5.241. nvalues_except_0

DESCRIPTIONLINKSGRAPH
Origin

Derived from πš—πšŸπšŠπš•πšžπšŽπšœ.

Constraint

πš—πšŸπšŠπš•πšžπšŽπšœ_πšŽπš‘πšŒπšŽπš™πš_0(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšπ™΄π™»π™Ύπ™Ώ,π™»π™Έπ™Όπ™Έπšƒ)

Arguments
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›-πšπšŸπšŠπš›)
πšπ™΄π™»π™Ύπ™ΏπšŠπšπš˜πš–
π™»π™Έπ™Όπ™ΈπšƒπšπšŸπšŠπš›
Restrictions
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
πšπ™΄π™»π™Ύπ™Ώβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

Let N be the number of distinct values, different from 0, assigned to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection. Enforce condition N πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ to hold.

Example
πšŸπšŠπš›-4,πšŸπšŠπš›-5,πšŸπšŠπš›-5,πšŸπšŠπš›-4,πšŸπšŠπš›-0,πšŸπšŠπš›-1,=,3

The πš—πšŸπšŠπš•πšžπšŽπšœ_πšŽπš‘πšŒπšŽπš™πš_0 constraint holds since the number of distinct values, different from 0, occurring within the collection 〈4,5,5,4,0,1βŒͺ is equal (i.e.,Β πšπ™΄π™»π™Ύπ™Ώ is set toΒ =) to its third argument π™»π™Έπ™Όπ™Έπšƒ=3.

Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • All occurrences of two distinct values of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› that are both different from 0 can be swapped; all occurrences of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› that is different from 0 can be renamed to any unused value that is also different from 0.

Reformulation

The πš—πšŸπšŠπš•πšžπšŽπšœ_πšŽπš‘πšŒπšŽπš™πš_0(〈V 1 ,V 2 ,...,V |πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚| βŒͺ, πšπ™΄π™»π™Ύπ™Ώ ,π™»π™Έπ™Όπ™Έπšƒ) constraint can be expressed in term of the conjunction πš—πšŸπšŠπš•πšžπšŽ(𝑁𝑉1,〈0,V 1 ,V 2 ,...,V |πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚| βŒͺ) ∧ 𝑁𝑉1-1 πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ.

Used in

πšŒπš’πšŒπš•πšŽ_πš˜πš›_πšŠπšŒπšŒπšŽπšœπšœπš’πš‹πš’πš•πš’πšπš’.

See also

common keyword: πšŠπšœπšœπš’πšπš—_πšŠπš—πš_πš—πšŸπšŠπš•πšžπšŽπšœΒ (number of distinct values),

πš—πšŸπšŠπš•πšžπšŽ, πš—πšŸπšŠπš•πšžπšŽπšœΒ (counting constraint,number of distinct values).

Keywords

characteristic of a constraint: joker value.

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component.

modelling: number of distinct values.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
β€’ πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›β‰ 0
β€’ πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
𝐍𝐒𝐂𝐂 πšπ™΄π™»π™Ύπ™Ώ π™»π™Έπ™Όπ™Έπšƒ

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.241.1 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐒𝐂𝐂 graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a value distinct from 0 that is assigned to some variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection. Beside value 0, the 3 following values 1, 4 and 5 are assigned to the variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection.

Figure 5.241.1. Initial and final graph of the πš—πšŸπšŠπš•πšžπšŽπšœ_πšŽπš‘πšŒπšŽπš™πš_0 constraint
ctrs/nvalues_except_0Actrs/nvalues_except_0B
(a) (b)