## 5.237. nset_of_consecutive_values

Origin

N.Β Beldiceanu

Constraint

$\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\right)$

Arguments
 $\mathrm{\pi ½}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$
Restrictions
 $\mathrm{\pi ½}\beta ₯1$ $\mathrm{\pi ½}\beta €|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\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

$\mathrm{\pi ½}$ is the number of set of consecutive values used by the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example
$\left(\begin{array}{c}2,β©\begin{array}{c}\mathrm{\pi \pi \pi }-3,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-7,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-1,\hfill \\ \mathrm{\pi \pi \pi }-2,\hfill \\ \mathrm{\pi \pi \pi }-8\hfill \end{array}βͺ\hfill \end{array}\right)$

In the example, the two parts $3,1,1,1,2$ and $7,8$ take respectively their values in the following sets of consecutive values $\left\{1,2,3\right\}$ and $\left\{7,8\right\}$. Consequently, the $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi ½}=2$ is set to the number of sets of consecutive values.

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

Usage

Used for specifying the fact that the values have to be used in a compact way is achieved by setting $\mathrm{\pi ½}$ to 1.

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)
$\mathrm{\pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }\right)\beta €1$
Graph property(ies)
$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½}$

Graph model

Since the arc constraint is symmetric each strongly connected component of the final graph corresponds exactly to one connected component of the final graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.237.1 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 two strongly connected components of the final graph.