5.171. inverse_offset

DESCRIPTIONLINKSGRAPH
Origin

Gecode

Constraint

πš’πš—πšŸπšŽπš›πšœπšŽ_𝚘𝚏𝚏𝚜𝚎𝚝(πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒ,π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒ,π™½π™Ύπ™³π™΄πš‚)

Synonym

πšŒπš‘πšŠπš—πš—πšŽπš•.

Arguments
πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒπš’πš—πš
π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒπš’πš—πš
π™½π™Ύπ™³π™΄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šŽπš‘-πš’πš—πš,𝚜𝚞𝚌𝚌-πšπšŸπšŠπš›,πš™πš›πšŽπš-πšπšŸπšŠπš›)
Restrictions
πš›πšŽπššπšžπš’πš›πšŽπš(π™½π™Ύπ™³π™΄πš‚,[πš’πš—πšπšŽπš‘,𝚜𝚞𝚌𝚌,πš™πš›πšŽπš])
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰₯1
π™½π™Ύπ™³π™΄πš‚.πš’πš—πšπšŽπš‘β‰€|π™½π™Ύπ™³π™΄πš‚|
πšπš’πšœπšπš’πš—πšŒπš(π™½π™Ύπ™³π™΄πš‚,πš’πš—πšπšŽπš‘)
π™½π™Ύπ™³π™΄πš‚.𝚜𝚞𝚌𝚌β‰₯1+πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒ
π™½π™Ύπ™³π™΄πš‚.πšœπšžπšŒπšŒβ‰€|π™½π™Ύπ™³π™΄πš‚|+πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒ
π™½π™Ύπ™³π™΄πš‚.πš™πš›πšŽπšβ‰₯1+π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒ
π™½π™Ύπ™³π™΄πš‚.πš™πš›πšŽπšβ‰€|π™½π™Ύπ™³π™΄πš‚|+π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒ
Purpose

Enforce each vertex of a digraph to have exactly one predecessor and one successor. In addition the following two statements are equivalent:

  1. The successor of the i th node minus π’πŽπ…π…π’π„π“ is equal to j.

  2. The predecessor of the j th node minus ππŽπ…π…π’π„π“ is equal to i.

I.e.,Β π™½π™Ύπ™³π™΄πš‚[i].𝚜𝚞𝚌𝚌-π’πŽπ…π…π’π„π“=jβ‡”π™½π™Ύπ™³π™΄πš‚[j].πš™πš›πšŽπš-ππŽπ…π…π’π„π“=i.

Example
-1,0,πš’πš—πšπšŽπš‘-1𝚜𝚞𝚌𝚌-4πš™πš›πšŽπš-3,πš’πš—πšπšŽπš‘-2𝚜𝚞𝚌𝚌-2πš™πš›πšŽπš-5,πš’πš—πšπšŽπš‘-3𝚜𝚞𝚌𝚌-0πš™πš›πšŽπš-2,πš’πš—πšπšŽπš‘-4𝚜𝚞𝚌𝚌-6πš™πš›πšŽπš-8,πš’πš—πšπšŽπš‘-5𝚜𝚞𝚌𝚌-1πš™πš›πšŽπš-1,πš’πš—πšπšŽπš‘-6𝚜𝚞𝚌𝚌-7πš™πš›πšŽπš-7,πš’πš—πšπšŽπš‘-7𝚜𝚞𝚌𝚌-5πš™πš›πšŽπš-4,πš’πš—πšπšŽπš‘-8𝚜𝚞𝚌𝚌-3πš™πš›πšŽπš-6

The πš’πš—πšŸπšŽπš›πšœπšŽ_𝚘𝚏𝚏𝚜𝚎𝚝 constraint holds since:

  • π™½π™Ύπ™³π™΄πš‚[1].𝚜𝚞𝚌𝚌-(-1)=5β‡”π™½π™Ύπ™³π™΄πš‚[5].πš™πš›πšŽπš-0=1,

  • π™½π™Ύπ™³π™΄πš‚[2].𝚜𝚞𝚌𝚌-(-1)=3β‡”π™½π™Ύπ™³π™΄πš‚[3].πš™πš›πšŽπš-0=2,

  • π™½π™Ύπ™³π™΄πš‚[3].𝚜𝚞𝚌𝚌-(-1)=1β‡”π™½π™Ύπ™³π™΄πš‚[1].πš™πš›πšŽπš-0=3,

  • π™½π™Ύπ™³π™΄πš‚[4].𝚜𝚞𝚌𝚌-(-1)=7β‡”π™½π™Ύπ™³π™΄πš‚[7].πš™πš›πšŽπš-0=4,

  • π™½π™Ύπ™³π™΄πš‚[5].𝚜𝚞𝚌𝚌-(-1)=2β‡”π™½π™Ύπ™³π™΄πš‚[2].πš™πš›πšŽπš-0=5.

