC 1 or 0 \NC goto next slot (idem)                     \NC \type {\column[slot]}   \NC \NR
\NC 5     \NC n      \NC next slot when more than n lines          \NC \type {\testroom[5]}    \NC \NR
\NC 6     \NC s      \NC next slot when more than s scaled points  \NC \type {\testroom[80pt]} \NC \NR
\stoptabulate

\stopsectionlevel

\startsectionlevel[title=Marks]

It is possible to synchronize the marks with those in the results of balanced
segments with a few \LUA\ helpers that do the same as the page builder does at
the start of a page, while packaging the page and when wrapping it up. So, instead
of split marks we can have real marks.

\stopsectionlevel

\startsectionlevel[title=Inserts]

Before we go into detail, we want to point out that when implementing a
(balancing) mechanism as introduced above, decisions have to be made. In
traditional \TEX\ there is for instance an approach to inserts that involves
splitting them over pages. In our case that is a bit harder to do but there are
ways to deal with it. When deciding on an approach it helps that we know a bit
what situations occur and where we can put some constraints. One can argue that
solutions should be very generic because (for instance) a publisher has some
specific demands but in practice those are not our audience. In decades of
developing \LUATEX\ and \LUAMETATEX\ it's (\CONTEXT) user demands and challenges
that drives what gets implemented. Publishers, their suppliers, and large scale
(commercial) users are pretty silent when it comes to development (and supporting
it) while users communicate via meetings and mailing lists. Also, rendering of
documents that have notes are often typeset kind of traditional.

Users on the other hand have come up with demands for columns, typesetting on the
grid, multiple notes, balancing, and parallel content streams. The picture we get
from that makes us confident that what we provide is generally enough and as
users understand the issues at hand (maybe as side effect of struggling with
solutions) it's not that hard to explain why constraints are in place. It makes
more sense to have a limited reliable mechanism that deals with the kind of
(foot)notes that known users need than to cook up some complex mechanism that
caters potential specific demands by potential users. Of course we have our own
challenges to deal with, even if the resulting features will probably not be used
that often. So here are the criteria that make sense:

\startitemize[packed]
\startitem We can assume a reasonable amount of notes. \stopitem
\startitem These are normally small with no (vertical) whitespace. \stopitem
\startitem Notes taking multiple lines may split. \stopitem
\startitem But we need to obey widow and club penalties. \stopitem
\startitem There can be math formulas but mostly inline. \stopitem
\startitem We need to keep them close to where they are referred from. \stopitem
\stopitemize

But,

\startitemize[packed]
\startitem We can ignore complex conflicting demands. \stopitem
\startitem As long as we get some result, we're fine. \stopitem
\startitem So users have to check what comes out. \stopitem
\startitem We don't assume fully automated unattended usage. \stopitem
\stopitemize

And of course:

\startitemize[packed]
\startitem Performance should be acceptable. \stopitem
\startitem User interfaces should be intuitive. \stopitem
\startitem Memory consumption should be reasonable. \stopitem
\stopitemize

We have users who use multiple note classes so that also has to be handled but
again we don't need to come up with solutions that solve all possible demands. We
can assume that when a book is published that needs them, the author will operate
within the constraints.

We mentioned footnotes being handled by the page builder so how about them in
these balanced slots? Given the above remarks, we assume sane usage, so for
instance columns that have a single slot with possibly fixed content at the top
or bottom (and maybe as part of the stream). The balancer handles notes by taking
their height into account and when a result is used one can request the embedded
inserts and deal with them. Again this is very macro package dependent. Among the
features dealt with are space above and between a set of notes, which means that
we need to identify the first and successive notes in a class. Given how the
routine works, this is a dynamic feature of a line: the amount of space needed
depends on how many inserts are within a slot. When we did some extreme tests
with several classes of notes and multiple per column we saw runtime increasing
because instead of a few passes we got a few hundred. In an extreme case of 800
passes to balance the result we noticed over four million checks for note related
spacing. We could bring that down to one tenth so in the end we are still slower
but less noticeable. Here are the helper primitives for inserts:

\starttyping
<state> = \boxinserts <box>
<box>   = \vbalancedinsert <box> <class>
<state> = \boxinserts <box>
\stoptyping

A (foot)note implementation is very macro package dependent so the next example
is just that: an example of using the available primitive. We start by populating
a mvl with a sample text and a single footnote.

