5.220. minimum_modulo

DESCRIPTIONLINKSGRAPH
Origin

Derived from πš–πš’πš—πš’πš–πšžπš–.

Constraint

πš–πš’πš—πš’πš–πšžπš–_πš–πš˜πšπšžπš•πš˜(𝙼𝙸𝙽,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,𝙼)

Arguments
π™Όπ™Έπ™½π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™Όπš’πš—πš
Restrictions
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>0
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

𝙼𝙸𝙽 is a minimum value of the collection of domain variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ according to the following partial ordering: (X mod 𝙼)<(Y mod 𝙼).

Example
(6,9,1,7,6,5,3)
(9,9,1,7,6,5,3)

The πš–πš’πš—πš’πš–πšžπš–_πš–πš˜πšπšžπš•πš˜ constraints hold since 𝙼𝙸𝙽 is respectively set to values 6 and 9, where 6 mod 3=0 and 9 mod 3=0 are both less than or equal to all the expressions 9 mod 3=0, 1 mod 3=1, 7 mod 3=1, 6 mod 3=0, and 5 mod 3=2.

Symmetry

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

See also

comparison swapped: πš–πšŠπš‘πš’πš–πšžπš–_πš–πš˜πšπšžπš•πš˜.

specialisation: πš–πš’πš—πš’πš–πšžπ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).

Keywords

characteristic of a constraint: modulo, maxint, minimum.

constraint type: order constraint.

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š”πšŽπš’=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πš”πšŽπš’,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)
πŽπ‘πƒπ„π‘(0,π™Όπ™°πš‡π™Έπ™½πšƒ,πšŸπšŠπš›)=𝙼𝙸𝙽

Graph model

We use a similar definition that the one that was utilised for the πš–πš’πš—πš’πš–πšžπš– constraint. Within the arc constraint we replace the condition X<Y by the condition (X mod 𝙼)<(Y mod 𝙼).

PartsΒ (A) andΒ (B) of FigureΒ 5.220.1 respectively show the initial and final graph associated with the second example of the Example slot. Since we use the πŽπ‘πƒπ„π‘ graph property, the vertex of rank 0 (without considering the loops) associated with value 9 is outlined with a thick circle.

Figure 5.220.1. Initial and final graph of the πš–πš’πš—πš’πš–πšžπš–_πš–πš˜πšπšžπš•πš˜ constraint
ctrs/minimum_moduloActrs/minimum_moduloB
(a) (b)