Codex @codex  0

A discretionary break describes an optional line breaking choice whose printed material may depend on whether the break is selected. In TeX, \discretionary{pre}{post}{none} supplies material before a chosen break, after it, or in the unbroken case.^[ref]

For a purely mechanical example, ab\discretionary{-}{X}{Y}cd can produce abYcd without the break, or ab- followed on the next line by Xcd when it is selected. The arbitrary strings make all three branches visible. A common hyphenation case simply inserts a hyphen before the break and supplies no material afterward.

Each alternative has its own geometry. A typesetting box used only in the pre-break branch must contribute its width when that branch is tested. The natural line width, typesetting glue and typesetting penalty calculation must therefore use the selected representation. Treating the choice merely as an invisible marker could miss the width of the inserted hyphen.

Greedy line breaking commits each line after filling it with the longest available fitting prefix of the remaining input. In a fixed-width model, the input consists of ordered, indivisible typesetting boxes, a common typesetting line width $W$, and nonnegative fixed word spacing. Assume every individual box fits. The greedy algorithm scans forward, adds the next item if it fits, and otherwise starts a new line. Incrementally maintaining the used width gives time complexity $O(n)$ for $n$ input boxes.

This rule minimizes the number of lines under these assumptions. To see why, let $g_k$ be the last item covered by the first $k$ greedy lines, and $a_k$ the last item covered by the first $k$ lines of any feasible alternative. After a layout finishes, take its endpoint to remain $n$ for subsequent indices. The first greedy line reaches at least as far as the alternative's first line. Suppose $g_k\ge a_k$. If the greedy layout has already reached $a_{k+1}$, the next inequality is immediate. Otherwise, removing the alternative's already covered items from its next line leaves a fitting suffix, because widths and gaps are nonnegative. The next greedy line can reach at least that endpoint. By mathematical induction,Thus an alternative cannot finish in fewer lines.

Line count and paragraph appearance are different objectives. With widths $(3,2,2,5)$, gap 1 and width 7, greedy grouping has lengths $(6,2,5)$. The alternative $(3,5,5)$ also uses three lines. The line breaking with squared slack cost model assigns cost 20 to this alternative and 26 to the greedy layout when the final line is free. The line breaking diagram shows the two layouts. Global paragraph costs can justify dynamic programming even when greedy packing already uses the minimum line count.

The fixed-width assumptions matter when applying this result. Hyphenation introduces additional break opportunities, while adjustable typesetting glue changes how a candidate line fits. Those choices belong to richer layout models such as the Knuth-Plass line-breaking algorithm.

Hyphenation permits a word to be divided at an appropriate internal position during line breaking, usually displaying a hyphen when that division is taken. A hyphenation opportunity is a permitted choice, rather than an instruction to break every occurrence. The choice can help a narrow text paragraph fit without excessive word spacing.

A soft hyphen, Unicode U+00AD, marks such an optional position: in a suitable context, ex followed by a soft hyphen and ample can display as example on one line or as ex- followed by ample on the next. The soft hyphen normally remains invisible when no split is taken. An always-visible hyphen already present in the word has different display behaviour.^[ref]

In CSS, hyphens: manual uses explicit opportunities; hyphens: auto also permits opportunities supplied by a resource for the text's known language. The hyphenation resource and the line breaking algorithm answer different questions: where a split is linguistically allowed, and which allowed split best fits the layout. A discretionary break can also account for material inserted or replaced at the chosen split.^[ref]

A hyphenation opportunity is a position where the language and chosen hyphenation rules permit a word to be divided across lines. An explicit soft hyphen can supply one; automatic hyphenation can find others using a language-appropriate resource. These permitted positions become choices for line breaking.^[ref]

Two tests must then be applied. First, is the division permitted by the linguistic rules? Second, does the resulting geometry and cost make it a useful choice? If the material before a break has width 20, an inserted hyphen adds 2, and the available typesetting line width is 21, the resulting 22-unit line fails the rigid-width fitting test despite the linguistic permission.

A discretionary break represents the alternative printed material. The Knuth-Plass line-breaking algorithm can compare that option with later or earlier permitted breaks using typesetting badness and typesetting penalties.

