TAOCP 7.2.2.1 Exercise 413
Let Algorithm C be the exact cover procedure derived from Algorithm X in Section 7.
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: 1m38s
Setup
Let Algorithm C be the exact cover procedure derived from Algorithm X in Section 7.2.2.1, applied to the formulation of exercise 412 in which a slitherlink or related loop puzzle is represented as an XCC instance whose solutions are weak solutions, meaning that selected options correspond to edges satisfying local degree constraints but may form multiple disjoint cycles.
In that construction, each option corresponds to selecting a local configuration that contributes edges to a grid graph. A weak solution is therefore a set of edges such that every vertex has degree $0$ or $2$. The required modification is to restrict attention to those weak solutions in which the edges form exactly one cycle containing every vertex of degree $2$.
The task is to modify Algorithm C so that all partial and completed solutions satisfy this single-loop condition during the backtracking process, without restricting the method to slitherlink alone.
Solution
Augment the XCC state maintained by Algorithm C with additional structure on the underlying graph induced by currently chosen options. Let $G$ denote the graph whose vertices are those incident to selected edges and whose edges correspond to selected options. Maintain, in addition to the dancing-links structure, two auxiliary data fields.
For each vertex $v$, maintain $\deg(v)$, the number of selected edges incident to $v$, together with the invariant $\deg(v)\in{0,1,2}$. For each vertex $v$ with $\deg(v)\ge 1$, maintain a union-find structure $\mathrm{Find}(v)$ representing connected components of $G$ induced by selected edges.
When an option $O$ is considered at a recursive node of Algorithm C, interpret $O$ as a set of edges, each contributing two endpoints. The modification concerns the acceptance test for $O$ before the cover operations are committed.
Let $O$ contain edges $(u_1,v_1),\dots,(u_t,v_t)$. For each $k$ with $1\le k\le t$, the algorithm performs a feasibility check before executing the standard cover and hide operations of Algorithm C.
First, for each endpoint $x\in{u_k,v_k}$, the tentative degree after selection is computed as $\deg(x)+1$. If this value exceeds $2$, the option $O$ is rejected immediately, since no completion of a simple loop can restore feasibility once a vertex exceeds degree $2$.
Second, consider the effect of joining endpoints in the union-find structure. If $\mathrm{Find}(u_k)=\mathrm{Find}(v_k)$ holds, then adding $(u_k,v_k)$ creates a cycle inside the currently selected edges. Such a cycle is permitted only in the case that the resulting partial solution is already complete in the sense of Algorithm C, meaning that no items remain uncovered and every vertex in the solution has degree exactly $2$. If any item remains uncovered at this stage, the option $O$ is rejected.
If the option passes all checks, Algorithm C proceeds exactly as in the original formulation: the cover operation is executed for all items in $O$, and the auxiliary structures are updated by incrementing degrees and performing union operations for each edge endpoint pair.
At backtracking time, when uncovering an item and restoring a previously removed option, the same updates are reversed: degrees are decremented, and union-find structure is not explicitly undone in its standard form; instead, a rollback mechanism is required, implemented by storing the sequence of union operations on a stack so that components can be restored in reverse order.
The recursion terminates successfully only when Algorithm C reaches a state in which all items are covered and the auxiliary condition $\deg(v)=2$ holds for every vertex participating in the solution. At that moment, the selected edges form a $2$-regular connected graph, hence a single cycle.
Verification
Consider any output produced by the modified algorithm. Every vertex in the selected configuration has degree exactly $2$, since any option violating the degree bound is rejected and the termination condition requires full saturation. Therefore the selected edges form a disjoint union of cycles in the underlying graph.
Suppose two distinct cycles occur in a completed solution. Then during construction there must exist a stage at which the second cycle becomes closed without connecting the remaining vertices. At the moment of closure, two endpoints of a newly added edge lie in the same union-find component while uncovered items still remain. The modification rejects exactly this situation, hence no completed search branch can contain more than one cycle.
Conversely, let a valid single-loop solution exist. In such a solution every vertex has degree $2$, and the graph is connected. Every prefix of the construction preserves degree at most $2$, and no edge ever connects two vertices already connected in the partial solution before the final step, except when the last edge closes the unique cycle after all vertices have been incorporated. The acceptance test permits this final closure because no items remain uncovered at that stage. Therefore every valid single-cycle solution survives the pruning.
The modified algorithm therefore enumerates exactly the weak solutions satisfying both degree constraints and global connectivity as a single cycle.
This completes the proof. ∎
Notes
The modification is not specific to slitherlink. Any loop-discovery problem encoded as XCC in which solutions correspond to $2$-regular subgraphs can be restricted to single-cycle solutions by adding a connectivity structure that forbids premature cycle closure. The union-find device is convenient, but any rollback-capable connectivity maintenance structure compatible with backtracking can be used within Algorithm C without altering the dancing-links core.