## 5.350. valley

 DESCRIPTION LINKS AUTOMATON
Origin
Constraint

$\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}\left(𝙽,\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}\right)$

Arguments
 $𝙽$ $\mathrm{𝚍𝚟𝚊𝚛}$ $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ $\mathrm{𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗}\left(\mathrm{𝚟𝚊𝚛}-\mathrm{𝚍𝚟𝚊𝚛}\right)$
Restrictions
 $𝙽\ge 0$ $2*𝙽\le \mathrm{𝚖𝚊𝚡}\left(|\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}|-1,0\right)$ $\mathrm{𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍}$$\left(\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂},\mathrm{𝚟𝚊𝚛}\right)$
Purpose

A variable ${V}_{k}$ $\left(1 of the sequence of variables $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}={V}_{1},...,{V}_{m}$ is a valley if and only if there exists an $i$ $\left(1 such that ${V}_{i-1}>{V}_{i}$ and ${V}_{i}={V}_{i+1}=...={V}_{k}$ and ${V}_{k}<{V}_{k+1}$. $𝙽$ is the total number of valleys of the sequence of variables $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$.

Example
$\left(\begin{array}{c}1,〈\begin{array}{c}\mathrm{𝚟𝚊𝚛}-1,\hfill \\ \mathrm{𝚟𝚊𝚛}-1,\hfill \\ \mathrm{𝚟𝚊𝚛}-4,\hfill \\ \mathrm{𝚟𝚊𝚛}-8,\hfill \\ \mathrm{𝚟𝚊𝚛}-8,\hfill \\ \mathrm{𝚟𝚊𝚛}-2,\hfill \\ \mathrm{𝚟𝚊𝚛}-7,\hfill \\ \mathrm{𝚟𝚊𝚛}-1\hfill \end{array}〉\hfill \end{array}\right)$

The $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ constraint holds since the sequence $11488271$ contains one valley that corresponds to the variable that is assigned to value 2.

##### Figure 5.350.1. The sequence and its unique valley Symmetries
• Items of $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ can be reversed.

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

Usage

Useful for constraining the number of valleys of a sequence of domain variables.

Remark

Since the arity of the arc constraint is not fixed, the $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ constraint cannot be currently described. However, this would not hold anymore if we were introducing a slot that specifies how to merge adjacent vertices of the final graph.

See also

specialisation: $\mathrm{𝚗𝚘}_\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ (the variable counting the number of valleys is set to 0 and removed).

Keywords
Automaton

Figure 5.350.2 depicts the automaton associated with the $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ constraint. To each pair of consecutive variables $\left({\mathrm{𝚅𝙰𝚁}}_{i},{\mathrm{𝚅𝙰𝚁}}_{i+1}\right)$ of the collection $\mathrm{𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂}$ corresponds a signature variable ${𝚂}_{i}$. The following signature constraint links ${\mathrm{𝚅𝙰𝚁}}_{i}$, ${\mathrm{𝚅𝙰𝚁}}_{i+1}$ and ${𝚂}_{i}$: $\left({\mathrm{𝚅𝙰𝚁}}_{i}<{\mathrm{𝚅𝙰𝚁}}_{i+1}⇔{𝚂}_{i}=0\right)\wedge \left({\mathrm{𝚅𝙰𝚁}}_{i}={\mathrm{𝚅𝙰𝚁}}_{i+1}⇔{𝚂}_{i}=1\right)\wedge \left({\mathrm{𝚅𝙰𝚁}}_{i}>{\mathrm{𝚅𝙰𝚁}}_{i+1}⇔{𝚂}_{i}=2\right)$.

##### Figure 5.350.2. Automaton of the $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ constraint ##### Figure 5.350.3. Hypergraph of the reformulation corresponding to the automaton of the $\mathrm{𝚟𝚊𝚕𝚕𝚎𝚢}$ constraint 