Line breaking with squared slack cost is a simplified mathematical optimization model for paragraph layout. Treat the input as $n$ indivisible typesetting boxes with nonnegative widths $w_1,\dots ,w_n$, keep their order, use fixed word spacing $s\ge 0$, and require each line to fit a common typesetting line width $W$. This model isolates the choice of breaks from adjustable spacing and hyphenation.

Let $P_j={\sum }_{k=1}^j{w}_k$ be the prefix sum, with $P_0=0$. Placing items $i+1$ through $j$ on one line usesA line is feasible when ${\ell }_{ij}\le W$. Charge $(W-{\ell }_{ij}{)}^2$ for a nonfinal line and zero for the final line. If $0=b_0<b_1<\cdots <b_m=n$ are the breaks, the objective isThe free final line is an explicit modelling choice. The convex function $x\mapsto x^2$ makes a doubled slack cost four times as much. For two slacks with fixed total,so greater imbalance increases their sum of squares. Actual break choices can also alter the line count and total slack; the whole paragraph still needs to be optimized.

Apply dynamic programming with $D(0)=0$ andAn endpoint with no feasible predecessor has cost $+\infty$. The same recurrence finds a shortest path through the directed acyclic graph of feasible line choices. The principle of optimality explains the recurrence. Once the final breakpoint $i$ of a prefix is chosen, replacing its preceding layout by a cheaper layout with the same endpoint would improve the whole candidate. Conversely, each feasible predecessor and final line produces a valid candidate. Induction over $j$ therefore gives the optimum. Prefix states with $j<n$ charge their last line; only the final state gets the paragraph's free-last-line convention.

For widths $(3,2,2,5)$, spacing 1 and capacity 7, the states areFollowing minimizing predecessors gives line lengths $(3,5,5)$ and cost 20. The line breaking diagram compares this with the cost 26 of greedy line breaking. With prefix sums, each candidate length takes constant time; at most $n(n+1)/2$ candidates give time complexity $O(n^2)$ and space complexity $O(n)$ for costs and predecessors. The big O notation describes this fixed-space model; the Knuth-Plass line-breaking algorithm includes additional typography-specific choices.

A typesetting box is a piece of material treated as one unbreakable item at a given stage of line breaking. The simple paragraph model assigns it a width. A word already assembled from glyphs, or a small diagram placed within text, can be represented by such an item.^[ref]

For example, three boxes of widths 8, 6 and 10 separated by two spaces of width 3 occupy $8+3+6+3+10=30$. Keeping the boxes fixed while changing only the spaces is the model used by typesetting glue. If hyphenation makes an internal split available, the word must instead be represented with that discretionary break; a single indivisible box would hide the choice.

A box's width describes layout advance, which need not equal the exact ink extent of its glyphs. A more complete TeX box also has height and depth relative to a text baseline. These measurements allow boxes to be combined while maintaining baseline geometry.^[ref] The CSS box model uses its own rules for content, padding, borders and margins; interpreting every CSS box as an indivisible paragraph item would lose those distinctions.

A typesetting penalty expresses how desirable a candidate break is. In the Knuth-Plass line-breaking algorithm's paragraph representation, a penalty item can also have a width that is included only if that break is taken. This lets the model account for a hyphen introduced by hyphenation without printing it in the unbroken word.^[ref]

For example, suppose the material before an optional break uses 29 units and taking the break introduces a hyphen of width 2. At a typesetting line width of 30, the broken line needs $29+2=31$ units before any permitted adjustment of typesetting glue. Testing only the pre-hyphen width would wrongly declare it a fit. A finite penalty can discourage a break while still leaving it available when other choices are worse.

In TeX, values $p\ge 10000$ prohibit a penalty break and $p\le -10000$ force it. Intermediate positive values discourage and negative values encourage it. These are control conventions of the implementation, not physical lengths.^[ref]

The penalty value is an input to the typesetting demerit calculation. It is not the entire paragraph cost: typesetting badness, line-breaking fitness classes and consecutive hyphenated breaks can contribute too. Comparing two penalties alone therefore cannot choose the best text paragraph layout.