5.65. common_partition

DESCRIPTIONLINKSGRAPH
Origin

Derived from πšŒπš˜πš–πš–πš˜πš—.

Constraint

πšŒπ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™½π™²π™Ύπ™Όπ™Όπ™Ύπ™½1,𝙽𝙲𝙾𝙼𝙼𝙾𝙽2,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2,π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½π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
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1πšπšŸπšŠπš›
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2πšπšŸπšŠπš›
πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1π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š…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2π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•)
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1β‰₯0
𝙽𝙲𝙾𝙼𝙼𝙾𝙽1≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2β‰₯0
𝙽𝙲𝙾𝙼𝙼𝙾𝙽2≀|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|
πš›πšŽπššπšžπš’πš›πšŽπš(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πšŸπšŠπ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‚2,πšŸπšŠπš›)
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™)
|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|β‰₯2
Purpose

𝙽𝙲𝙾𝙼𝙼𝙾𝙽1 is the number of variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 collection taking a value in a partition derived from the values assigned to the variables of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 and from π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

𝙽𝙲𝙾𝙼𝙼𝙾𝙽2 is the number of variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 collection taking a value in a partition derived from the values assigned to the variables of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and from π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

Example
3,4,2,3,6,0,πšŸπšŠπš›-0,πšŸπšŠπš›-6,πšŸπšŠπš›-3,πšŸπšŠπš›-3,πšŸπšŠπš›-7,πšŸπšŠπš›-1,πš™-1,3,πš™-4,πš™-2,6

In the example, the last argument π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚ defines the partitions πš™-〈1,3βŒͺ, πš™-〈4βŒͺ and πš™-〈2,6βŒͺ. As a consequence the first three items of collection 〈2,3,6,0βŒͺ respectively correspond to the partitions πš™-〈2,6βŒͺ, πš™-〈1,3βŒͺ, and πš™-〈2,6βŒͺ. Similarly the items of collection 〈0,6,3,3,7,1βŒͺ (from which we remove items 0 and 7 since they do not belong to any partition) respectively correspond to the partitions πš™-〈2,6βŒͺ, πš™-〈1,3βŒͺ, πš™-〈1,3βŒͺ, and πš™-〈1,3βŒͺ. The πšŒπ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— constraint holds since:

  • Its first argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1=3 is the number of partitions associated with the items of collection 〈2,3,6,0βŒͺ that also correspond to partitions associated with 〈0,6,3,3,7,1βŒͺ.

  • Its second argument 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2=4 is the number of partitions associated with the items of collection 〈0,6,3,3,7,1βŒͺ that also correspond to partitions associated with 〈2,3,6,0βŒͺ.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš›)>1
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|>1
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš›)>1
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|>|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|>|π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚|
Symmetries
  • Arguments are permutable w.r.t. permutation (𝙽𝙲𝙾𝙼𝙼𝙾𝙽1,𝙽𝙲𝙾𝙼𝙼𝙾𝙽2) (πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1,πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2) (π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚).

  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 are permutable.

  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 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.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.πšŸπšŠπš› can be replaced by any other value that also belongs to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

  • An occurrence of a value of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.πšŸπšŠπš› can be replaced by any other value that also belongs to the same partition of π™Ώπ™°πšπšƒπ™Έπšƒπ™Έπ™Ύπ™½πš‚.

See also

specialisation: πšŒπš˜πš–πš–π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.

constraint arguments: constraint between two collections of variables.

final graph structure: acyclic, bipartite, no loop.

Arc input(s)

πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2

Arc generator
π‘ƒπ‘…π‘‚π·π‘ˆπΆπ‘‡β†¦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)
β€’ ππ’πŽπ”π‘π‚π„=𝙽𝙲𝙾𝙼𝙼𝙾𝙽1
β€’ ππ’πˆππŠ=𝙽𝙲𝙾𝙼𝙼𝙾𝙽2

Graph class
β€’ π™°π™²πšˆπ™²π™»π™Έπ™²
β€’ π™±π™Έπ™Ώπ™°πšπšƒπ™Έπšƒπ™΄
β€’ 𝙽𝙾_𝙻𝙾𝙾𝙿

Graph model

PartsΒ (A) andΒ (B) of FigureΒ 5.65.1 respectively show the initial and final graph associated with the Example slot. Since we use the ππ’πŽπ”π‘π‚π„ and ππ’πˆππŠ graph properties, the source and sink vertices of the final graph are stressed with a double circle. Since the graph has only 3 sources and 4 sinks the variables 𝙽𝙲𝙾𝙼𝙼𝙾𝙽1 and 𝙽𝙲𝙾𝙼𝙼𝙾𝙽2 are respectively equal to 3 and 4. Note that the vertices corresponding to the variables that take values 0 or 7 were removed from the final graph since there is no arc for which the associated 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 holds.

Figure 5.65.1. Initial and final graph of the πšŒπ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— constraint
ctrs/common_partitionActrs/common_partitionB
(a) (b)