Global Constraint Catalog: Cnequivalence

5.274. nequivalence

DESCRIPTION

LINKS

GRAPH

Origin

Derived from $𝚗𝚟𝚊𝚕𝚞𝚎$ .

Constraint

$𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎 (𝙽𝙴𝚀𝚄𝙸𝚅, 𝙼, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$

Arguments

$𝙽𝙴𝚀𝚄𝙸𝚅$	$𝚍𝚟𝚊𝚛$
$𝙼$	$𝚒𝚗𝚝$
$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)$

Restrictions

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

𝙽𝙴𝚀𝚄𝙸𝚅 \geq 𝚖𝚒𝚗 (1, | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |)

𝙽𝙴𝚀𝚄𝙸𝚅 \leq 𝚖𝚒𝚗 (𝙼, | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |)

𝙽𝙴𝚀𝚄𝙸𝚅 \leq

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛)

𝙼 > 0

Purpose

$𝙽𝙴𝚀𝚄𝙸𝚅$ is the number of distinct rests obtained by dividing the variables of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ by $𝙼$ .

Example

(2, 3, 〈3, 2, 5, 6, 15, 3, 3〉)

Since the expressions $3 \mod 3 = 0$ , $2 \mod 3 = 2$ , $5 \mod 3 = 2$ , $6 \mod 3 = 0$ , $15 \mod 3 = 0$ , $3 \mod 3 = 0$ , and $3 \mod 3 = 0$ involve two distinct values (values 0 and 2), the first argument $𝙽𝙴𝚀𝚄𝙸𝚅$ of the $𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎$ constraint is set to value 2.

Typical

𝙽𝙴𝚀𝚄𝙸𝚅 > 1

𝙽𝙴𝚀𝚄𝙸𝚅 < | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |

𝙽𝙴𝚀𝚄𝙸𝚅 <

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛)

𝙼 > 1

𝙼 <

𝚖𝚊𝚡𝚟𝚊𝚕

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛)

Symmetries

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ are permutable.
An occurrence of a value $u$ of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛$ can be replaced by any other value $v$ such that $v$ is congruent to $u$ modulo $𝙼$ .

Arg. properties

Functional dependency: $𝙽𝙴𝚀𝚄𝙸𝚅$ determined by $𝙼$ and $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ .
Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝙽𝙴𝚀𝚄𝙸𝚅 = 1$ and $| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 0$ .
Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝙽𝙴𝚀𝚄𝙸𝚅 = | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |$ .
Extensible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝙽𝙴𝚀𝚄𝙸𝚅 = 𝙼$ .

Algorithm

Since constraints $X = Y$ and $X \equiv Y (\mod M)$ are similar, one should also use a similar algorithm as the one [Beldiceanu01], [BeldiceanuCarlssonThiel02] provided for constraint $𝚗𝚟𝚊𝚕𝚞𝚎$ .

See also

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

specialisation: $𝚗𝚟𝚊𝚕𝚞𝚎$ ( $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎 \mod 𝚌𝚘𝚗𝚜𝚝𝚊𝚗𝚝$ replaced by $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎$ ).

Keywords

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)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝐶𝐿𝐼𝑄𝑈𝐸

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 \mod 𝙼 = 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛 \mod 𝙼

Graph property(ies)

𝐍𝐒𝐂𝐂

= 𝙽𝙴𝚀𝚄𝙸𝚅

Graph model

Parts (A) and (B) of Figure 5.274.1 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 one equivalence class: We have two equivalence classes that respectively correspond to values ${3, 6, 15}$ and ${2, 5}$ .

Figure 5.274.1. Initial and final graph of the $𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎$ constraint


(a)	(b)

Global Constraint Catalog

5. Global Constraint Catalogue

5.274. nequivalence

Figure 5.274.1. Initial and final graph of the 𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎 constraint

Figure 5.274.1. Initial and final graph of the $𝚗𝚎𝚚𝚞𝚒𝚟𝚊𝚕𝚎𝚗𝚌𝚎$ constraint