  • π™½π™Ύπ™³π™΄πš‚[6].𝚜𝚞𝚌𝚌-(-1)=8β‡”π™½π™Ύπ™³π™΄πš‚[8].πš™πš›πšŽπš-0=6.

  • π™½π™Ύπ™³π™΄πš‚[7].𝚜𝚞𝚌𝚌-(-1)=6β‡”π™½π™Ύπ™³π™΄πš‚[6].πš™πš›πšŽπš-0=7.

  • π™½π™Ύπ™³π™΄πš‚[8].𝚜𝚞𝚌𝚌-(-1)=4β‡”π™½π™Ύπ™³π™΄πš‚[4].πš™πš›πšŽπš-0=8.

FigureΒ 5.171.1 shows the board that can be associated with this example.

Figure 5.171.1. Board associated with the example of the Example slot
ctrs/inverse_offset1
Typical
πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒβ‰₯-1
πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒβ‰€1
π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒβ‰₯-1
π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒβ‰€1
|π™½π™Ύπ™³π™΄πš‚|>1
Symmetry

Items of π™½π™Ύπ™³π™΄πš‚ are permutable.

Remark

The πš’πš—πšŸπšŽπš›πšœπšŽ_𝚘𝚏𝚏𝚜𝚎𝚝 constraint is called πšŒπš‘πšŠπš—πš—πšŽπš• in Gecode (http://www.gecode.org/). Having two offsets was motivated by the fact that it is possible to declare arrays at any position in the MiniZinc modelling language.

Systems

inverseChanneling in Choco, channel in Gecode.

See also

specialisation: πš’πš—πšŸπšŽπš›πšœπšŽΒ (assume that πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒ and π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒ are both equal to 0).

Keywords

constraint type: graph constraint.

filtering: arc-consistency.

heuristics: heuristics.

modelling: channelling constraint, dual model.

Arc input(s)

π™½π™Ύπ™³π™΄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š—πš˜πšπšŽπšœ1,πš—πš˜πšπšŽπšœ2)

Arc arity
Arc constraint(s)
β€’ πš—πš˜πšπšŽπšœ1.𝚜𝚞𝚌𝚌-πš‚π™Ύπ™΅π™΅πš‚π™΄πšƒ=πš—πš˜πšπšŽπšœ2.πš’πš—πšπšŽπš‘
β€’ πš—πš˜πšπšŽπšœ2.πš™πš›πšŽπš-π™Ώπ™Ύπ™΅π™΅πš‚π™΄πšƒ=πš—πš˜πšπšŽπšœ1.πš’πš—πšπšŽπš‘
Graph property(ies)
𝐍𝐀𝐑𝐂=|π™½π™Ύπ™³π™΄πš‚|

Graph model

In order to express the binary constraint that links two vertices one has to make explicit the identifier of the vertices. This is why the πš’πš—πšŸπšŽπš›πšœπšŽ_𝚘𝚏𝚏𝚜𝚎𝚝 constraint considers objects that have three attributes:

  • One fixed attribute πš’πš—πšπšŽπš‘ that is the identifier of the vertex,

  • One variable attribute 𝚜𝚞𝚌𝚌 that is the successor of the vertex,

  • One variable attribute πš™πš›πšŽπš that is the predecessor of the vertex.

PartsΒ (A) andΒ (B) of FigureΒ 5.171.2 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.

Figure 5.171.2. Initial and final graph of the πš’πš—πšŸπšŽπš›πšœπšŽ_𝚘𝚏𝚏𝚜𝚎𝚝 constraint
ctrs/inverse_offsetActrs/inverse_offsetB
(a) (b)