TAOCP 7.2.2.1 Exercise 253

Exercise 253. ▶ [21] [21] Extend Algorithm Z so that it reports the total number of solutions.

Verified: yes
Solve time: 1m32s

Solution

Let $Z$ denote Algorithm Z as in Section 7.2.2.1, with step $Z3$ performing a recursive exploration over the set of remaining items and active options. Let $\mathcal{S}(P)$ denote the number of solutions of a residual exact cover problem $P$ defined by the current active items and options.

The extension required is a transformation of $Z$ from a decision procedure into an enumerative procedure that accumulates contributions from all successful terminal states. The invariant structure of the search tree is unchanged; only the return behavior at terminal and branching points is modified so that every complete covering contributes exactly one unit to a global total.

At a residual problem $P$, if no items remain to be covered, the construction has produced a complete exact cover. The recursion therefore contributes $\mathcal{S}(P)=1$ in this case, since exactly one completed solution corresponds to the current branch of choices leading to $P$.

If some item $i$ has no active options, the covering constraint for $i$ cannot be satisfied in any continuation of the current partial solution. Every extension of the current branch fails, so $\mathcal{S}(P)=0$.

In the nonterminal case, Algorithm Z selects an item $i=\texttt{SLINK}(0)$ or by the original selection rule of step $Z1$ and $Z2$, and iterates over all options $O$ that contain $i$. Each such option is temporarily chosen, and all items $j \in O$ are covered in sequence using the covering mechanism defined in (12). This produces a residual problem $P_{O}$ for each option $O$.

Each branch contributes independently to the total number of solutions because the covering operation removes all conflicts involving $i$, and different choices of $O$ produce disjoint search subtrees. The total number of solutions is therefore the sum of contributions of all branches.

The recursive definition becomes

$$ \mathcal{S}(P) = \begin{cases} 1, & \text{if no items remain}, \ 0, & \text{if some item has no active options}, \ \sum_{O \ni i} \mathcal{S}(P_{O}), & \text{otherwise}, \end{cases} $$

where $i$ is the item chosen at step $Z3$ and $P_{O}$ denotes the residual problem obtained after selecting option $O$ and performing all cover operations required by (9) and (12).

To implement this inside Algorithm Z, introduce a variable $t$ that accumulates the number of solutions returned by recursive calls. At each invocation of $Z$, initialize $t \leftarrow 0$. When the terminal success condition is reached, return $1$ to the caller. When the failure condition is reached, return $0$. In the branching case, for each option $O$ involving the chosen item $i$, perform the standard cover sequence, then evaluate the recursive call and add its return value to $t$, then restore the structure using uncover operations so that subsequent options are evaluated in the same original state.

Each iteration over an option $O$ contributes exactly one recursive evaluation after the cover operations and exactly one restoration via uncover, preserving correctness of subsequent iterations. The value returned by the invocation of Algorithm Z is the accumulated sum $t$, which equals the total number of exact covers of the original instance.

The modification preserves the correctness of the original search because every solution corresponds to exactly one maximal path from the root call to a terminal state where all items have been covered, and every such path is counted exactly once when it reaches the base case. This completes the construction of an exact counting version of Algorithm Z. ∎