Global Constraint Catalog: Cchange

5.62. change_continuity

DESCRIPTION

LINKS

GRAPH

AUTOMATON

Origin

N. Beldiceanu

Constraint

$𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 (\begin{matrix} 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴, \\ 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈, \\ 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴, \\ 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴, \\ 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈, \\ 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈, \\ 𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴, \\ 𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈, \\ 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, \\ 𝙲𝚃𝚁 \end{matrix})$

Arguments

$𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$	$𝚍𝚟𝚊𝚛$
$𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$	$𝚍𝚟𝚊𝚛$
$𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$	$𝚍𝚟𝚊𝚛$
$𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$	$𝚍𝚟𝚊𝚛$
$𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$	$𝚍𝚟𝚊𝚛$
$𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$	$𝚍𝚟𝚊𝚛$
$𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$	$𝚍𝚟𝚊𝚛$
$𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$	$𝚍𝚟𝚊𝚛$
$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$	$𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)$
$𝙲𝚃𝚁$	$𝚊𝚝𝚘𝚖$

Restrictions

𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴 \geq 0

𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 \geq 0

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴 \geq 0

𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴 \geq 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 \geq 0

𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 \geq 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈

𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴 \geq 0

𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 \geq 0

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

𝙲𝚃𝚁 \in [=, \neq, <, \geq, >, \leq]

Purpose

On the one hand a change is defined by the fact that constraint $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i] . 𝚟𝚊𝚛 𝙲𝚃𝚁 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i + 1] . 𝚟𝚊𝚛$ holds.

On the other hand a continuity is defined by the fact that constraint $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i] . 𝚟𝚊𝚛 𝙲𝚃𝚁 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i + 1] . 𝚟𝚊𝚛$ does not hold.

A period of change on variables

𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i] . 𝚟𝚊𝚛, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i + 1] . 𝚟𝚊𝚛, \dots, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [j] . 𝚟𝚊𝚛 (i < j)

is defined by the fact that all constraints $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [k] . 𝚟𝚊𝚛 𝙲𝚃𝚁 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [k + 1] . 𝚟𝚊𝚛$ hold for $k \in [i, j - 1]$ .

A period of continuity on variables

𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i] . 𝚟𝚊𝚛, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [i + 1] . 𝚟𝚊𝚛, \dots, 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [j] . 𝚟𝚊𝚛 (i < j)

is defined by the fact that all constraints $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [k] . 𝚟𝚊𝚛 𝙲𝚃𝚁 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 [k + 1] . 𝚟𝚊𝚛$ do not hold for $k \in [i, j - 1]$ .

The constraint $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ holds if and only if:

$𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to the number of periods of change,
$𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to the number of periods of continuity,
$𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to the number of variables of the smallest period of change,
$𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to the number of variables of the largest period of change,
$𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to the number of variables of the smallest period of continuity,
$𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to the number of variables of the largest period of continuity,
$𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to the total number of changes,
$𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to the total number of continuities.

Example

(3, 2, 2, 4, 2, 4, 6, 4, 〈1, 3, 1, 8, 8, 4, 7, 7, 7, 7, 2〉, \neq)

Figure 5.62.1 makes clear the different parameters that are associated with the given example for the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 = 〈 1, 3, 1, 8, 8, 4, 7, 7, 7, 7, 2 〉$ . We place character | for representing a change and a blank for a continuity. On top of the solution we represent the different periods of change, while below we show the different periods of continuity. The $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint holds since:

Its number of periods of change $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to 3 (i.e., the 3 periods depicted on top of Figure 5.62.1),
Its number of periods of continuity $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to 2 (i.e., the 2 periods depicted below Figure 5.62.1),
The number of variables of its smallest period of change $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to 2 (i.e., the number of variables involved in the third period of change $72$ depicted on top of Figure 5.62.1),
The number of variables of the largest period of change $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to 4 (i.e., the number of variables involved in the first period of change $1318$ depicted on top of Figure 5.62.1),
The number of variables of the smallest period of continuity $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to 2 (i.e., the number of variables involved in the first period $88$ depicted below Figure 5.62.1),
The number of variables of the largest period of continuity $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to 4 (i.e., the number of variables involved in the second period $7777$ depicted below Figure 5.62.1),
The total number of changes $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ is equal to 6 (i.e., the number of occurrences of character | in Figure 5.62.1),
The total number of continuities $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ is equal to 4.

