5.262. orths_are_connected

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

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(π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚)

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›)
Argument
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™΄)
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£>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‘)
Purpose

There should be one single group of connected orthotopes. Two orthotopes touch each other (i.e.,Β are connected) if they overlap in all dimensions except one, and if, for the dimension where they do not overlap, the distance between the two orthotopes is equal to 0.

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

FigureΒ 5.262.1 shows the rectangles associated with the example. One can note that:

  • Rectangle 2 touch rectangle 1,

  • Rectangle 1 touch rectangle 2 and rectangle 4,

  • Rectangle 4 touch rectangle 1 and rectangle 3,

  • Rectangle 3 touch rectangle 4.

Consequently, since we have one single group of connected rectangles, the 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 constraint holds.

Figure 5.262.1. Four connected rectangles
ctrs/orths_are_connected1
Symmetries
  • Items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚ are permutable.

  • Items of π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚.πš˜πš›πšπš‘ are permutable (same permutation used).

  • 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‘.

Usage

In floor planning problem there is a typical constraint, that states that one should be able to access every room from any room.

See also

implies: πšπš’πšπšπš—.

used in graph description: 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, 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.

Keywords

geometry: geometrical constraint, touch, contact, non-overlapping, orthotope.

Arc input(s)

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

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šŽπšœ)

Arc arity
Arc constraint(s)
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š˜πš›πšπš‘πš˜πšπš˜πš™πšŽπšœ.πš˜πš›πšπš‘)
Graph property(ies)
𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|

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‰ )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šŠπš›πšŽ_πš’πš—_πšŒπš˜πš—πšπšŠπšŒπ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‘)
Graph property(ies)
β€’ 𝐍𝐕𝐄𝐑𝐓𝐄𝐗=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|
β€’ 𝐍𝐂𝐂=1

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.262.2 respectively show the initial and final graph associated with the Example slot.Since we use the 𝐍𝐕𝐄𝐑𝐓𝐄𝐗 graph property the vertices of the final graph are stressed in bold. Since we also use the 𝐍𝐂𝐂 graph property we show the unique connected component of the final graph. An arc between two vertices indicates that two rectangles are in contact.

Figure 5.262.2. Initial and final graph of the 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 constraint
ctrs/orths_are_connectedActrs/orths_are_connectedB
(a) (b)
Signature

Since the first graph constraint uses the 𝑆𝐸𝐿𝐹 arc generator on the π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚ collection the corresponding initial graph contains |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| arcs. Therefore the final graph of the first graph constraint contains at most |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| arcs and we can rewrite 𝐍𝐀𝐑𝐂=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| to 𝐍𝐀𝐑𝐂β‰₯|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|. So we can simplify 𝐍𝐀𝐑𝐂 Β― Μ² to 𝐍𝐀𝐑𝐂 Β―.

Consider now the second graph constraint. Since its corresponding initial graph contains |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| vertices, its final graph has a maximum number of vertices also equal to |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|. Therefore we can rewrite 𝐍𝐕𝐄𝐑𝐓𝐄𝐗=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| to 𝐍𝐕𝐄𝐑𝐓𝐄𝐗β‰₯|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| and simplify 𝐍𝐕𝐄𝐑𝐓𝐄𝐗 Β― Μ² to 𝐍𝐕𝐄𝐑𝐓𝐄𝐗 Β―. From the graph property 𝐍𝐕𝐄𝐑𝐓𝐄𝐗=|π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚| and from the restriction |π™Ύπšπšƒπ™·π™Ύπšƒπ™Ύπ™Ώπ™΄πš‚|>0 the final graph is not empty. Therefore it contains at least one connected component. So we can rewrite 𝐍𝐂𝐂=1 to 𝐍𝐂𝐂≀1 and simplify 𝐍𝐂𝐂 Β― Μ² to 𝐍𝐂𝐂 Μ².