## 5.203. longest_change

Origin
Constraint

$\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi Έ\pi \pi ΄},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi ²\pi \pi }\right)$

Arguments
 $\mathrm{\pi \pi Έ\pi \pi ΄}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi ²\pi \pi }$ $\mathrm{\pi \pi \pi \pi }$
Restrictions
 $\mathrm{\pi \pi Έ\pi \pi ΄}\beta ₯0$ $\mathrm{\pi \pi Έ\pi \pi ΄}<|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|$ $\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

$\mathrm{\pi \pi Έ\pi \pi ΄}$ is the maximum number of consecutive variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ for which constraint $\mathrm{\pi ²\pi \pi }$ holds in an uninterrupted way. We count a change when $X\mathrm{\pi ²\pi \pi }Y$ holds; $X$ and $Y$ are two consecutive variables of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example

The $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint holds since its first argument $\mathrm{\pi \pi Έ\pi \pi ΄}=4$ is fixed to the length of the longest subsequence of consecutive values of the collection $\beta ©8,8,3,4,1,1,5,5,2\beta ͺ$ such that two consecutive values are distinct (i.e.,Β subsequence $8341$).

Symmetry

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

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)
$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$$=\mathrm{\pi \pi Έ\pi \pi ΄}$

Graph model

In order to specify the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint, we use $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$, which is the number of vertices of the largest connected component. Since the initial graph corresponds to a path, this will be the length of the longest path in the final graph.

PartsΒ (A) andΒ (B) of FigureΒ 5.203.1 respectively show the initial and final graph associated with the Example slot. Since we use the $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi }$ graph property we show the largest connected component of the final graph. It corresponds to the longest period of uninterrupted changes: sequence $8,3,4,1$ that involves 4 consecutive variables.

Automaton

FigureΒ 5.203.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi \pi \pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi }$ constraint. To each pair of consecutive variables $\left({\mathrm{\pi  \pi °\pi }}_{i},{\mathrm{\pi  \pi °\pi }}_{i+1}\right)$ of the collection $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ corresponds a 0-1 signature variable ${\mathrm{\pi }}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${\mathrm{\pi }}_{i}$: ${\mathrm{\pi  \pi °\pi }}_{i}\mathrm{\pi ²\pi \pi }{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {\mathrm{\pi }}_{i}$.