Figure 5.62.1. Illustration of the Example slot: periods of changes and periods of continuities wrt the constraint $𝙲𝚃𝚁$ equal to $\neq$

Typical

𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴 > 0

𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 > 0

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴 > 0

𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 > 0

𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴 > 0

𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 > 0

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 1

𝚛𝚊𝚗𝚐𝚎

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 . 𝚟𝚊𝚛) > 1

𝙲𝚃𝚁 \in [\neq]

Symmetry

One and the same constant can be added to the $𝚟𝚊𝚛$ attribute of all items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ .

Arg. properties

Functional dependency: $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .
Functional dependency: $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ determined by $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ and $𝙲𝚃𝚁$ .

Remark

If the variables of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ have to take distinct values between 1 and the total number of variables, we have what is called a permutation. In this case, if we choose the binary constraint $<$ , then $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ gives the size of the longest run of the permutation; A run is a maximal increasing contiguous subsequence in a permutation.

Keywords

characteristic of a constraint: automaton, automaton with counters, automaton with same input symbol.

combinatorial object: sequence, run of a permutation, permutation.

constraint arguments: reverse of a constraint.

constraint network structure: sliding cyclic(1) constraint network(2), sliding cyclic(1) constraint network(3).

constraint type: timetabling constraint.

filtering: glue matrix.

final graph structure: connected component, apartition, acyclic, bipartite, no loop.

modelling: functional dependency.

Arc input(s)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝑃𝐴𝑇𝐻

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 𝙲𝚃𝚁 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛

Graph property(ies)

•

𝐍𝐂𝐂

= 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴

•

𝐌𝐈𝐍_𝐍𝐂𝐂

= 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴

•

𝐌𝐀𝐗_𝐍𝐂𝐂

= 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴

•

𝐍𝐀𝐑𝐂

= 𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴

Graph class

•

𝙰𝙲𝚈𝙲𝙻𝙸𝙲

•

𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴

•

𝙽𝙾_𝙻𝙾𝙾𝙿

Arc input(s)

$𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$

Arc generator

𝑃𝐴𝑇𝐻

\mapsto 𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1, 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2)

Arc arity

Arc constraint(s)

𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 1 . 𝚟𝚊𝚛 \neg 𝙲𝚃𝚁 𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜 2 . 𝚟𝚊𝚛

Graph property(ies)

•

𝐍𝐂𝐂

= 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈

•

𝐌𝐈𝐍_𝐍𝐂𝐂

= 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈

•

𝐌𝐀𝐗_𝐍𝐂𝐂

= 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈

•

𝐍𝐀𝐑𝐂

= 𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈

Graph class

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𝙰𝙲𝚈𝙲𝙻𝙸𝙲

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𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴

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𝙽𝙾_𝙻𝙾𝙾𝙿

Graph model

We use two graph constraints to respectively catch the constraints on the period of changes and of the period of continuities. In both case each period corresponds to a connected component of the final graph.

Parts (A) and (B) of Figure 5.62.2 respectively show the initial and final graph associated with the first graph constraint of the Example slot.

Figure 5.62.2. Initial and final graph of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint


(a)	(b)

Automaton: Figures 5.62.3 , 5.62.4 , 5.62.7 , 5.62.8 , 5.62.11 , 5.62.12 and 5.62.15 depict the automata associated with the different graph parameters of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint. For the automata that respectively compute $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ , $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ , $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ , $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ and $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ we have a 0-1 signature variable $S_{i}$ for each pair of consecutive variables $({𝚅𝙰𝚁}_{i}, {𝚅𝙰𝚁}_{i + 1})$ of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ . The following signature constraint links ${𝚅𝙰𝚁}_{i}$ , ${𝚅𝙰𝚁}_{i + 1}$ and $S_{i}$ : ${𝚅𝙰𝚁}_{i} 𝙲𝚃𝚁 {𝚅𝙰𝚁}_{i + 1} \Leftrightarrow S_{i}$ .

