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Line breaking selects where a paragraph is divided into successive lines. The choice combines the widths of words and spaces with a measure of acceptable layout. A glyph advance supplies part of the underlying geometry, while the chosen breaks determine which content shares a CSS line box in a web layout.

A greedy algorithm can put as much content as possible on each line before proceeding. Greedy line breaking is easy to compute, but an early choice can leave an awkward short line later. Line breaking with squared slack cost gives a small, explicit example. Suppose four word widths are $3,2,2,5$, each internal space has width 1, and the permitted line width is 7, all in arbitrary consistent units. Greedy grouping produces line lengths $6,2,5$. An alternative grouping produces $3,5,5$. Charging squared unused width on every line except the last givesAn immediately fuller first line can therefore produce a worse paragraph under this particular cost.

Dynamic programming makes the global choice systematic. Let $D(j)$ be the minimum cost for the first $j$ words, and let ${\ell }_{ij}$ include words $i+1$ through $j$ and their internal spaces. For feasible lines, set $c(i,j)=(W-{\ell }_{ij}{)}^2$, except that $c(i,n)=0$ for the final line. ThenPossible break positions form a directed acyclic graph; a feasible line is an edge, so the same calculation finds a shortest path from the start to the end of the paragraph.

Donald E. Knuth and Michael F. Plass developed a richer paragraph-wide method using boxes, adjustable glue and penalties. Their Knuth-Plass line-breaking algorithm accommodates typographic choices beyond the fixed spaces and squared costs in the illustrative model above.^[ref]

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.