5.226. nequivalence

DESCRIPTIONLINKSGRAPH
Origin

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

Constraint

πš—πšŽπššπšžπš’πšŸπšŠπš•πšŽπš—πšŒπšŽ(π™½π™΄πš€πš„π™Έπš…,𝙼,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)

Arguments
π™½π™΄πš€πš„π™Έπš…πšπšŸπšŠπš›
π™Όπš’πš—π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›)
Restrictions
π™½π™΄πš€πš„π™Έπš…β‰₯πš–πš’πš—(1,|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|)
π™½π™΄πš€πš„π™Έπš…β‰€πš–πš’πš—(𝙼,|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|)
𝙼>0
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›)
Purpose

π™½π™΄πš€πš„π™Έπš… is the number of distinct rests obtained by dividing the variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ by 𝙼.

Example
2,3,πšŸπšŠπš›-3,πšŸπšŠπš›-2,πšŸπšŠπš›-5,πšŸπšŠπš›-6,πšŸπšŠπš›-15,πšŸπšŠπš›-3,πšŸπšŠπš›-3

Since the expressions 3 mod 3=0, 2 mod 3=2, 5 mod 3=2, 6 mod 3=0, 15 mod 3=0, 3 mod 3=0, and 3 mod 3=0 involve two distinct values (values 0 and 2), the first argument π™½π™΄πš€πš„π™Έπš… of the πš—πšŽπššπšžπš’πšŸπšŠπš•πšŽπš—πšŒπšŽ constraint is set to value 2.

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

  • An occurrence of a value u of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be replaced by any other value v such that v is congruent to u modulo 𝙼.

Algorithm

Since constraints X=Y and X≑Y( mod M) are similar, one should also use a similar algorithm as the one [Beldiceanu01], [BeldiceanuCarlssonThiel02] provided for constraint πš—πšŸπšŠπš•πšžπšŽ.

See also

related: πš—πšŒπš•πšŠπšœπšœΒ (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 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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šŽ mod πšŒπš˜πš—πšœπšπšŠπ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šŽ mod πšŒπš˜πš—πšœπšπšŠπ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šŽ mod πšŒπš˜πš—πšœπšπšŠπš—πš replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

Keywords

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šŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš› mod 𝙼=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš› mod 𝙼
Graph property(ies)
𝐍𝐒𝐂𝐂=π™½π™΄πš€πš„π™Έπš…

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.226.1 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 one equivalence class: We have two equivalence classes that respectively correspond to values {3,6,15} and {2,5}.

Figure 5.226.1. Initial and final graph of the πš—πšŽπššπšžπš’πšŸπšŠπš•πšŽπš—πšŒπšŽ constraint
ctrs/nequivalenceActrs/nequivalenceB
(a) (b)