Figure 5.62.3. Automaton for the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint and its glue matrix; note that the reverse of $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ with $𝙲𝚃𝚁 \in {=, \neq}$ is the same constraint, while the reverse with $𝙲𝚃𝚁 \in {<}$ (resp. $𝙲𝚃𝚁 \in {\leq}$ ) is $𝙲𝚃𝚁 \in {>}$ (resp. $𝙲𝚃𝚁 \in {\geq}$ ).

Figure 5.62.4. Automaton for the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint and its glue matrix; note that the reverse of $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ with $𝙲𝚃𝚁 \in {=, \neq}$ is the same constraint, while the reverse with $𝙲𝚃𝚁 \in {<}$ (resp. $𝙲𝚃𝚁 \in {\leq}$ ) is $𝙲𝚃𝚁 \in {>}$ (resp. $𝙲𝚃𝚁 \in {\geq}$ ).

Figure 5.62.5. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.6. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.7. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.8. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.9. Hypergraph of the reformulation corresponding to the automaton of the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint where $𝙽$ stands for $| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |$ (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n - 1}$ )

Figure 5.62.10. Hypergraph of the reformulation corresponding to the automaton of the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint where $𝙽$ stands for $| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 |$ (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n - 1}$ )

Figure 5.62.11. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.12. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.13. Hypergraph of the reformulation corresponding to the automaton of the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n - 1}$ )

Figure 5.62.14. Hypergraph of the reformulation corresponding to the automaton of the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint (since all states of the automaton are accepting there is no restriction on the last variable $Q_{n - 1}$ )

Figure 5.62.15. Automata for the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ and $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ arguments of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; their common glue matrix when $arg C T R \in {=, \neq}$ .

Figure 5.62.16. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.17. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Global Constraint Catalog

5. Global Constraint Catalogue

5.62. change_continuity

Figure 5.62.1. Illustration of the Example slot: periods of changes and periods of continuities wrt the constraint $𝙲𝚃𝚁$ equal to $\neq$

Figure 5.62.2. Initial and final graph of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.5. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.6. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.7. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.8. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.11. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.12. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.15. Automata for the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ and $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ arguments of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; their common glue matrix when $arg C T R \in {=, \neq}$ .

Figure 5.62.16. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.17. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Global Constraint Catalog

5. Global Constraint Catalogue

5.62. change_continuity

Figure 5.62.1. Illustration of the Example slot: periods of changes and periods of continuities wrt the constraint 𝙲𝚃𝚁 equal to ≠

Figure 5.62.2. Initial and final graph of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint

Figure 5.62.5. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint

Figure 5.62.6. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint

Figure 5.62.7. Automaton for the 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint; its glue matrix when 𝙲𝚃𝚁∈{=,≠}.

Figure 5.62.8. Automaton for the 𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint; its glue matrix when 𝙲𝚃𝚁∈{=,≠}.

Figure 5.62.11. Automaton for the 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint; its glue matrix when 𝙲𝚃𝚁∈{=,≠}.

Figure 5.62.12. Automaton for the 𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint; its glue matrix when 𝙲𝚃𝚁∈{=,≠}.

Figure 5.62.15. Automata for the 𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴 and 𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 arguments of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint; their common glue matrix when argCTR∈{=,≠}.

Figure 5.62.16. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint

Figure 5.62.17. Hypergraph of the reformulation corresponding to the automaton of the 𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈 argument of the 𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢 constraint

Figure 5.62.1. Illustration of the Example slot: periods of changes and periods of continuities wrt the constraint $𝙲𝚃𝚁$ equal to $\neq$

Figure 5.62.2. Initial and final graph of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.5. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.6. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙿𝙴𝚁𝙸𝙾𝙳_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.7. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.8. Automaton for the $𝙼𝙸𝙽_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.11. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.12. Automaton for the $𝙼𝙰𝚇_𝚂𝙸𝚉𝙴_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; its glue matrix when $𝙲𝚃𝚁 \in {=, \neq}$ .

Figure 5.62.15. Automata for the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ and $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ arguments of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint; their common glue matrix when $arg C T R \in {=, \neq}$ .

Figure 5.62.16. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙷𝙰𝙽𝙶𝙴$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint

Figure 5.62.17. Hypergraph of the reformulation corresponding to the automaton of the $𝙽𝙱_𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈$ argument of the $𝚌𝚑𝚊𝚗𝚐𝚎_𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢$ constraint