5.110. distance_between

DESCRIPTIONLINKSGRAPH
Origin

N.Β Beldiceanu

Constraint

πšπš’πšœπšπšŠπš—πšŒπšŽ_πš‹πšŽπšπš πšŽπšŽπš—(π™³π™Έπš‚πšƒ,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™²πšƒπš)

Synonym

πšπš’πšœπšπšŠπš—πšŒπšŽ.

Arguments
π™³π™Έπš‚πšƒπ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™²πšƒπšπšŠπšπš˜πš–
Restrictions
π™³π™Έπš‚πšƒβ‰₯0
π™³π™Έπš‚πšƒβ‰€|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š…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
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™°πšπ™Έπ™°π™±π™»π™΄πš‚1|=|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2|
π™²πšƒπšβˆˆ[=,β‰ ,<,β‰₯,>,≀]
Purpose

Let U i and V i be respectively the i th and j th variables (iβ‰ j) of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1. In a similar way, let X i and Y i be respectively the i th and j th variables (iβ‰ j) of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2. π™³π™Έπš‚πšƒ is equal to the number of times one of the following mutually incompatible conditions are true:

  • U i π™²πšƒπš V i holds and X i π™²πšƒπš Y i does not hold,

  • X i π™²πšƒπš Y i holds and U i π™²πšƒπš V i does not hold.

Example
2,3,4,6,2,4,2,6,9,3,6,<

The πšπš’πšœπšπšŠπš—πšŒπšŽ_πš‹πšŽπšπš πšŽπšŽπš— constraint holds since the following π™³π™Έπš‚πšƒ=2 conditions are verified:

  • πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1[4].πšŸπšŠπš›=2<πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1[1].πšŸπšŠπš›=3 βˆ§πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2[4].πšŸπšŠπš›=3β‰₯πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2[1].πšŸπšŠπš›=2

  • πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2[1].πšŸπšŠπš›=2<πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2[4].πšŸπšŠπš›=3 βˆ§πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1[1].πšŸπšŠπš›=3β‰₯πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1[4].πšŸπšŠπš›=2

Typical
π™³π™Έπš‚πšƒ>0
π™³π™Έπš‚πšƒ<|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š…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1|>1
Symmetries
  • Arguments are permutable w.r.t. permutation (π™³π™Έπš‚πšƒ) (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™²πšƒπš).

  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1 and πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2 are permutable (same permutation used).

  • One and the same constant can be added to the πšŸπšŠπš› attribute of all items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚1.

  • One and the same constant can be added to the πšŸπšŠπš› attribute of all items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚2.

Usage

Measure the distance between two sequences in term of the number of constraint changes. This should be put in contrast to the number of value changes that is sometimes superficial.

See also

common keyword: πšπš’πšœπšπšŠπš—πšŒπšŽ_πšŒπš‘πšŠπš—πšπšŽΒ (proximity constraint).

Keywords

constraint type: proximity constraint.

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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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šŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ1.πšŸπšŠπš› π™²πšƒπš πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›
Graph property(ies)
πƒπˆπ’π“π€ππ‚π„=π™³π™Έπš‚πšƒ

Graph model

Within the Arc input(s) slot, the character / indicates that we generate two distinct graphs. The graph property πƒπˆπ’π“π€ππ‚π„ measures the distance between two digraphs G 1 and G 2 . This distance is defined as the sum of the following quantities:

  • The number of arcs of G 1 that do not belong to G 2 ,

  • The number of arcs of G 2 that do not belong to G 1 .

PartΒ (A) of FigureΒ 5.110.1 gives the final graph associated with the sequence πšŸπšŠπš›-3,πšŸπšŠπš›-4,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-4 (i.e.,Β the second argument of the constraint of the Example slot), while partΒ (B) shows the final graph corresponding to πšŸπšŠπš›-2,πšŸπšŠπš›-6,πšŸπšŠπš›-9,πšŸπšŠπš›-3,πšŸπšŠπš›-6 (i.e.,Β the third argument of the constraint of the Example slot). The two arc constraints that differ from one graph to the other are marked by a dotted line. The πšπš’πšœπšπšŠπš—πšŒπšŽ_πš‹πšŽπšπš πšŽπšŽπš— constraint holds since between sequence πšŸπšŠπš›-3,πšŸπšŠπš›-4,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-4 and sequence πšŸπšŠπš›-2,πšŸπšŠπš›-6,πšŸπšŠπš›-9,πšŸπšŠπš›-3,πšŸπšŠπš›-6 there are π™³π™Έπš‚πšƒ=2 changes that respectively correspond to:

  • Within the final graph associated with sequence πšŸπšŠπš›-3,πšŸπšŠπš›-4,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-4 the arc 4β†’1 (i.e.,Β values 2β†’3) does not occur in the final graph associated with πšŸπšŠπš›-2,πšŸπšŠπš›-6,πšŸπšŠπš›-9,πšŸπšŠπš›-3,πšŸπšŠπš›-6,

  • Within the final graph associated with sequence πšŸπšŠπš›-2,πšŸπšŠπš›-6,πšŸπšŠπš›-9,πšŸπšŠπš›-3,πšŸπšŠπš›-6 the arc 1β†’4 (i.e.,Β values 2β†’3) does not occur in the final graph associated with πšŸπšŠπš›-3,πšŸπšŠπš›-4,πšŸπšŠπš›-6,πšŸπšŠπš›-2,πšŸπšŠπš›-4.

Figure 5.110.1. Final graphs of the πšπš’πšœπšπšŠπš—πšŒπšŽ_πš‹πšŽπšπš πšŽπšŽπš— constraint
ctrs/distance_betweenActrs/distance_betweenB
(a) (b)