\startbuffer[populate]
\begingroup
    \forgetall
    \beginmvl
        index 5
        options \numexpr
            \ignoreprevdepthmvloptioncode
          + \discardtopmvloptioncode
        \relax
    \relax
        \hsize .4tw
        Line 1 \par Line 2 \footnote {Note 1} \par Line 3 \par
        Line 4 \footnote {Note 2} \par Line 5 \par Line 6 \par
    \endmvl
\endgroup
\stopbuffer

\typebuffer[populate]

We fetch the footnote number, which is one of many possible defined
inserts

\startbuffer[whatever]
\cdef\currentnote{footnote}%
\scratchcounter\currentnoteinsertionnumber
\stopbuffer

\typebuffer[whatever]

The quick and dirty balancer uses a simple shape of 5 lines with normal strut
properties. From the balanced result we take two columns. We test if there is an
insert and take action when there is. Here we just filter the footnotes but there
can of course be more. We overlay these notes over (under) the column that has
them. So we work per column.

\startbuffer[balance]
\begingroup
    \setbox\scratchboxone\flushmvl 5
    \balanceshape 1
        vsize      5lh
        topskip    1sh
        bottomskip 1sd
    \relax
    \setbox\scratchboxtwo\vbalance\scratchboxone
    \ruledhbox \bgroup
        \localcontrolledrepeat 2 {
          \ifnum\currentloopiterator > 1
            \hskip2\emwidth
          \fi
          \setbox\scratchboxthree\vbalancedbox\scratchboxtwo \relax
          \ifnum\boxinserts\scratchboxthree > 3
            \setbox\scratchboxfour\vbalancedinsert
                \scratchboxthree\scratchcounter
            \wd\scratchboxfour 0pt
            \box\scratchboxfour
          \fi
          \box\scratchboxthree
        }\unskip
    \egroup
\endgroup
\stopbuffer

\typebuffer[balance]

The result is:

\start
    \getbuffer[populate] % outside next vbox
    \startlinecorrection
        \getbuffer[whatever]
        \automigrationmode 0
        \getbuffer[balance]
    \stoplinecorrection
\stop

As we progressed we realized that the \quote {balancer} used in column sets can
also be used for single columns and we can even support a mix of single and multi
columns. There is however a problem: within a mvl we can deal with spacing but we
can't do that reliable across mvl's and especially when we cross a page it
becomes hard to identify if some (vertical) spacing is needed; we don't want it
at the bottom or top of a page. This feature is too experimental to be discussed
right now.

We assumed reasonable notes to be used but even if a user tries to keep notes
small and avoid too many, there are cases where they might look like a paragraph
and when there are more in a row, it might be that a column overflows. This is
why we have some support for split notes. This is accomplished by two additional
commands:

\starttyping
\setbox\scratchboxone\vbalance\scratchboxone\relax
\vbalanceddeinsert\scratchboxone\relax
\stoptyping

Here we convert inserts in such a way that they are taken into account by the
balancer so that multi|-|slot optimization takes place. Afterwards, when we loop
over the result we can reconstruct the inserts:

\starttyping
\setbox\scratchboxtwo\vbalancedbox\scratchboxone
\vbalancedreinsert\scratchboxtwo\relax
\stoptyping

Among the reasons that these are explicit actions, is that we want to experiment
but also be able to see the effect by selectively enabling it. You can get better
results by forcing depth correction.

\starttyping
\setbox\scratchboxone\vbalance\scratchboxone
\vbalanceddeinsert\scratchboxone forcedepth\relax
\stoptyping

This will use the depth as defined by \type {\insertlinedepth} which is an insert
class specific parameter, but discussing details of inserts is not what we do
here. The reason for using a \type {\relax} in the above examples is that we want
to stress that when keywords are involved, you need to prevent look|-|ahead,
especially when an \type {\if...} or expandable loop follows, which is not
uncommon when we balance.

It is possible to define top and bottom inserts but of course these need to be
filtered and placed at the \TEX\ end, so this is macro package specific. Here we
just mention that it is possible to set \type {\insertstretch} and \type
{\insertshrink} which will be taken into account. However, this can result in
overlap so if indeed stretch or shrink is applied, the \type {handle_uinsert}
callback should be used for bringing what actually gets inserted to the right
dimensions. For now we consider this an experimental feature.

\stopsectionlevel

\startsectionlevel[title=Discardables]

This is a preliminary explanation.

