5.101. diffn_column

DESCRIPTIONLINKSGRAPH
Origin

CHIP: option guillotine cut (column) of πšπš’πšπšπš—.

Constraint

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™΄πš‚,𝙳𝙸𝙼)

Type
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›)
Arguments
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚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™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄)
π™³π™Έπ™Όπš’πš—π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
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚,πš˜πš›πšπš‘)
πšœπšŠπš–πšŽ_πšœπš’πš£πšŽ(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚,πš˜πš›πšπš‘)
𝙳𝙸𝙼>0
𝙳𝙸𝙼≀|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|
πšπš’πšπšπš—(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚)
Purpose

Extension of the generalised multi -dimensional non -overlapping diffn constraint. Holds if, for each pair of orthotopes (O 1 ,O 2 ) the following conditions hold:

  • O 1 and O 2 do not overlap. Two orthotopes do not overlap if there exists at least one dimension where their projections do not overlap.

  • Let P 1 and P 2 respectively denote the projections of O 1 and O 2 in dimension 𝙳𝙸𝙼. If P 1 and P 2 overlap then the size of their intersection is equal to the size of O 1 in dimension 𝙳𝙸𝙼, as well as to the size of O 2 in dimension 𝙳𝙸𝙼.

Example
πš˜πš›πšπš‘-πš˜πš›πš’-1πšœπš’πš£-3πšŽπš—πš-4,πš˜πš›πš’-3πšœπš’πš£-2πšŽπš—πš-5,πš˜πš›πšπš‘-πš˜πš›πš’-9πšœπš’πš£-1πšŽπš—πš-10,πš˜πš›πš’-4πšœπš’πš£-3πšŽπš—πš-7,πš˜πš›πšπš‘-πš˜πš›πš’-4πšœπš’πš£-2πšŽπš—πš-6,πš˜πš›πš’-3πšœπš’πš£-4πšŽπš—πš-7,πš˜πš›πšπš‘-πš˜πš›πš’-1πšœπš’πš£-3πšŽπš—πš-4,πš˜πš›πš’-6πšœπš’πš£-1πšŽπš—πš-7,πš˜πš›πšπš‘-πš˜πš›πš’-6πšœπš’πš£-2πšŽπš—πš-8,πš˜πš›πš’-1πšœπš’πš£-4πšŽπš—πš-5,πš˜πš›πšπš‘-πš˜πš›πš’-10πšœπš’πš£-1πšŽπš—πš-11,πš˜πš›πš’-1πšœπš’πš£-1πšŽπš—πš-2,πš˜πš›πšπš‘-πš˜πš›πš’-9πšœπš’πš£-1πšŽπš—πš-10,πš˜πš›πš’-1πšœπš’πš£-1πšŽπš—πš-2,πš˜πš›πšπš‘-πš˜πš›πš’-6πšœπš’πš£-2πšŽπš—πš-8,πš˜πš›πš’-6πšœπš’πš£-1πšŽπš—πš-7,1

FigureΒ 5.101.1 represents the respective position of the eight rectangles of the example. The coordinates of the leftmost lowest corner of each rectangle are stressed in bold. The πšπš’πšπšπš—_πšŒπš˜πš•πšžπš–πš— constraint holds since (1)Β the eight rectangles do not overlap and since (2)Β when their projection in dimension 𝙳𝙸𝙼=1 overlap the size of their intersection is equal to the size of the corresponding rectangles in dimension 𝙳𝙸𝙼=1.

Figure 5.101.1. Eight non-overlapping rectangles such that, for each pair of rectangles R i , R j (1≀i<j≀12), if the projections in dimension 1 of rectangles R i and R j intersect then the size of their intersection is equal to the size of R i in dimension 1 and to the size of R j in dimension 1
ctrs/diffn_column1
Typical
|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄|>1
π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄.πšœπš’πš£>0
|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|>1
Symmetries
  • Items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚ 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™΄πš‚.πš˜πš›πšπš‘.

See also

common keyword: πšπš’πšπšπš—Β (geometrical constraint,orthotope), πšπš’πšπšπš—_πš’πš—πšŒπš•πšžπšπšŽΒ (geometrical constraint,orthotope,positioning constraint).

used in graph description: 𝚝𝚠𝚘_πš˜πš›πšπš‘_πšŒπš˜πš•πšžπš–πš—.

Keywords

constraint type: decomposition.

geometry: geometrical constraint, positioning constraint, orthotope, guillotine cut.

Arc input(s)

π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚

Arc generator
πΆπΏπΌπ‘„π‘ˆπΈ(<)β†¦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šŽπšœ1,πš˜πš›πšπš‘πš˜πšπš˜πš™πšŽπšœ2)

Arc arity
Arc constraint(s)
𝚝𝚠𝚘_πš˜πš›πšπš‘_πšŒπ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šŽπšœ1.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šŽπšœ2.πš˜πš›πšπš‘,𝙳𝙸𝙼)
Graph property(ies)
𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|*(|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|-1)/2

Graph model

Since showing all items produces too big graphs, partsΒ (A) andΒ (B) of FigureΒ 5.101.2 respectively show the initial and final graph associated with the first three items of the Example slot. Since we use the 𝐍𝐀𝐑𝐂 graph property, the arcs of the final graph are stressed in bold.

Figure 5.101.2. Initial and final graph of the πšπš’πšπšπš—_πšŒπš˜πš•πšžπš–πš— constraint
ctrs/diffn_columnActrs/diffn_columnB
(a) (b)