Global Constraint Catalog: Cnvalues

5.288. nvalues

DESCRIPTION

LINKS

GRAPH

Origin

Inspired by $𝚗𝚟𝚊𝚕𝚞𝚎$ and $𝚌𝚘𝚞𝚗𝚝$ .

Constraint

$𝚗𝚟𝚊𝚕𝚞𝚎𝚜 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚁𝙴𝙻𝙾𝙿, 𝙻𝙸𝙼𝙸𝚃)$

Arguments

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)$
$𝚁𝙴𝙻𝙾𝙿$	$𝚊𝚝𝚘𝚖$
$𝙻𝙸𝙼𝙸𝚃$	$𝚍𝚟𝚊𝚛$

Restrictions

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

𝚁𝙴𝙻𝙾𝙿 \in [=, \neq, <, \geq, >, \leq]

Purpose

Let $N$ be the number of distinct values assigned to the variables of the $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ collection. Enforce condition $N 𝚁𝙴𝙻𝙾𝙿 𝙻𝙸𝙼𝙸𝚃$ to hold.

Example

(〈4, 5, 5, 4, 1, 5〉, =, 3)

The $𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ constraint holds since the number of distinct values occurring within the collection $〈 4, 5, 5, 4, 1, 5 〉$ is equal (i.e., $𝚁𝙴𝙻𝙾𝙿$ is set to $=$ ) to its third argument $𝙻𝙸𝙼𝙸𝚃 = 3$ .

Typical

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 1

𝙻𝙸𝙼𝙸𝚃 > 1

𝙻𝙸𝙼𝙸𝚃 < | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |

𝚁𝙴𝙻𝙾𝙿 \in [=, <, \geq, >, \leq]

Symmetries

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ are permutable.
All occurrences of two distinct values of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛$ can be swapped; all occurrences of a value of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛$ can be renamed to any unused value.

Arg. properties

Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝚁𝙴𝙻𝙾𝙿 \in [<, \leq]$ .
Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝚁𝙴𝙻𝙾𝙿 \in [=]$ , $𝙻𝙸𝙼𝙸𝚃 = 1$ and $| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 0$ .
Contractible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝚁𝙴𝙻𝙾𝙿 \in [=]$ and $𝙻𝙸𝙼𝙸𝚃 = | 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |$ .
Extensible wrt. $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ when $𝚁𝙴𝙻𝙾𝙿 \in [\geq, >]$ .

Usage

Used in the Constraint(s) on sets slot for defining some constraints like $𝚊𝚜𝚜𝚒𝚐𝚗_𝚊𝚗𝚍_𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ , $𝚌𝚒𝚛𝚌𝚞𝚒𝚝_𝚌𝚕𝚞𝚜𝚝𝚎𝚛$ or $𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍_𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎$ .

Reformulation

The $𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚁𝙴𝙻𝙾𝙿, 𝙻𝙸𝙼𝙸𝚃)$ constraint can be expressed in term of the conjunction $𝚗𝚟𝚊𝚕𝚞𝚎$ $(𝑁𝑉, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$ $\land$ $𝑁𝑉 𝚁𝙴𝙻𝙾𝙿 𝙻𝙸𝙼𝙸𝚃$ .

Systems

nvalues in Gecode.

Used in

$𝚊𝚜𝚜𝚒𝚐𝚗_𝚊𝚗𝚍_𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ , $𝚌𝚒𝚛𝚌𝚞𝚒𝚝_𝚌𝚕𝚞𝚜𝚝𝚎𝚛$ , $𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍_𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎$ , $𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍_𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜$ .

common keyword: $𝚗𝚟𝚊𝚕𝚞𝚎𝚜_𝚎𝚡𝚌𝚎𝚙𝚝_0$ (counting constraint,number of distinct values).

specialisation: $𝚗𝚟𝚊𝚕𝚞𝚎$ (replace a comparison with the number of distinct values by an equality with the number of distinct values).

Keywords

constraint type: counting constraint, value partitioning constraint.

final graph structure: strongly connected component, equivalence.

modelling: number of distinct equivalence classes, number of distinct values.

problems: domination.

Cond. implications

$𝚗𝚟𝚊𝚕𝚞𝚎𝚜 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚁𝙴𝙻𝙾𝙿, 𝙻𝙸𝙼𝙸𝚃)$

with $𝚖𝚒𝚗𝚟𝚊𝚕$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛) > 0$

implies $𝚗𝚟𝚊𝚕𝚞𝚎𝚜_𝚎𝚡𝚌𝚎𝚙𝚝_0$ $(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚁𝙴𝙻𝙾𝙿, 𝙻𝙸𝙼𝙸𝚃)$ .

Arc input(s)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝐶𝐿𝐼𝑄𝑈𝐸

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 = 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛

Graph property(ies)

𝐍𝐒𝐂𝐂

𝚁𝙴𝙻𝙾𝙿 𝙻𝙸𝙼𝙸𝚃

Graph class

𝙴𝚀𝚄𝙸𝚅𝙰𝙻𝙴𝙽𝙲𝙴

Graph model

Parts (A) and (B) of Figure 5.288.1 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 value that is assigned to some variables of the $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ collection. The 3 following values 1, 4 and 5 are used by the variables of the $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ collection.

Figure 5.288.1. Initial and final graph of the $𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ constraint


(a)	(b)

Global Constraint Catalog

5. Global Constraint Catalogue

5.288. nvalues

Figure 5.288.1. Initial and final graph of the 𝚗𝚟𝚊𝚕𝚞𝚎𝚜 constraint

Figure 5.288.1. Initial and final graph of the $𝚗𝚟𝚊𝚕𝚞𝚎𝚜$ constraint