## 5.330. range_ctr

Origin

Arithmetic constraint.

Constraint

$\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi },\mathrm{\pi }\right)$

Arguments
 $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi ²\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi }$ $\mathrm{\pi \pi \pi \pi }$
Restrictions
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>0$ $\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)$
Purpose

Constraint the difference between the maximum value and the minimum value of a set of domain variables. More precisely, let $\mathrm{\pi \pi °\pi ½\pi Ά\pi ΄}$ denote the difference between the largest and the smallest variables of the $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ collection plus one. Enforce the following constraint to hold: $\mathrm{\pi \pi °\pi ½\pi Ά\pi ΄}\mathrm{\pi ²\pi \pi }\mathrm{\pi }$.

Example
$\left(β©1,9,4βͺ,=,9\right)$

The $\mathrm{\pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi }$ constraint holds since $max\left(1,9,4\right)-min\left(1,9,4\right)+1$ is equal (i.e.,Β $\mathrm{\pi ²\pi \pi }$ is set to $=$) to its last argument $\mathrm{\pi }=9$.

Typical
 $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi ²\pi \pi }\beta \left[=,<,\beta ₯,>,\beta €\right]$
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.

• One and the same constant can be added to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Arg. properties
• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi ²\pi \pi }\beta \left[<,\beta €\right]$.

• Extensible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi ²\pi \pi }\beta \left[\beta ₯,>\right]$.

Used in
Keywords
Arc input(s)

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

Arc generator
$\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi ΄}$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)\mathrm{\pi ²\pi \pi }\mathrm{\pi }$

Graph model

Since we want to keep all the vertices of the initial graph we use the $\mathrm{\pi \pi Έ\pi Ώ\pi Ή}$ arc generator together with the $\mathrm{\pi \pi \pi \pi ΄}$ arc constraint. This predefined arc constraint always holds.

PartsΒ (A) andΒ (B) of FigureΒ 5.330.2 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi \pi ΄}$ arc constraint both graphs are identical.