## 5.285. nset_of_consecutive_values

 DESCRIPTION LINKS GRAPH
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
 $\left(2,β©3,1,7,1,1,2,8βͺ\right)$ $\left(7,β©3,1,5,7,9,11,13βͺ\right)$ $\left(1,β©3,3,3,3,3,3,3βͺ\right)$

In the first 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 corresponding $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi ½}=2$ is set to the number of sets of consecutive values.

Typical
 $\mathrm{\pi ½}>1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
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

Functional dependency: $\mathrm{\pi ½}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

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.

Counting
 Length ($n$) 2 3 4 5 6 7 8 Solutions 9 64 625 7776 117649 2097152 43046721

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

Length ($n$)2345678
Total9646257776117649209715243046721
 Parameter value

17342171716161591763662187637
223037247406501096906615695624
3--3613203492084252019989900
4----15601092005047560
5------126000

Solution count for $\mathrm{\pi \pi \pi \pi }_\mathrm{\pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$: domains $0..n$

See also
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 5.285.1 respectively show the initial and final graph associated with the first example of 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.