5.282. not_all_equal

Origin

CHIP

Constraint

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Argument
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$
Purpose

The variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ should take more than a single value.

Example
$\left(β©3,1,3,3,3βͺ\right)$

The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint holds since the collection $\beta ©3,1,3,3,3\beta ͺ$ involves more than one value (i.e.,Β values 1 and 3).

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>2$ $\mathrm{\pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>2$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ are permutable.

• All occurrences of two distinct values of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be swapped; all occurrences of a value of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }$ can be renamed to any unused value.

Arg. properties

Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Algorithm

If the intersection of the domains of the variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection is empty the $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint is entailed. Otherwise, when only a single variable $V$ remains not fixed, remove the unique value (unique since the constraint is not entailed) taken by the other variables from the domain of $V$.

Reformulation

The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$ constraint can be expressed as $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$$\left(2,\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 6 60 620 7770 117642 2097144 43046712

Number of solutions for $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$: domains $0..n$

Systems

rel in Gecode.

generalisation: $\mathrm{\pi \pi \pi \pi \pi \pi }$Β (introduce a variable for counting the number of distinct values).

specialisation: $\mathrm{\pi \pi \pi }$Β (when go down to two variables).

Keywords
Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi Ά\pi Ώ\pi Ό\pi \pi \pi Έ}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }=\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$>1$

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.282.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi }$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a value that is assigned to some variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection. The $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ holds since the final graph contains more than one strongly connected component.

Automaton

FigureΒ 5.282.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi }$ constraint. To each pair of consecutive variables $\left({\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi  \pi °\pi }}_{i+1}\right)$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a signature variable ${S}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${S}_{i}$: ${\mathrm{\pi  \pi °\pi }}_{i}={\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}$.