TAOCP 7.2.2.1 Exercise 413

Section 7.2.2.1: Dancing Links

Exercise 413. [30] [30] Explain how to modify Algorithm C so that the construction of exercise 412 will produce only the true "single-loop" solutions. Your modified algorithm shouldn't be specific to slitherlink; it should apply also to masyu and other loop-discovery puzzles.

Verified: yes
Solve time: 1m30s

Setup

In the construction of exercise 412, the vertices of the slitherlink grid are represented by items that enforce the local degree condition. A selected option corresponds to a tile centered at a vertex, and an exact cover solution gives a collection of tiles whose union is a weak solution: every used vertex has degree $2$, and every unused vertex has degree $0$. The remaining possibility is that the selected edges form several disjoint cycles.

Algorithm C enumerates the exact covers of the constructed XCC instance. The required modification is to reject partial exact covers that already contain a completed loop while some required edges remain unaccounted for. The modification must depend only on the current partial solution, not on the particular puzzle, so that it also applies to masyu and other loop-discovery problems.

Let the currently chosen options in Algorithm C be $S$. Regard the union of the edges represented by $S$ as a graph $G(S)$. Every vertex of $G(S)$ has degree $0$, $1$, or $2$ during the search. A vertex of degree $1$ is an open endpoint of the unfinished path construction, and a component whose vertices all have degree $2$ is already a closed loop.

The goal is to alter Algorithm C so that a solution is accepted only when $G(S)$ consists of one cycle containing all selected edges.

Solution

Add to Algorithm C a test immediately before the successful termination condition. In the notation of Algorithm C, the original termination occurs when there are no remaining primary items. At that point, instead of appending the current choice to the list of solutions immediately, perform the following additional test.

Maintain, together with the current partial solution, the connected components of $G(S)$. For each component, maintain whether it contains a vertex of degree $1$. A component is called closed if it contains no vertex of degree $1$. Since every vertex in a closed component has degree $2$, every closed component is a cycle.

When Algorithm C reaches a state in which all required items have been covered, accept the solution if and only if exactly one component of $G(S)$ is closed and that component contains every edge of $G(S)$. Otherwise reject the solution.

The test can be incorporated earlier in the recursion. After each option is appended to the partial solution, inspect the component containing the newly added edges. If this component becomes closed while some uncovered item still requires future choices, abandon that branch. A closed component can never be joined to another component later, because all of its vertices already have degree $2$, so no extension can turn several loops into one loop.

The correctness of the modification follows from two directions. Suppose Algorithm C accepts a solution. The termination test guarantees that every required item is covered, so the original XCC construction guarantees that every vertex has degree $0$ or $2$. The acceptance condition permits exactly one closed component containing all selected edges. Hence the selected edges form one cycle, which is a true single-loop solution.

Conversely, suppose a true single-loop solution exists. Every partial choice along the path to this solution is contained in that single cycle. Before the final option of the cycle is selected, the selected edges form a path or a collection of partial paths, so no closed component is detected. After the final option is selected, the graph has exactly one closed component containing all selected edges, so the acceptance test succeeds. Therefore every true single-loop solution generated by the original XCC search is retained.

The same argument does not depend on the meaning of the items or options. It uses only the property that the desired objects are connected cycles with no premature closed components. Therefore the modification applies equally to slitherlink, masyu, and other loop-discovery puzzles.

Verification

A weak solution with two loops produces two closed components at the termination point. The acceptance test rejects it because the number of closed components is not one.

A weak solution containing one loop together with isolated unused vertices produces exactly one closed component in the graph of selected edges. The acceptance test accepts it, because the unused vertices are not part of the selected edge set.

A partial branch that has already completed a loop but still has uncovered items cannot become a single-loop solution after further choices, because later choices can only add edges to other components or violate the degree restrictions. The early rejection therefore removes only branches that cannot lead to valid answers.

Thus the modified Algorithm C enumerates precisely the single-loop solutions.

This completes the proof.

∎

Notes

The modification is a general connectivity constraint added on top of the exact-cover constraints. The XCC formulation enforces local conditions, while the additional component test enforces the global condition that all selected edges belong to one connected cycle. This separation allows the same search mechanism to solve different loop puzzles without changing the exact-cover encoding.