5.259. orth_link_ori_siz_end

DESCRIPTIONLINKSGRAPH
Origin

Used by several constraints between orthotopes

Constraint

πš˜πš›πšπš‘_πš•πš’πš—πš”_πš˜πš›πš’_πšœπš’πš£_πšŽπš—πš(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄)

Argument
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš˜πš›πš’-πšπšŸπšŠπš›,πšœπš’πš£-πšπšŸπšŠπš›,πšŽπš—πš-πšπšŸπšŠπš›)
Restrictions
|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|>0
πš›πšŽπššπšžπš’πš›πšŽ_𝚊𝚝_πš•πšŽπšŠπšœπš(2,π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄,[πš˜πš›πš’,πšœπš’πš£,πšŽπš—πš])
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πšœπš’πš£β‰₯0
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πš˜πš›πš’β‰€π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πšŽπš—πš
Purpose

Enforce for each item of the π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄ collection the constraint πš˜πš›πš’+πšœπš’πš£=πšŽπš—πš.

Example
πš˜πš›πš’-2 πšœπš’πš£-2 πšŽπš—πš-4,πš˜πš›πš’-1 πšœπš’πš£-3 πšŽπš—πš-4

The πš˜πš›πšπš‘_πš•πš’πš—πš”_πš˜πš›πš’_πšœπš’πš£_πšŽπš—πš constraint holds since the two items βŒ©πš˜πš›πš’-2 πšœπš’πš£-2 πšŽπš—πš-4βŒͺ and βŒ©πš˜πš›πš’-1 πšœπš’πš£-3 πšŽπš—πš-4βŒͺ respectively verify the conditions 2+2=4 and 1+3=4.

Symmetries
  • Items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄ are permutable.

  • One and the same constant can be added to the πš˜πš›πš’ and πšŽπš—πš attributes of all items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.

  • One and the same constant can be added to the πšœπš’πš£ and πšŽπš—πš attributes of all items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.

Usage

Used in the Arc constraint(s) slot for defining some constraints like πšπš’πšπšπš—, πš™πš•πšŠπšŒπšŽ_πš’πš—_πš™πš’πš›πšŠπš–πš’πš or πš˜πš›πšπš‘πšœ_πšŠπš›πšŽ_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš.

Used in

πšπš’πšπšπš—, πš˜πš›πšπš‘_πš˜πš—_πšπš‘πšŽ_πšπš›πš˜πšžπš—πš, πš˜πš›πšπš‘_πš˜πš—_πšπš˜πš™_𝚘𝚏_πš˜πš›πšπš‘, πš˜πš›πšπš‘πšœ_πšŠπš›πšŽ_πšŒπš˜πš—πš—πšŽπšŒπšπšŽπš, 𝚝𝚠𝚘_πš˜πš›πšπš‘_πšŠπš›πšŽ_πš’πš—_πšŒπš˜πš—πšπšŠπšŒπš, 𝚝𝚠𝚘_πš˜πš›πšπš‘_πšŒπš˜πš•πšžπš–πš—, 𝚝𝚠𝚘_πš˜πš›πšπš‘_𝚍𝚘_πš—πš˜πš_πš˜πšŸπšŽπš›πš•πšŠπš™, 𝚝𝚠𝚘_πš˜πš›πšπš‘_πš’πš—πšŒπš•πšžπšπšŽ.

Keywords

constraint type: decomposition.

geometry: orthotope.

Arc input(s)

π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄

Arc generator
π‘†πΈπΏπΉβ†¦πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ)

Arc arity
Arc constraint(s)
πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ.πš˜πš›πš’+πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ.πšœπš’πš£=πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽ.πšŽπš—πš
Graph property(ies)
𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.259.1 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐀𝐑𝐂 graph property, the loops of the final graph are stressed in bold.

Figure 5.259.1. Initial and final graph of the πš˜πš›πšπš‘_πš•πš’πš—πš”_πš˜πš›πš’_πšœπš’πš£_πšŽπš—πš constraint
ctrs/orth_link_ori_siz_endActrs/orth_link_ori_siz_endB
(a) (b)
Signature

Since we use the 𝑆𝐸𝐿𝐹 arc generator on the π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄ collection the number of arcs of the initial graph is equal to |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|. Therefore the maximum number of arcs of the final graph is also equal to |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|. For this reason we can rewrite the graph property 𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄| to 𝐍𝐀𝐑𝐂β‰₯|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄| and simplify 𝐍𝐀𝐑𝐂 Β― Μ² to 𝐍𝐀𝐑𝐂 Β―.