TAOCP 7.2.2.1 Exercise 412

Section 7.2.2.1: Dancing Links

Exercise 412. ▶ [22] [22] A "weak solution" to a slitherlink diagram is a set of edges that obeys the numeric constraints, and touches every vertex of the grid either twice or not at all; but it may form arbitrarily many loops. For example, the diagram of exercise 410(i) has six weak solutions, three of which are shown in 410(iii), (iv), and (v).

Show that there's a nice way to obtain all the weak solutions of a given diagram, by formulating a suitable XCC problem. Hint: Think of the edges as constructed from tiles centered at the vertices, and use even/odd coordinates as in answer 133.

Verified: no
Solve time: 2m

Edit

Solution

Use the coordinate convention suggested by the hint. Put the grid vertices at points with even coordinates, so the center of a cell has odd coordinates. A tile centered at a grid vertex describes the local choice of edges incident with that vertex.

At a vertex there are exactly seven possible choices that satisfy the weak-solution condition: either no incident edge is chosen, or exactly two of the four incident edges are chosen. Thus the possible tiles are

[ \varnothing , ]

and the six two-edge choices

[ {N,E},\quad {E,S},\quad {S,W},\quad {W,N}, \quad {N,S},\quad {E,W}. ]

We construct an XCC instance whose options are these vertex-centered tiles.

The items of the XCC instance are of three kinds.

First, for every grid vertex (v), introduce a primary item (V_v). Every tile centered at (v) contains (V_v). Therefore any exact cover must choose exactly one tile at every vertex.

Second, for every grid edge (e), introduce an item (E_e) with required multiplicity (2). A tile contains (E_e) exactly when the tile uses that edge. Since every grid edge has two endpoints, an edge is selected precisely when both endpoint tiles contain (E_e). Hence the edge constraints ensure that the local tile choices combine to form a well-defined set of grid edges.

Third, for every cell (c) with clue (k), introduce an item (C_c) with required multiplicity

[ 2k . ]

A tile contains (C_c) exactly when one of its incident edges is a boundary edge of (c). If a boundary edge of (c) is selected, then both endpoint tiles of that edge contain (C_c). Consequently, every selected boundary edge contributes exactly two occurrences of (C_c). Therefore the requirement that (C_c) occur (2k) times is exactly equivalent to requiring that exactly (k) boundary edges of (c) be selected.

This gives a finite XCC instance, because the grid has finitely many vertices, edges, and cells, and each vertex contributes only seven possible options.

Now suppose a weak solution of the slitherlink diagram is given. At every vertex, choose the tile consisting of the two incident solution edges if the vertex is touched, and choose the empty tile otherwise. The vertex items are covered exactly once. Each selected edge has two endpoints, so its edge item is covered exactly twice, while an unselected edge item is not covered. Finally, a cell with (k) solution edges has (2k) endpoint incidences from those edges, so its cell item is covered exactly (2k) times. Hence the chosen tiles form an XCC solution.

Conversely, suppose an XCC solution is given. The vertex items force exactly one tile to be chosen at each grid vertex. The edge-item constraints imply that an edge is either chosen at both endpoints or at neither endpoint. Therefore the chosen tiles define a set of grid edges. Because every nonempty vertex tile contains exactly two incident edges, every vertex in this edge set has degree (2), and every other vertex has degree (0). Thus the vertex condition of a weak solution is satisfied.

For any cell (c), the cell item (C_c) occurs (2k) times, where (k) is the clue of (c). Each selected boundary edge contributes exactly two occurrences, one from each endpoint tile. Therefore (c) has exactly (k) selected boundary edges. Hence all numeric constraints are satisfied.

The two transformations are inverse to each other: a weak solution uniquely determines the chosen vertex tiles, and an XCC solution uniquely determines the corresponding set of edges. Therefore the weak solutions of the slitherlink diagram are in one-to-one correspondence with the solutions of this XCC instance.

[ \boxed{\text{The XCC formulation enumerates exactly all weak solutions.}} ]