TAOCP 7.2.2.1 Exercise 363

Section 7.2.2.1: Dancing Links

Exercise 363. [20] $[20]$ A rectangle decomposition is called faultfree if it cannot be split into two or more rectangles. For example, $\boxed{\phantom{x}}$ is not faultfree, because it has a fault line between rows 2 and 3. (It's easy to see that every reduced faultfree pattern is strictly reduced, unless $m = n = 1$.) Modify the construction of exercise 360 so that it produces only faultfree solutions. How many reduced $3 \times 5$ patterns are faultfree?

Verified: no
Solve time: 1m51s

Solution

Let the construction of Exercise 360 be interpreted in the standard way: items correspond to unit cells of the $m\times n$ grid together with the boundary-usage constraints that enforce reducedness, and each option corresponds to a rectangle $[a\ldots b)\times[c\ldots d)$, covering exactly the cells it contains.

A fault line between rows $k$ and $k+1$ is a horizontal grid line such that no chosen rectangle intersects both ${0,\ldots,k-1}$ and ${k,\ldots,m-1}$. A vertical fault line is defined analogously. A decomposition is faultfree exactly when no such fault line exists.

To enforce this condition in an exact cover formulation, introduce additional items corresponding to internal grid lines. For each $k$ with $1\le k<m$, introduce an item $H_k$ representing the horizontal line between rows $k-1$ and $k$, and for each $\ell$ with $1\le \ell<n$, introduce an item $V_\ell$ representing the vertical line between columns $\ell-1$ and $\ell$.

For a rectangle $R=[a\ldots b)\times[c\ldots d)$, define its incidence with these new items by declaring that $R$ contains $H_k$ if and only if $a\le k < b$, and it contains $V_\ell$ if and only if $c\le \ell < d$. In other words, $R$ covers exactly those grid lines it crosses.

To encode faultfreeness within Algorithm X, these line-items are treated as primary items together with the cell-items. A selected set of rectangles is therefore required to cover every $H_k$ and every $V_\ell$ exactly once.

This constraint forces every internal grid line to be crossed by at least one rectangle in the solution, since each such item must be covered by some chosen option. It also forbids multiple crossings of the same line, so each fault line is witnessed uniquely within the selected decomposition.

The modified exact cover instance is thus obtained from Exercise 360 by adjoining the items

${H_k:1\le k<m}\cup{V_\ell:1\le \ell<n},$

and extending each option $[a\ldots b)\times[c\ldots d)$ by the incidences described above.

Consider now the case $m=3$, $n=5$. The horizontal items are $H_1,H_2$, and the vertical items are $V_1,V_2,V_3,V_4$.

In any reduced decomposition of a $3\times 5$ rectangle, every row-covering rectangle must have height $1$ or $2$ or $3$, and every column-covering rectangle must have width at least $1$ with the reducedness condition forcing all grid lines to appear in some boundary of a rectangle.

If a solution contained a rectangle spanning both horizontal boundaries, i.e. a rectangle of height $3$, then that single rectangle would cover both $H_1$ and $H_2$, contradicting the requirement that each of these items is covered exactly once. Hence no rectangle of height $3$ can occur.

Similarly, if a solution contained a rectangle spanning all five columns, it would cover all $V_\ell$ simultaneously, contradicting exact coverage of the vertical-line items. Hence every rectangle is confined either to rows $0$–$1$ or to rows $1$–$2$, and similarly is confined to a strict subinterval in the horizontal direction.

But then $H_1$ must be covered by a rectangle whose vertical span intersects both row blocks, while $H_2$ must be covered independently, which forces two disjoint vertical behaviors. This separation forces a decomposition into two independent subproblems split along a horizontal fault line, contradicting the requirement that $H_1$ and $H_2$ each be covered exactly once by rectangles that must simultaneously respect the reduced boundary structure.

Hence no selection of rectangles satisfies all cell-items together with all line-items.

Therefore the modified exact cover instance for $3\times 5$ has no solution.

The number of reduced $3\times 5$ patterns that are faultfree is therefore

$$ \boxed{0}. $$$