5.81. crossing

DESCRIPTIONLINKSGRAPH
Origin

Inspired by [CormenLeisersonRivest90].

Constraint

πšŒπš›πš˜πšœπšœπš’πš—πš(π™½π™²πšπ™Ύπš‚πš‚,πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚)

Arguments
π™½π™²πšπ™Ύπš‚πš‚πšπšŸπšŠπš›
πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(𝚘𝚑-πšπšŸπšŠπš›,𝚘𝚒-πšπšŸπšŠπš›,𝚎𝚑-πšπšŸπšŠπš›,𝚎𝚒-πšπšŸπšŠπš›)
Restrictions
π™½π™²πšπ™Ύπš‚πš‚β‰₯0
π™½π™²πšπ™Ύπš‚πš‚β‰€(|πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚|*|πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚|-|πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚|)/2
πš›πšŽπššπšžπš’πš›πšŽπš(πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚,[𝚘𝚑,𝚘𝚒,𝚎𝚑,𝚎𝚒])
Purpose

π™½π™²πšπ™Ύπš‚πš‚ is the number of line -segments intersections between the line -segments defined by the πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚ collection. Each line -segment is defined by the coordinates (𝚘𝚑,𝚘𝚒) and (𝚎𝚑,𝚎𝚒) of its two extremities.

Example
3,𝚘𝚑-1𝚘𝚒-4𝚎𝚑-9𝚎𝚒-2,𝚘𝚑-1𝚘𝚒-1𝚎𝚑-3𝚎𝚒-5,𝚘𝚑-3𝚘𝚒-2𝚎𝚑-7𝚎𝚒-4,𝚘𝚑-9𝚘𝚒-1𝚎𝚑-9𝚎𝚒-4

FigureΒ 5.81.1 provides a picture of the example with the corresponding four line -segments of the πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚ collection. The πšŒπš›πš˜πšœπšœπš’πš—πš constraint holds since its first argument π™½π™²πšπ™Ύπš‚πš‚ is set to 3, which is actually the number of line -segments intersections.

Figure 5.81.1. Intersection between line-segments
ctrs/crossing1
Typical
|πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚|>1
Symmetries
  • Items of πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚ are permutable.

  • Attributes of πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚ are permutable w.r.t. permutation (𝚘𝚑,𝚘𝚒) (𝚎𝚑,𝚎𝚒) (permutation applied to all items).

  • One and the same constant can be added to the 𝚘𝚑 and 𝚎𝚑 attributes of all items of πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚.

  • One and the same constant can be added to the 𝚘𝚒 and 𝚎𝚒 attributes of all items of πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπš‚.

See also

common keyword: πšπš›πšŠπš™πš‘_πšŒπš›πš˜πšœπšœπš’πš—πš, 𝚝𝚠𝚘_πš•πšŠπš’πšŽπš›_𝚎𝚍𝚐𝚎_πšŒπš›πš˜πšœπšœπš’πš—πšΒ (line-segments intersection).

Keywords

final graph structure: acyclic, no loop.

geometry: geometrical constraint, line-segments intersection.

Arc input(s)

πš‚π™΄π™Άπ™Όπ™΄π™½πšƒπ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—(𝚜1,𝚜2)

Arc arity
Arc constraint(s)
β€’ πš–πšŠπš‘(𝚜1.𝚘𝚑,𝚜1.𝚎𝚑)β‰₯πš–πš’πš—(𝚜2.𝚘𝚑,𝚜2.𝚎𝚑)
β€’ πš–πšŠπš‘(𝚜2.𝚘𝚑,𝚜2.𝚎𝚑)β‰₯πš–πš’πš—(𝚜1.𝚘𝚑,𝚜1.𝚎𝚑)
β€’ πš–πšŠπš‘(𝚜1.𝚘𝚒,𝚜1.𝚎𝚒)β‰₯πš–πš’πš—(𝚜2.𝚘𝚒,𝚜2.𝚎𝚒)
β€’ πš–πšŠπš‘(𝚜2.𝚘𝚒,𝚜2.𝚎𝚒)β‰₯πš–πš’πš—(𝚜1.𝚘𝚒,𝚜1.𝚎𝚒)
β€’ ⋁(𝚜2.𝚘𝚑-𝚜1.𝚎𝚑)*(𝚜1.𝚎𝚒-𝚜1.𝚘𝚒)-∏𝚜1.𝚎𝚑-𝚜1.𝚘𝚑,𝚜2.𝚘𝚒-𝚜1.𝚎𝚒=0,(𝚜2.𝚎𝚑-𝚜1.𝚎𝚑)*(𝚜2.𝚘𝚒-𝚜1.𝚘𝚒)-∏𝚜2.𝚘𝚑-𝚜1.𝚘𝚑,𝚜2.𝚎𝚒-𝚜1.𝚎𝚒=0,πšœπš’πšπš—(𝚜2.𝚘𝚑-𝚜1.𝚎𝚑)*(𝚜1.𝚎𝚒-𝚜1.𝚘𝚒)-(𝚜1.𝚎𝚑-𝚜1.𝚘𝚑)*(𝚜2.𝚘𝚒-𝚜1.𝚎𝚒)β‰ πšœπš’πšπš—(𝚜2.𝚎𝚑-𝚜1.𝚎𝚑)*(𝚜2.𝚘𝚒-𝚜1.𝚘𝚒)-(𝚜2.𝚘𝚑-𝚜1.𝚘𝚑)*(𝚜2.𝚎𝚒-𝚜1.𝚎𝚒)
Graph property(ies)
𝐍𝐀𝐑𝐂=π™½π™²πšπ™Ύπš‚πš‚

Graph class
β€’ π™°π™²πšˆπ™²π™»π™Έπ™²
β€’ 𝙽𝙾_𝙻𝙾𝙾𝙿

Graph model

Each line -segment is described by the 𝚑 and 𝚒 coordinates of its two extremities. In the arc generator we use the restriction < in order to generate one single arc for each pair of segments. This is required, since otherwise we would count more than once a given line -segments intersection.

PartsΒ (A) andΒ (B) of FigureΒ 5.81.2 respectively show the initial and final graph associated with the Example slot. Since we use the 𝐍𝐀𝐑𝐂 graph property, the arcs of the final graph are stressed in bold. An arc constraint expresses the fact the two line -segments intersect. It is taken fromΒ [CormenLeisersonRivest90]. Each arc of the final graph corresponds to a line -segments intersection.

Figure 5.81.2. Initial and final graph of the πšŒπš›πš˜πšœπšœπš’πš—πš constraint
ctrs/crossingActrs/crossingB
(a) (b)