\startbuffer[populate]
\begingroup
    \beginmvl
        index 5
        options \numexpr
            \ignoreprevdepthmvloptioncode
          + \discardtopmvloptioncode
        \relax
    \relax
        \hsize .4tw
        \par
        \vskip0pt
        {\darkred \hrule discardable height 1sh depth 1sd width 1em}
        \par
        % we need the strut because the rule obscures it .. todo
        \dorecurse{8}{\strut Line #1 \par}
        \vskip\zeropoint
        {\darkblue \hrule discardable height 1sh depth 1sd width 1em}
        \par
    \endmvl
\endgroup
\stopbuffer

\typebuffer[populate]

\startbuffer[balance]
\setbox\scratchboxone\flushmvl 5
\balanceshape 1
    vsize       5lh
    topskip     1sh % see comment above
    bottomskip  1sd
    options     3
\relax
\setbox\scratchboxtwo\vbalance\scratchboxone\relax % lookhead
\stopbuffer

\startbuffer[flush]
\hpack \bgroup
    \localcontrolledrepeat 3 {
        \ifvoid\scratchboxtwo\else
            \setbox\scratchboxthree\vbalancedbox\scratchboxtwo
            \ifvoid\scratchboxthree\else
                \dontleavehmode\llap{[\the\currentloopiterator]\quad}%
                \ruledhpack{\box\scratchboxthree}\par
            \fi
            \hskip 4em
        \fi
    }\unskip
\egroup
\stopbuffer

\typebuffer[balance,flush]

\start
    \forgetall
    \getbuffer[populate] % outside next vbox
    \blank[2*line]
    \startlinecorrection
        \getbuffer[balance]
        \getbuffer[flush]
    \stoplinecorrection
%     \blank[2*line]
\stop

When at the top, the rule will be ignored and basically sticks out. When at the
bottom the rule might end up in a zero dimension box. With \typ
{\vbalanceddiscard \scratchboxtwo} they will become an \type {\nohrule}.
Basically we're talking of optional content. The \type {options} bitset in the
shape definition tells if we have a top (1) and|/| or bottom (2), here we have
both (3) but in for instance column sets it depends.

\start
    \forgetall
    \showmakeup[vglue]
    \getbuffer[populate] % outside next vbox
    \showmakeup[reset]
    \blank
    \startlinecorrection
    \showmakeup[vglue]
        \getbuffer[balance]
        \vbalanceddiscard\scratchboxtwo
        \getbuffer[flush]
    \stoplinecorrection
%     \blank[2*line]
\stop

Here we actually still have the rule but marked as invisible. So, topskip has a
negative amount. In the next case the \type {remove} keyword makes the rule go
away in which case we also adapt the topskip accordingly.

\start
    \forgetall
    \showmakeup[vglue]
    \getbuffer[populate] % outside next vbox
    \showmakeup[reset]
    \blank
    \startlinecorrection
        \getbuffer[balance]
        \vbalanceddiscard \scratchboxtwo remove\relax
        \getbuffer[flush]
    \stoplinecorrection
%     \blank[2*line]
\stop

You need to juggle a bit with skips and penalties to get this working as you
like. Instead of rules you can also use boxes, for example before:

\starttyping
\vskip\zeropoint
\ruledvbox discardable {\hpack{\strut BEFORE}}
\par
\stoptyping

and after:

\starttyping
\forgetall \par \vskip\zeropoint
\ruledvbox discardable {\hpack{\strut AFTER}}%
\penalty\minusone % !
\par
\stoptyping

It currently is a playground so it might (and probably will) evolve. Although it
was also made for a specific issue it might have other usage.

\stopsectionlevel

\startsectionlevel[title=Passes]

{\em todo}

\starttyping
\showmakeup[vpenalty,line]
\balancefinalpenalties 6 10000 9000 8000 7000 6000 5000\relax
\balancevsize 5\lineheight
\setbox\scratchbox\vbox{\dorecurse{1}{\samplefile{tufte}\footnote{!}\par}}
\vbalance\scratchbox
\stoptyping

\stopsectionlevel

\startsectionlevel[title=Passes]

In \LUAMETATEX\ the par builder has been extended with additional features (like
orphan, toddler and twin control) and the ability to define and apply multiple
passes over the paragraph to get the best result. The balancer has a similar
feature: \type {\balancepasses}. As with \type {\parpasses} we have an
infrastructure for tracing.

\starttyping
% threshold
% tolerance
% looseness
% adjdemerits
% originalstretch
% emergencystretch
% emergencyfactor
% emergencypercentage
\stoptyping

\stopsectionlevel

% tests/mkiv/typesetting/balancing-001.tex

\stopdocument

% (Re)written mixed with watching Talk Talk in Montreux DVD and energetic The
% Warning live concerts on YT, just to get a positive constructive vibe. As with
% the mechanisms discussed here, it's all about cooperation and subtle (and honest)
% quality. It's often music that drives this development.

