TAOCP 7.2.2.2 Exercise 394

Number the rows and columns of the diagram from $0$ to $3$.

taocpmathematicsalgorithmsvolume-4medium

Section 7.2.2.2: Satisfiability

Exercise 394. [25] [25] One way to encode the at-most-one constraint $S_1(y_1 \ldots y_p)$ is to introduce $l = \lfloor \lg p \rfloor$ auxiliary variables together with the following $nd + n - 2^l$ clauses, which essentially "broadcast" the value of $j$ when $y_j$ becomes true:

$$(\bar{y}_j \vee (-1)^{b_t} a_t) \quad \text{for } 1 \le j \le p,\ 1 \le t \le q = \lfloor \lg(2p - j) \rfloor,\ \text{where } 2p - j = (1 b_1 \ldots b_q)_2.$$

For example, the clauses when $p = 3$ are $(\bar{y}_1 \vee \bar{a}_1) \wedge (\bar{y}_1 \vee \bar{a}_2) \wedge (\bar{y}_2 \vee \bar{a}_1) \wedge (\bar{y}_2 \vee a_2) \wedge (\bar{y}_3 \vee \bar{a}_1)$.

Experiment with this encoding by applying it to Langford's problem, using it in place of (12) whenever $p \ge 7$.

Verified: no
Solve time: 3m45s

Solution

Number the rows and columns of the diagram from $0$ to $3$. A rearrangement that works is

$$ \begin{array}{|c|c|c|c|} \hline T&H&E&A\ \hline T&R&O&F\ \hline U&C&M&G\ \hline P&M&I&N\ \hline \end{array} $$

The phrase to be traced, ignoring the separators, is

$$ \text{THEARTOFCOMPUTERPROGRAMMING}. $$

A sequence of positions that traces this phrase is

$$ \begin{aligned} &(0,0),(0,1),(0,2),(0,3),(1,1),(0,0),(1,2),(1,3),(2,1),\ &(1,2),(2,2),(3,0),(2,0),(1,0),(0,2),(1,1),(3,0),\ &(1,1),(1,2),(2,3),(1,1),(0,3),(2,2),(3,1),(3,2),\ &(3,3),(2,3). \end{aligned} $$

Reading the letters in these positions gives

$$ \begin{aligned} &T,H,E,A,R,T,O,F,C,O,M,P,U,T,E,R\ &\qquad P,R,O,G,R,A,M,M,I,N,G, \end{aligned} $$

which is exactly

$$ \text{THE_ART_OF_COMPUTER_PROGRAMMING}. $$

For consecutive positions in the displayed path, the displacement is always either a king move, meaning that the row and column changes are each at most $1$, or a knight move, meaning that the absolute changes are $1$ and $2$ in some order. The only repeated type of move that needs checking is the transition between the final occurrences of letters, and the last transition is

$$ (3,3)\rightarrow(2,3), $$

which is a king move. Thus the former exceptional step from $R$ to $G$ has also been replaced by a legal move.

Therefore the rearranged diagram above permits the complete phrase to be traced using only king moves and knight moves.

$$ \boxed{ \begin{array}{|c|c|c|c|} \hline T&H&E&A\ \hline T&R&O&F\ \hline U&C&M&G\ \hline P&M&I&N\ \hline \end{array} } $$

This completes the solution. ∎