Global Constraint Catalog: Cminimum

5.265. minimum_modulo

DESCRIPTION

LINKS

GRAPH

Origin

Derived from $𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ .

Constraint

$𝚖𝚒𝚗𝚒𝚖𝚞𝚖_𝚖𝚘𝚍𝚞𝚕𝚘 (𝙼𝙸𝙽, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝙼)$

Arguments

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

Restrictions

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 0

𝙼 > 0

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

Purpose

$𝙼𝙸𝙽$ is a minimum value of the collection of domain variables $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ according to the following partial ordering: $(X \mod 𝙼) < (Y \mod 𝙼)$ .

Example

(6, 〈9, 1, 7, 6, 5〉, 3)

(9, 〈9, 1, 7, 6, 5〉, 3)

The $𝚖𝚒𝚗𝚒𝚖𝚞𝚖_𝚖𝚘𝚍𝚞𝚕𝚘$ constraints hold since $𝙼𝙸𝙽$ is respectively set to values 6 and 9, where $6 \mod 3 = 0$ and $9 \mod 3 = 0$ are both less than or equal to all the expressions $9 \mod 3 = 0$ , $1 \mod 3 = 1$ , $7 \mod 3 = 1$ , $6 \mod 3 = 0$ , and $5 \mod 3 = 2$ .

Typical

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 1

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛) > 1

𝙼 > 1

𝙼 <

𝚖𝚊𝚡𝚟𝚊𝚕

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛)

Symmetry

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ are permutable.

Arg. properties

Functional dependency: $𝙼𝙸𝙽$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙼$ .

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

Keywords

characteristic of a constraint: modulo, maxint, minimum.

constraint arguments: pure functional dependency.

constraint type: order constraint.

modelling: functional dependency.

Arc input(s)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝐶𝐿𝐼𝑄𝑈𝐸

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

⋁ (\begin{matrix} 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚔𝚎𝚢 = 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚔𝚎𝚢, \\ 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 \mod 𝙼 < 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛 \mod 𝙼 \end{matrix})

Graph property(ies)

𝐎𝐑𝐃𝐄𝐑

(0, 𝙼𝙰𝚇𝙸𝙽𝚃, 𝚟𝚊𝚛) = 𝙼𝙸𝙽

Graph model

We use a similar definition that the one that was utilised for the $𝚖𝚒𝚗𝚒𝚖𝚞𝚖$ constraint. Within the arc constraint we replace the condition $X < Y$ by the condition $(X \mod 𝙼) < (Y \mod 𝙼)$ .

Parts (A) and (B) of Figure 5.265.1 respectively show the initial and final graph associated with the second example of the Example slot. Since we use the $𝐎𝐑𝐃𝐄𝐑$ graph property, the vertex of rank 0 (without considering the loops) associated with value 9 is outlined with a thick circle.

Figure 5.265.1. Initial and final graph of the $𝚖𝚒𝚗𝚒𝚖𝚞𝚖_𝚖𝚘𝚍𝚞𝚕𝚘$ constraint


(a)	(b)

Global Constraint Catalog

5. Global Constraint Catalogue

5.265. minimum_modulo

Figure 5.265.1. Initial and final graph of the 𝚖𝚒𝚗𝚒𝚖𝚞𝚖_𝚖𝚘𝚍𝚞𝚕𝚘 constraint

Figure 5.265.1. Initial and final graph of the $𝚖𝚒𝚗𝚒𝚖𝚞𝚖_𝚖𝚘𝚍𝚞𝚕𝚘$ constraint