5.8. alldifferent_cst

DESCRIPTIONLINKSGRAPH
Origin

CHIP

Constraint

πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_𝚌𝚜𝚝(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚)

Synonyms

πšŠπš•πš•πšπš’πšπš_𝚌𝚜𝚝, πšŠπš•πš•πšπš’πšœπšπš’πš—πšŒπš_𝚌𝚜𝚝.

Argument
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›,𝚌𝚜𝚝-πš’πš—πš)
Restriction
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›,𝚌𝚜𝚝])
Purpose

For all pairs of items (πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i],πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j]) (iβ‰ j) of the collection πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ enforce πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŸπšŠπš›+πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[i].πšŒπšœπšβ‰ πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].πšŸπšŠπš›+πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚[j].𝚌𝚜𝚝.

Example
πšŸπšŠπš›-5𝚌𝚜𝚝-0,πšŸπšŠπš›-1𝚌𝚜𝚝-1,πšŸπšŠπš›-9𝚌𝚜𝚝-0,πšŸπšŠπš›-3𝚌𝚜𝚝-4

The πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_𝚌𝚜𝚝 constraint holds since all the expressions 5+0=5, 1+1=2, 9+0=9 and 3+4=7 correspond to distinct values.

Typical
|πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚|>2
πš›πšŠπš—πšπšŽ(πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚.𝚌𝚜𝚝)>0
Symmetries
  • Items of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable.

  • Attributes of πš…π™°πšπ™Έπ™°π™±π™»π™΄πš‚ are permutable w.r.t. permutation (πšŸπšŠπš›,𝚌𝚜𝚝) (permutation not necessarily applied to all items).

  • 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‚.

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

Usage

The πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_𝚌𝚜𝚝 constraint was originally introduced in CHIP in order to express the n-queen problem with 3 global constraints (see the Usage slot of the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint).

Algorithm

See the filtering algorithms of the πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš constraint.

Systems

distinct in Gecode.

See also

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šŽ+πšŒπš˜πš—πšœπšπšŠπš—πš replaced by πšŸπšŠπš›πš’πšŠπš‹πš•πšŽ).

Keywords

characteristic of a constraint: all different, disequality.

constraint type: value constraint.

filtering: bipartite matching, bipartite matching in convex bipartite graphs, convex bipartite graph, arc-consistency.

final graph structure: one_succ.

modelling exercises: n-Amazon.

puzzles: n-Amazon, n-queen.

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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šœ1.𝚌𝚜𝚝=πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.πšŸπšŠπš›+πšŸπšŠπš›πš’πšŠπš‹πš•πšŽπšœ2.𝚌𝚜𝚝
Graph property(ies)
πŒπ€π—_𝐍𝐒𝐂𝐂≀1

Graph class
𝙾𝙽𝙴_πš‚πš„π™²π™²

Graph model

We generate a clique with an equality 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 one.

PartsΒ (A) andΒ (B) of FigureΒ 5.8.1 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. The πšŠπš•πš•πšπš’πšπšπšŽπš›πšŽπš—πš_𝚌𝚜𝚝 holds since all the strongly connected components have at most one vertex: a value is used at most once.

Figure 5.8.1. 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_𝚌𝚜𝚝 constraint
ctrs/alldifferent_cstActrs/alldifferent_cstB
(a) (b)