Global Constraint Catalog: Corth_link_ori_siz

5.310. orth_link_ori_siz_end

DESCRIPTION

LINKS

GRAPH

Origin

Used by several constraints between orthotopes

Constraint

$𝚘𝚛𝚝𝚑_𝚕𝚒𝚗𝚔_𝚘𝚛𝚒_𝚜𝚒𝚣_𝚎𝚗𝚍 (𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴)$

Argument

𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴

𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚘𝚛𝚒 - 𝚍𝚟𝚊𝚛, 𝚜𝚒𝚣 - 𝚍𝚟𝚊𝚛, 𝚎𝚗𝚍 - 𝚍𝚟𝚊𝚛)

Restrictions

| 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 | > 0

𝚛𝚎𝚚𝚞𝚒𝚛𝚎_𝚊𝚝_𝚕𝚎𝚊𝚜𝚝

(2, 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴, [𝚘𝚛𝚒, 𝚜𝚒𝚣, 𝚎𝚗𝚍])

𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚜𝚒𝚣 \geq 0

𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚘𝚛𝚒 \leq 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚎𝚗𝚍

Purpose

Enforce for each item of the $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ collection the constraint $𝚘𝚛𝚒 + 𝚜𝚒𝚣 = 𝚎𝚗𝚍$ .

Example

(〈𝚘𝚛𝚒 - 2 𝚜𝚒𝚣 - 2 𝚎𝚗𝚍 - 4, 𝚘𝚛𝚒 - 1 𝚜𝚒𝚣 - 3 𝚎𝚗𝚍 - 4〉)

The $𝚘𝚛𝚝𝚑_𝚕𝚒𝚗𝚔_𝚘𝚛𝚒_𝚜𝚒𝚣_𝚎𝚗𝚍$ constraint holds since the two items $〈 𝚘𝚛𝚒 - 2 𝚜𝚒𝚣 - 2 𝚎𝚗𝚍 - 4 〉$ and $〈 𝚘𝚛𝚒 - 1 𝚜𝚒𝚣 - 3 𝚎𝚗𝚍 - 4 〉$ respectively verify the conditions $2 + 2 = 4$ and $1 + 3 = 4$ .

Typical

| 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 | > 1

𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚜𝚒𝚣 > 0

Symmetries

Items of $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ are permutable.
One and the same constant can be added to the $𝚘𝚛𝚒$ and $𝚎𝚗𝚍$ attributes of all items of $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ .
One and the same constant can be added to the $𝚜𝚒𝚣$ and $𝚎𝚗𝚍$ attributes of all items of $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ .

Arg. properties

Functional dependency: $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚘𝚛𝚒$ determined by $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚜𝚒𝚣$ and $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚎𝚗𝚍$ .
Functional dependency: $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚜𝚒𝚣$ determined by $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚘𝚛𝚒$ and $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚎𝚗𝚍$ .
Functional dependency: $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚎𝚗𝚍$ determined by $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚘𝚛𝚒$ and $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 . 𝚜𝚒𝚣$ .
Contractible wrt. $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ .

Usage

Used in the Arc constraint(s) slot for defining some constraints like $𝚍𝚒𝚏𝚏𝚗$ , $𝚙𝚕𝚊𝚌𝚎_𝚒𝚗_𝚙𝚢𝚛𝚊𝚖𝚒𝚍$ or $𝚘𝚛𝚝𝚑𝚜_𝚊𝚛𝚎_𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍$ .

Used in

$𝚍𝚒𝚏𝚏𝚗$ , $𝚘𝚛𝚝𝚑_𝚘𝚗_𝚝𝚑𝚎_𝚐𝚛𝚘𝚞𝚗𝚍$ , $𝚘𝚛𝚝𝚑_𝚘𝚗_𝚝𝚘𝚙_𝚘𝚏_𝚘𝚛𝚝𝚑$ , $𝚘𝚛𝚝𝚑𝚜_𝚊𝚛𝚎_𝚌𝚘𝚗𝚗𝚎𝚌𝚝𝚎𝚍$ , $𝚝𝚠𝚘_𝚘𝚛𝚝𝚑_𝚊𝚛𝚎_𝚒𝚗_𝚌𝚘𝚗𝚝𝚊𝚌𝚝$ , $𝚝𝚠𝚘_𝚘𝚛𝚝𝚑_𝚌𝚘𝚕𝚞𝚖𝚗$ , $𝚝𝚠𝚘_𝚘𝚛𝚝𝚑_𝚍𝚘_𝚗𝚘𝚝_𝚘𝚟𝚎𝚛𝚕𝚊𝚙$ , $𝚝𝚠𝚘_𝚘𝚛𝚝𝚑_𝚒𝚗𝚌𝚕𝚞𝚍𝚎$ .

Keywords

constraint arguments: pure functional dependency.

constraint type: decomposition.

geometry: orthotope.

modelling: functional dependency.

Arc input(s)

$𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$

Arc generator

𝑆𝐸𝐿𝐹

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎)

Arc arity

Arc constraint(s)

𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎 . 𝚘𝚛𝚒 + 𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎 . 𝚜𝚒𝚣 = 𝚘𝚛𝚝𝚑𝚘𝚝𝚘𝚙𝚎 . 𝚎𝚗𝚍

Graph property(ies)

𝐍𝐀𝐑𝐂

= | 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 |

Graph model

Parts (A) and (B) of Figure 5.310.1 respectively show the initial and final graph associated with the Example slot. Since we use the $𝐍𝐀𝐑𝐂$ graph property, the loops of the final graph are stressed in bold.

Figure 5.310.1. Initial and final graph of the $𝚘𝚛𝚝𝚑_𝚕𝚒𝚗𝚔_𝚘𝚛𝚒_𝚜𝚒𝚣_𝚎𝚗𝚍$ constraint


(a)	(b)

Signature

Since we use the $𝑆𝐸𝐿𝐹$ arc generator on the $𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴$ collection the number of arcs of the initial graph is equal to $| 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 |$ . Therefore the maximum number of arcs of the final graph is also equal to $| 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 |$ . For this reason we can rewrite the graph property $𝐍𝐀𝐑𝐂 = | 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 |$ to $𝐍𝐀𝐑𝐂 \geq | 𝙾𝚁𝚃𝙷𝙾𝚃𝙾𝙿𝙴 |$ and simplify $\underset{̲}{\bar{𝐍𝐀𝐑𝐂}}$ to $\bar{𝐍𝐀𝐑𝐂}$ .

Global Constraint Catalog

5. Global Constraint Catalogue

5.310. orth_link_ori_siz_end

Figure 5.310.1. Initial and final graph of the 𝚘𝚛𝚝𝚑_𝚕𝚒𝚗𝚔_𝚘𝚛𝚒_𝚜𝚒𝚣_𝚎𝚗𝚍 constraint

Figure 5.310.1. Initial and final graph of the $𝚘𝚛𝚝𝚑_𝚕𝚒𝚗𝚔_𝚘𝚛𝚒_𝚜𝚒𝚣_𝚎𝚗𝚍$ constraint