5.223. nclass

DESCRIPTIONLINKSGRAPH
Origin

Derived from πš—πšŸπšŠπš•πšžπšŽ.

Constraint

πš—πšŒπš•πšŠπšœπšœ(π™½π™²π™»π™°πš‚πš‚,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)

Type
πš…π™°π™»πš„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š—πš)
Arguments
π™½π™²π™»π™°πš‚πš‚πšπšŸπšŠπš›
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™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš™-πš…π™°π™»πš„π™΄πš‚)
Restrictions
|πš…π™°π™»πš„π™΄πš‚|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
πšπš’πšœπšπš’πš—πšŒπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
π™½π™²π™»π™°πš‚πš‚β‰₯0
π™½π™²π™»π™°πš‚πš‚β‰€πš–πš’πš—(|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|,|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|)
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šŽπššπšžπš’πš›πšŽπš(π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚,πš™)
|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|β‰₯2
Purpose

Number of partitions of the collection π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ such that at least one value is assigned to at least one variable of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.

Example
2,3,2,7,2,6,πš™-1,3,πš™-4,πš™-2,6

Note that the values of 〈3,2,7,2,6βŒͺ occur within partitions πš™-〈1,3βŒͺ and πš™-〈2,6βŒͺ but not within πš™-〈4βŒͺ. Consequently, the πš—πšŒπš•πšŠπšœπšœ constraint holds since its first argument π™½π™²π™»π™°πš‚πš‚ is set to value 2.

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

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ are permutable.

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.πš™ are permutable.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be replaced by any other value that also belongs to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

  • All occurrences of two distinct tuples of values in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› or π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.πš™.πšŸπšŠπš• can be swapped; all occurrences of a tuple of values in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› or π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.πš™.πšŸπšŠπš• can be renamed to any unused tuple of values.

Algorithm

[Beldiceanu01], [BeldiceanuCarlssonThiel02].

See also

related: πš—πšŽπššπšžπš’πšŸπšŠπš•πšŽπš—πšŒπšŽΒ (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ mod πšŒπš˜πš—πšœπšπšŠπš—πš), πš—πš’πš—πšπšŽπš›πšŸπšŠπš•Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ/πšŒπš˜πš—πšœπšπšŠπš—πš), πš—πš™πšŠπš’πš›Β (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πš™πšŠπš’πš› of πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ).

specialisation: πš—πšŸπšŠπš•πšžπšŽΒ (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

used in graph description: πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

Keywords

characteristic of a constraint: partition.

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes.

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)
πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’π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šŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)
Graph property(ies)
𝐍𝐒𝐂𝐂=π™½π™²π™»π™°πš‚πš‚

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.223.1 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 class of values that was assigned to some variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection. je effeL">π1 at je effeL">π1𝙱𝙻πbc_eq_tupleeL">π1𝙱𝙻ection. je effeL">π1 at je effeL">π1π™i>πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚π1π™i>πš…π™°πl>πšπš’πšœππš‚ππš.

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