## 5.62. change_continuity

Origin

N.Β Beldiceanu

Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}\left(\begin{array}{c}\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄},\hfill \\ \mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi },\hfill \\ \mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄},\hfill \\ \mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄},\hfill \\ \mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi },\hfill \\ \mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi },\hfill \\ \mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄},\hfill \\ \mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi },\hfill \\ \mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\hfill \\ \mathrm{\pi ²\pi \pi }\hfill \end{array}\right)$

Arguments
 $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi ²\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$
Restrictions
 $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}\beta ₯0$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }\beta ₯0$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}\beta ₯0$ $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}\beta ₯\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }\beta ₯0$ $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }\beta ₯\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}\beta ₯0$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }\beta ₯0$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi \pi }\right)$
Purpose

On the one hand a change is defined by the fact that constraint $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i+1\right].\mathrm{\pi \pi \pi }$ holds.

On the other hand a continuity is defined by the fact that constraint $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i+1\right].\mathrm{\pi \pi \pi }$ does not hold.

A period of change on variables

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i+1\right].\mathrm{\pi \pi \pi },\beta ―,\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[j\right].\mathrm{\pi \pi \pi }\left(i

is defined by the fact that all constraints $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[k\right].\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[k+1\right].\mathrm{\pi \pi \pi }$ hold for $k\beta \left[i,j-1\right]$.

A period of continuity on variables

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i\right].\mathrm{\pi \pi \pi },\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[i+1\right].\mathrm{\pi \pi \pi },\beta ―,\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[j\right].\mathrm{\pi \pi \pi }\left(i

is defined by the fact that all constraints $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[k\right].\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }\left[k+1\right].\mathrm{\pi \pi \pi }$ do not hold for $k\beta \left[i,j-1\right]$.

The constraint $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}$ holds if and only if:

• $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to the number of periods of change,

• $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to the number of periods of continuity,

• $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to the number of variables of the smallest period of change,

• $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to the number of variables of the largest period of change,

• $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to the number of variables of the smallest period of continuity,

• $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to the number of variables of the largest period of continuity,

• $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to the total number of changes,

• $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to the total number of continuities.

Example

FigureΒ 5.62.1 makes clear the different parameters that are associated with the given example for the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }=\beta ©1,3,1,8,8,4,7,7,7,7,2\beta ͺ$. 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 $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}$ constraint holds since:

• Its number of periods of change $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to 3 (i.e.,Β the 3 periods depicted on top of FigureΒ 5.62.1),

• Its number of periods of continuity $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to 2 (i.e.,Β the 2 periods depicted below FigureΒ 5.62.1),

• The number of variables of its smallest period of change $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to 2 (i.e.,Β the number of variables involved in the third period of change $72$ depicted on top of FigureΒ 5.62.1),

• The number of variables of the largest period of change $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to 4 (i.e.,Β the number of variables involved in the first period of change $1318$ depicted on top of FigureΒ 5.62.1),

• The number of variables of the smallest period of continuity $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to 2 (i.e.,Β the number of variables involved in the first period $88$ depicted below FigureΒ 5.62.1),

• The number of variables of the largest period of continuity $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to 4 (i.e.,Β the number of variables involved in the second period $7777$ depicted below FigureΒ 5.62.1),

• The total number of changes $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ is equal to 6 (i.e.,Β the number of occurrences of character | in FigureΒ 5.62.1),

• The total number of continuities $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ is equal to 4.

Typical
 $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}>0$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }>0$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}>0$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }>0$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}>0$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }>0$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>1$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$
Symmetry

One and the same constant can be added to the $\mathrm{\pi \pi \pi }$ attribute of all items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Arg. properties
• Functional dependency: $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

• Functional dependency: $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ determined by $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ and $\mathrm{\pi ²\pi \pi }$.

Remark

If the variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ 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 $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ 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)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi \pi ΄\pi \pi »}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }\mathrm{\pi ²\pi \pi }\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
 $\beta ’$$\mathrm{\pi \pi \pi }$$=\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ $\beta ’$$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

Arc input(s)

$\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$

Arc generator
$\mathrm{\pi \pi ΄\pi \pi »}$$\beta ¦\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1},\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}\right)$

Arc arity
Arc constraint(s)
$\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{1}.\mathrm{\pi \pi \pi }Β¬\mathrm{\pi ²\pi \pi }\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi }\mathtt{2}.\mathrm{\pi \pi \pi }$
Graph property(ies)
 $\beta ’$$\mathrm{\pi \pi \pi }$$=\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\beta ’$$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\beta ’$$\mathrm{\pi \pi \pi \pi }$$=\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$

Graph class
 $\beta ’$$\mathrm{\pi °\pi ²\pi \pi ²\pi »\pi Έ\pi ²}$ $\beta ’$$\mathrm{\pi ±\pi Έ\pi Ώ\pi °\pi \pi \pi Έ\pi \pi ΄}$ $\beta ’$$\mathrm{\pi ½\pi Ύ}_\mathrm{\pi »\pi Ύ\pi Ύ\pi Ώ}$

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.

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

Automaton

FiguresΒ 5.62.3 ,Β 5.62.4 ,Β 5.62.7 ,Β 5.62.8 ,Β 5.62.11 ,Β 5.62.12 andΒ 5.62.15 depict the automata associated with the different graph parameters of the $\mathrm{\pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi ’}$ constraint. For the automata that respectively compute $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$, $\mathrm{\pi ½\pi ±}_\mathrm{\pi Ώ\pi ΄\pi \pi Έ\pi Ύ\pi ³}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$, $\mathrm{\pi Ό\pi Έ\pi ½}_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$, $\mathrm{\pi Ό\pi °\pi }_\mathrm{\pi \pi Έ\pi \pi ΄}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi ·\pi °\pi ½\pi Ά\pi ΄}$ and $\mathrm{\pi ½\pi ±}_\mathrm{\pi ²\pi Ύ\pi ½\pi \pi Έ\pi ½\pi \pi Έ\pi \pi }$ we have a 0-1 signature variable ${S}_{i}$ for each pair of consecutive variables $\left({\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi  \pi °\pi }}_{i+1}\right)$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${S}_{i}$: ${\mathrm{\pi  \pi °\pi }}_{i}\mathrm{\pi ²\pi \pi }{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}$.