5.236. npair

DESCRIPTIONLINKSGRAPH
Origin

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

Constraint

πš—πš™πšŠπš’πš›(π™½π™Ώπ™°π™Έπšπš‚,π™Ώπ™°π™Έπšπš‚)

Arguments
π™½π™Ώπ™°π™Έπšπš‚πšπšŸπšŠπš›
π™Ώπ™°π™Έπšπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚑-πšπšŸπšŠπš›,𝚒-πšπšŸπšŠπš›)
Restrictions
π™½π™Ώπ™°π™Έπšπš‚β‰₯πš–πš’πš—(1,|π™Ώπ™°π™Έπšπš‚|)
π™½π™Ώπ™°π™Έπšπš‚β‰€|π™Ώπ™°π™Έπšπš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ώπ™°π™Έπšπš‚,[𝚑,𝚒])
Purpose

π™½π™Ώπ™°π™Έπšπš‚ is the number of distinct pairs of values assigned to the pairs of variables of the collection π™Ώπ™°π™Έπšπš‚.

Example
2,𝚑-3𝚒-1,𝚑-1𝚒-5,𝚑-3𝚒-1,𝚑-3𝚒-1,𝚑-1𝚒-5

The πš—πš™πšŠπš’πš› constraint holds since its first argument π™½π™Ώπ™°π™Έπšπš‚=2 is set to the number of distinct pairs 〈𝚑-3 𝚒-1βŒͺ and 〈𝚑-1 𝚒-5βŒͺ of its second argument π™Ώπ™°π™Έπšπš‚.

Symmetries
  • Items of π™Ώπ™°π™Έπšπš‚ are permutable.

  • Attributes of π™Ώπ™°π™Έπšπš‚ are permutable w.r.t. permutation (𝚑,𝚒) (permutation applied to all items).

  • All occurrences of two distinct tuples of values of π™½π™Ώπ™°π™Έπšπš‚ can be swapped; all occurrences of a tuple of values of π™½π™Ώπ™°π™Έπšπš‚ can be renamed to any unused tuple of values.

Remark

This is an example of a number of distinct values constraint where there is more than one attribute that is associated with each vertex of the final graph.

See also

related: πš—πšŒπš•πšŠπšœπšœΒ (πš™πšŠπš’πš› of πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ replaced by 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—), πš—πšŽπššπšžπš’πšŸπšŠπš•πšŽπš—πšŒπšŽΒ (πš™πšŠπš’πš› of πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ 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š™πšŠπš’πš› of πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ 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).

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

Keywords

characteristic of a constraint: pair.

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes.

Arc input(s)

π™Ώπ™°π™Έπš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š™πšŠπš’πš›πšœ1,πš™πšŠπš’πš›πšœ2)

Arc arity
Arc constraint(s)
β€’ πš™πšŠπš’πš›πšœ1.𝚑=πš™πšŠπš’πš›πšœ2.𝚑
β€’ πš™πšŠπš’πš›πšœ1.𝚒=πš™πšŠπš’πš›πšœ2.𝚒
Graph property(ies)
𝐍𝐒𝐂𝐂=π™½π™Ώπ™°π™Έπšπš‚

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.236.1 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 pair of values that is assigned to some pairs of variables of the π™Ώπ™°π™Έπšπš‚ collection. In our example we have the following pairs of values: 〈𝚑-3 𝚒-1βŒͺ and 〈𝚑-1 𝚒-5βŒͺ.

Figure 5.236.1. Initial and final graph of the πš—πš™πšŠπš’πš› constraint
ctrs/npairActrs/npairB
(a) (b)