TAOCP 7.2.2.2 Exercise 397

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

taocpmathematicsalgorithmsvolume-4medium

Section 7.2.2.2: Satisfiability

Exercise 397. [22] [22] (N. Tamura, 2014.) Suppose $x_0, x_1, \ldots, x_{p-1}$ are integer variables with the range $0 \le x_i < d$, represented in order encoding by Boolean variables $x_i^j = [x_i \ge j]$ for $0 \le i < p$ and $1 \le j < d$. Show that the all-different constraint, "$x_i \ne x_j$ for $0 \le i < j < p$," can be nicely encoded by introducing auxiliary integer variables $y_0$, $y_1, \ldots, y_{d-1}$ with the range $0 \le y_i < p$ representing the order encoding by Boolean variables $y_i^j = [y_i \ge j]$ for $1 \le i \le p$ and $0 \le i < d$, and by including clauses to enforce the condition $x_i = j \implies y_j = i$. Hints analogous to (162) can be given.

Verified: no
Solve time: 3m51s

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 Thus the former exceptional step from $R$ to $G$ has also been replaced by a legal move.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