5.319. stretch_path_partition

DESCRIPTIONLINKS
Origin

Derived from πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπšπš‘.

Constraint

πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπ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™»π™Έπ™Όπ™Έπšƒπš‚)

Synonym

πšœπšπš›πšŽπšπšŒπ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™»π™Έπ™Όπ™Έπšƒπ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‚,πš•πš–πš’πš—-πš’πš—πš,πš•πš–πšŠπš‘-πš’πš—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‚|>0
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™»π™Έπ™Όπ™Έπšƒπš‚|>0
πš›πšŽπššπšžπš’πš›πšŽπš(π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚,[πš™,πš•πš–πš’πš—,πš•πš–πšŠπš‘])
π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚.πš•πš–πš’πš—β‰€π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚.πš•πš–πšŠπš‘
Purpose

In order to define the meaning of the πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπšπš‘_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint, we first introduce the notions of stretch and span. Let n be the number of variables of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚. Let X i ,...,X j (1≀i≀j≀n) be consecutive variables of the collection of variables πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ such that the following conditions apply:

  • All variables X i ,...,X j take their values in the same partition of the π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚ collection (i.e.,Β βˆƒl∈[1,|π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚|] such that βˆ€k∈[i,j]:X k βˆˆπ™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚[l].p),

  • i=1 or X i-1 is different from X i ,

  • j=n or X j+1 is different from X j .

We call such a set of variables a stretch. The span of the stretch is equal to j-i+1, while the value of the stretch is l. We now define the condition enforced by the πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπšπš‘_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint.

Each item π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚[l]=(πš™-π‘£π‘Žπ‘™π‘’π‘’π‘ ,πš•πš–πš’πš—-s,πš•πš–πšŠπš‘-t) of the π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚ collection enforces the minimum value s as well as the maximum value t for the span of a stretch of value l over consecutive variables of the πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ collection.

Note that:

  1. Having an item π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚[l]=(πš™-π‘£π‘Žπ‘™π‘’π‘’π‘ ,πš•πš–πš’πš—-s,πš•πš–πšŠπš‘-t) with s strictly greater than 0 does not mean that values of π‘£π‘Žπ‘™π‘’π‘’π‘  should be assigned to one of the variables of collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚. It rather means that, when a value of π‘£π‘Žπ‘™π‘’π‘’π‘  is used, all stretches of value l must have a span that belong to interval [s,t].

  2. A variable of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ may be assigned a value that is not defined in the attribute πš™ of the π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚ collection.

Example
πšŸπšŠπš›-1,πšŸπšŠπš›-2,πšŸπšŠπš›-0,πšŸπšŠπš›-0,πšŸπšŠπš›-2,πšŸπšŠπš›-2,πšŸπšŠπš›-2,πšŸπšŠπš›-0,πš™-1,2πš•πš–πš’πš—-2πš•πš–πšŠπš‘-4,πš™-3πš•πš–πš’πš—-0πš•πš–πšŠπš‘-2

The πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπšπš‘_πš™πšŠπš›πšπš’πšπš’πš˜πš— constraint holds since the sequence 1 2 0 0 2 2 2 0 contains two stretches 1 2, and 2 2 2 respectively verifying the following conditions:

  • The span of the first stretch 1 2 is located within interval [2,4] (i.e.,Β the limit associated with item π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚[1]).

  • The span of the second stretch 2 2 2 is located within interval [2,4] (i.e.,Β the limit associated with item π™Ώπ™°πšπšƒπ™»π™Έπ™Όπ™Έπšƒπš‚[1]).

Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ can be reversed.

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

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

  • All occurrences of two distinct tuples of values in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› or π™Ώπ™°πšπ™»π™Έπ™Όπ™Έπšƒπš‚.πš™.πšŸπšŠπš• can be swapped; all occurrences of a tuple of values in πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.πšŸπšŠπš› or π™Ώπ™°πšπ™»π™Έπ™Όπ™Έπšƒπš‚.πš™.πšŸπšŠπš• can be renamed to any unused tuple of values.

See also

common keyword: πš™πšŠπšπšπšŽπš›πš—Β (sliding sequence constraint).

specialisation: πšœπšπš›πšŽπšπšŒπš‘_πš™πšŠπ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).

Keywords

characteristic of a constraint: automaton, automaton without counters, reified automaton constraint, partition.

combinatorial object: sequence.

constraint network structure: Berge-acyclic constraint network.

constraint type: timetabling constraint, sliding sequence constraint.

filtering: arc-consistency.

final graph structure: consecutive loops are connected.