## 5.62. change_continuity

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Origin

N. Beldiceanu

Constraint

$\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢}\left(\begin{array}{c}\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴},\hfill \\ \mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈},\hfill \\ \mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴},\hfill \\ \mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴},\hfill \\ \mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈},\hfill \\ \mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈},\hfill \\ \mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴},\hfill \\ \mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈},\hfill \\ \mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\hfill \\ \mathrm{𝙲𝚃𝚁}\hfill \end{array}\right)$

Arguments
 $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}$ $\mathrm{𝚊𝚝𝚘𝚖}$
Restrictions
 $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}\ge 0$ $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}\ge 0$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}\ge 0$ $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}\ge \mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}\ge 0$ $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}\ge \mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}\ge 0$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}\ge 0$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$ $\mathrm{𝙲𝚃𝚁}\in \left[=,\ne ,<,\ge ,>,\le \right]$
Purpose

On the one hand a change is defined by the fact that constraint $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛}\mathrm{𝙲𝚃𝚁}\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i+1\right].\mathrm{𝚟𝚊𝚛}$ holds.

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

A period of change on variables

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i+1\right].\mathrm{𝚟𝚊𝚛},\cdots ,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[j\right].\mathrm{𝚟𝚊𝚛}\left(i

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

A period of continuity on variables

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i\right].\mathrm{𝚟𝚊𝚛},\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[i+1\right].\mathrm{𝚟𝚊𝚛},\cdots ,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\left[j\right].\mathrm{𝚟𝚊𝚛}\left(i

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

The constraint $\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢}$ holds if and only if:

• $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to the number of periods of change,

• $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ is equal to the number of periods of continuity,

• $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to the number of variables of the smallest period of change,

• $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to the number of variables of the largest period of change,

• $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ is equal to the number of variables of the smallest period of continuity,

• $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ is equal to the number of variables of the largest period of continuity,

• $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to the total number of changes,

• $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ is equal to the total number of continuities.

Example
$\left(3,2,2,4,2,4,6,4,〈1,3,1,8,8,4,7,7,7,7,2〉,\ne \right)$

Figure 5.62.1 makes clear the different parameters that are associated with the given example for the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}=〈1,3,1,8,8,4,7,7,7,7,2〉$. 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 $\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢}$ constraint holds since:

• Its number of periods of change $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to 3 (i.e., the 3 periods depicted on top of Figure 5.62.1),

• Its number of periods of continuity $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ 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 $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ 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 $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ 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 $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ 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 $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ 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 $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ is equal to 6 (i.e., the number of occurrences of character | in Figure 5.62.1),

• The total number of continuities $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ is equal to 4.

Typical
 $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}>0$ $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}>0$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}>0$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}>0$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}>0$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}>0$ $|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|>1$ $\mathrm{𝚛𝚊𝚗𝚐𝚎}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}.\mathrm{𝚟𝚊𝚛}\right)>1$ $\mathrm{𝙲𝚃𝚁}\in \left[\ne \right]$
Symmetry

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

Arg. properties
• Functional dependency: $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

• Functional dependency: $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ determined by $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ and $\mathrm{𝙲𝚃𝚁}$.

Remark

If the variables of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ 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 $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ gives the size of the longest run of the permutation; A run is a maximal increasing contiguous subsequence in a permutation.

Keywords
Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝑃𝐴𝑇𝐻}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}\mathrm{𝙲𝚃𝚁}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}$
Graph property(ies)
 $•$$\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $•$$\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $•$$\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ $•$$\mathrm{𝐍𝐀𝐑𝐂}$$=\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$

Graph class
 $•$$\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}$ $•$$\mathrm{𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴}$ $•$$\mathrm{𝙽𝙾}_\mathrm{𝙻𝙾𝙾𝙿}$

Arc input(s)

$\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$

Arc generator
$\mathrm{𝑃𝐴𝑇𝐻}$$↦\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1},\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{1}.\mathrm{𝚟𝚊𝚛}¬\mathrm{𝙲𝚃𝚁}\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎𝚜}\mathtt{2}.\mathrm{𝚟𝚊𝚛}$
Graph property(ies)
 $•$$\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $•$$\mathrm{𝐌𝐈𝐍}_\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $•$$\mathrm{𝐌𝐀𝐗}_\mathrm{𝐍𝐂𝐂}$$=\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $•$$\mathrm{𝐍𝐀𝐑𝐂}$$=\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$

Graph class
 $•$$\mathrm{𝙰𝙲𝚈𝙲𝙻𝙸𝙲}$ $•$$\mathrm{𝙱𝙸𝙿𝙰𝚁𝚃𝙸𝚃𝙴}$ $•$$\mathrm{𝙽𝙾}_\mathrm{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.

Automaton

Figures 5.62.35.62.45.62.75.62.85.62.115.62.12 and 5.62.15 depict the automata associated with the different graph parameters of the $\mathrm{𝚌𝚑𝚊𝚗𝚐𝚎}_\mathrm{𝚌𝚘𝚗𝚝𝚒𝚗𝚞𝚒𝚝𝚢}$ constraint. For the automata that respectively compute $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$, $\mathrm{𝙽𝙱}_\mathrm{𝙿𝙴𝚁𝙸𝙾𝙳}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$, $\mathrm{𝙼𝙸𝙽}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$, $\mathrm{𝙼𝙰𝚇}_\mathrm{𝚂𝙸𝚉𝙴}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙷𝙰𝙽𝙶𝙴}$ and $\mathrm{𝙽𝙱}_\mathrm{𝙲𝙾𝙽𝚃𝙸𝙽𝚄𝙸𝚃𝚈}$ we have a 0-1 signature variable ${S}_{i}$ for each pair of consecutive variables $\left({\mathrm{𝚅𝙰𝚁}}_{i},{\mathrm{𝚅𝙰𝚁}}_{i+1}\right)$ of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$. The following signature constraint links ${\mathrm{𝚅𝙰𝚁}}_{i}$, ${\mathrm{𝚅𝙰𝚁}}_{i+1}$ and ${S}_{i}$: ${\mathrm{𝚅𝙰𝚁}}_{i}\mathrm{𝙲𝚃𝚁}{\mathrm{𝚅𝙰𝚁}}_{i+1}⇔{S}_{i}$.