5.13. alldifferent_partition

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš.

Constraint

πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πš™πšŠπš›πšπš’πšπš’πš˜πš—(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)

Synonyms

πšŠπš•πš•πšπš’πšπš_πš™πšŠπš›πšπš’πšπš’πš˜πš—, πšŠπš•πš•πšπš’πšœπšπš’πš—πšŒπš_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

Type
πš…π™°π™»πš„π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš•-πš’πš—πš)
Arguments
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πšŸπšŠπš›-πšπšŸπšŠπš›)
π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚πšŒπš˜πš•πš•πšŽπšŒπšπš’πš˜πš—(πš™-πš…π™°π™»πš„π™΄πš‚)
Restrictions
|πš…π™°π™»πš„π™΄πš‚|β‰₯1
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
πšπš’πšœπšπš’πš—πšŒπš(πš…π™°π™»πš„π™΄πš‚,πšŸπšŠπš•)
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|≀|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚,πšŸπšŠπš›)
|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|β‰₯2
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚,πš™)
Purpose

Enforce all variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ to take values that belong to distinct partitions.

Example
6,3,4,πš™-1,3,πš™-4,πš™-2,6

Since all variables take values that are located within distinct partitions the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint holds.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>2
Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ are permutable.

  • Items of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.πš™ are permutable.

  • A value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› can be renamed to any value that belongs to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

  • Two distinct values of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› that do not belong to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ can be swapped.

See also

common keyword: πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—Β (partition).

specialisation: πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πšΒ (πšŸπšŠπš›πš’πšŠπš‹πš•πšŽβˆˆπš™πšŠπš›πšπš’πšπš’πš˜πš— replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

used in graph description: πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—.

Keywords

characteristic of a constraint: partition, all different.

constraint type: value constraint.

filtering: arc-consistency.

final graph structure: one_succ.

modelling: incompatible pairs of values.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄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šŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2)

Arc arity
Arc constraint(s)
πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš—(πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš›,πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚)
Graph property(ies)
πŒπ€π—_𝐍𝐒𝐂𝐂≀1

Graph class
𝙾𝙽𝙴_πš‚πš„π™²π™²

Graph model

Similar to the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint, but we replace the binary equality constraint of the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint by the fact that two variables are respectively assigned to two values that belong to the same partition. We generate a clique with a πš’πš—_πšœπšŠπš–πšŽ_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint between each pair of vertices (including a vertex and itself) and state that the size of the largest strongly connected component should not exceed 1.

PartsΒ (A) andΒ (B) of FigureΒ 5.13.1 respectively show the initial and final graph associated with the Example slot. Since we use the πŒπ€π—_𝐍𝐒𝐂𝐂 graph property we show one of the largest strongly connected component of the final graph.

Figure 5.13.1. Initial and final graph of the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint
ctrs/alldifferent_partitionActrs/alldifferent_partitionB
(a) (b)