Global Constraint Catalog: Cnpair

5.284. npair

DESCRIPTION

LINKS

GRAPH

Origin

Derived from $𝚗𝚟𝚊𝚕𝚞𝚎$ .

Constraint

$𝚗𝚙𝚊𝚒𝚛 (𝙽𝙿𝙰𝙸𝚁𝚂, 𝙿𝙰𝙸𝚁𝚂)$

Arguments

$𝙽𝙿𝙰𝙸𝚁𝚂$	$𝚍𝚟𝚊𝚛$
$𝙿𝙰𝙸𝚁𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚡 - 𝚍𝚟𝚊𝚛, 𝚢 - 𝚍𝚟𝚊𝚛)$

Restrictions

𝙽𝙿𝙰𝙸𝚁𝚂 \geq 𝚖𝚒𝚗 (1, | 𝙿𝙰𝙸𝚁𝚂 |)

𝙽𝙿𝙰𝙸𝚁𝚂 \leq | 𝙿𝙰𝙸𝚁𝚂 |

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝙿𝙰𝙸𝚁𝚂, [𝚡, 𝚢])

Purpose

$𝙽𝙿𝙰𝙸𝚁𝚂$ is the number of distinct pairs of values assigned to the pairs of variables of the collection $𝙿𝙰𝙸𝚁𝚂$ .

Example

(\begin{matrix} 2, 〈\begin{matrix} 𝚡 - 3 & 𝚢 - 1, \\ 𝚡 - 1 & 𝚢 - 5, \\ 𝚡 - 3 & 𝚢 - 1, \\ 𝚡 - 3 & 𝚢 - 1, \\ 𝚡 - 1 & 𝚢 - 5 \end{matrix}〉 \end{matrix})

The $𝚗𝚙𝚊𝚒𝚛$ constraint holds since its first argument $𝙽𝙿𝙰𝙸𝚁𝚂 = 2$ is set to the number of distinct pairs $〈 𝚡 - 3 𝚢 - 1 〉$ and $〈 𝚡 - 1 𝚢 - 5 〉$ of its second argument $𝙿𝙰𝙸𝚁𝚂$ .

Typical

𝙽𝙿𝙰𝙸𝚁𝚂 > 1

𝙽𝙿𝙰𝙸𝚁𝚂 < | 𝙿𝙰𝙸𝚁𝚂 |

| 𝙿𝙰𝙸𝚁𝚂 | > 1

𝚛𝚊𝚗𝚐𝚎

(𝙿𝙰𝙸𝚁𝚂 . 𝚡) > 1

𝚛𝚊𝚗𝚐𝚎

(𝙿𝙰𝙸𝚁𝚂 . 𝚢) > 1

Symmetries

Items of $𝙿𝙰𝙸𝚁𝚂$ are permutable.
Attributes of $𝙿𝙰𝙸𝚁𝚂$ are permutable w.r.t. permutation $(𝚡, 𝚢)$ (permutation applied to all items).
All occurrences of two distinct tuples of values of $𝙽𝙿𝙰𝙸𝚁𝚂$ can be swapped; all occurrences of a tuple of values of $𝙽𝙿𝙰𝙸𝚁𝚂$ can be renamed to any unused tuple of values.

Arg. properties

Functional dependency: $𝙽𝙿𝙰𝙸𝚁𝚂$ determined by $𝙿𝙰𝙸𝚁𝚂$ .
Contractible wrt. $𝙿𝙰𝙸𝚁𝚂$ when $𝙽𝙿𝙰𝙸𝚁𝚂 = 1$ and $| 𝙿𝙰𝙸𝚁𝚂 | > 0$ .
Contractible wrt. $𝙿𝙰𝙸𝚁𝚂$ when $𝙽𝙿𝙰𝙸𝚁𝚂 = | 𝙿𝙰𝙸𝚁𝚂 |$ .

Remark

This is an example of a number of distinct values constraint where there is more than one attribute that is associated with each vertex of the final graph.

See also

related: $𝚗𝚌𝚕𝚊𝚜𝚜$ ( $𝚙𝚊𝚒𝚛$ of $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜$ replaced by $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎 \in 𝚙𝚊𝚛𝚝𝚒𝚝𝚒𝚘𝚗$ ), $𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎$ ( $𝚙𝚊𝚒𝚛$ of $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜$ replaced by $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎 \mod 𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝$ ), $𝚗𝚒𝚗𝚝𝚎𝚛𝚟𝚊𝚕$ ( $𝚙𝚊𝚒𝚛$ of $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜$ replaced by $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎 / 𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝$ ).

specialisation: $𝚗𝚟𝚊𝚕𝚞𝚎$ ( $𝚙𝚊𝚒𝚛$ of $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜$ replaced by $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎$ ).

Keywords

characteristic of a constraint: pair.

constraint arguments: pure functional dependency.

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes, functional dependency.

Arc input(s)

$𝙿𝙰𝙸𝚁𝚂$

Arc generator

𝐶𝐿𝐼𝑄𝑈𝐸

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚙𝚊𝚒𝚛𝚜 1, 𝚙𝚊𝚒𝚛𝚜 2)

Arc arity

Arc constraint(s)

• 𝚙𝚊𝚒𝚛𝚜 1 . 𝚡 = 𝚙𝚊𝚒𝚛𝚜 2 . 𝚡

• 𝚙𝚊𝚒𝚛𝚜 1 . 𝚢 = 𝚙𝚊𝚒𝚛𝚜 2 . 𝚢

Graph property(ies)

𝐍𝐒𝐂𝐂

= 𝙽𝙿𝙰𝙸𝚁𝚂

Graph model

Parts (A) and (B) of Figure 5.284.1 respectively show the initial and final graph associated with the Example slot. Since we use the $𝐍𝐒𝐂𝐂$ graph property we show the different strongly connected components of the final graph. Each strongly connected component corresponds to a pair of values that is assigned to some pairs of variables of the $𝙿𝙰𝙸𝚁𝚂$ collection. In our example we have the following pairs of values: $〈 𝚡 - 3 𝚢 - 1 〉$ and $〈 𝚡 - 1 𝚢 - 5 〉$ .

Figure 5.284.1. Initial and final graph of the $𝚗𝚙𝚊𝚒𝚛$ constraint


(a)	(b)

Global Constraint Catalog

5. Global Constraint Catalogue

5.284. npair

Figure 5.284.1. Initial and final graph of the 𝚗𝚙𝚊𝚒𝚛 constraint

Figure 5.284.1. Initial and final graph of the $𝚗𝚙𝚊𝚒𝚛$ constraint