## 5.52. big_valley

Origin
Constraint

$\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}\left(\mathrm{\pi ½},\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi },\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}\right)$

Arguments
 $\mathrm{\pi ½}$ $\mathrm{\pi \pi \pi \pi }$ $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ $\mathrm{\pi \pi \pi \pi \pi \pi \pi \pi \pi \pi }\left(\mathrm{\pi \pi \pi }-\mathrm{\pi \pi \pi \pi }\right)$ $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}$ $\mathrm{\pi \pi \pi }$
Restrictions
 $\mathrm{\pi ½}\beta ₯0$ $2*\mathrm{\pi ½}\beta €\mathrm{\pi \pi \pi ‘}\left(|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|-1,0\right)$ $\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)$ $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}\beta ₯0$
Purpose

A variable ${V}_{v}$ ($1) is a valley if and only if there exists an $i$ $\left(1 such that ${V}_{i-1}>{V}_{i}$ and ${V}_{i}={V}_{i+1}=\beta ―={V}_{v}$ and ${V}_{v}<{V}_{v+1}$. Similarly a variable ${V}_{p}$ ($1) of the sequence of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }={V}_{1},\beta ―,{V}_{m}$ is a peak if and only if there exists an $i$ ($1) such that ${V}_{i-1}<{V}_{i}$ and ${V}_{i}={V}_{i+1}=\beta ―={V}_{p}$ and ${V}_{p}>{V}_{p+1}$. A valley variable ${V}_{v}$ ($1) is a potential big valley wrt a non-negative integer $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}$ if and only if:

1. ${V}_{v}$ is a valley,

2. $\beta i,j\beta \left[1,m\right]|i, ${V}_{i}$ is a peak (or $i=1$ if there is no peak before position $p$), ${V}_{j}$ is a peak (or $i=m$ if there is no peak after position $p$), ${V}_{i}-{V}_{v}>\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}$, and ${V}_{j}-{V}_{v}>\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}$.

Let ${i}_{v}$ and ${j}_{v}$ be the largest $i$ and the smallest $j$ satisfying conditionΒ 2. Now a potential big valley ${V}_{v}$ ($1) is a big valley if and only if the interval $\left[i,j\right]$ does not contain any potential big valley that is strictly less than ${V}_{v}$. The constraint $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ holds if and only if $\mathrm{\pi ½}$ is the total number of big valleys of the sequence of variables $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$.

Example
 $\left(7,β©9,11,11,9,10,5,7,6,6,4,8,7,10,1,1,7,7,5,9,8,12βͺ,0\right)$ $\left(4,β©9,11,11,9,10,5,7,6,6,4,8,7,10,1,1,7,7,5,9,8,12βͺ,1\right)$

As shown part PartΒ (A) of FigureΒ 5.52.1, the first $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint holds since the sequence $91111910576648710117759812$ contains seven big valleys wrt a tolerance of 0 (i.e., we consider standard valleys).

As shown part PartΒ (B) of FigureΒ 5.52.1, the second $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ constraint holds since the same sequence $91111910576648710117759812$ contains only four big valleys wrt a tolerance of 1.

Typical
 $\mathrm{\pi ½}\beta ₯1$ $|\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }|>6$ $\mathrm{\pi \pi \pi \pi \pi }$$\left(\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }.\mathrm{\pi \pi \pi }\right)>1$ $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}>1$
Symmetries
• Items of $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ can be reversed.

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

• Contractible wrt. $\mathrm{\pi  \pi °\pi \pi Έ\pi °\pi ±\pi »\pi ΄\pi }$ when $\mathrm{\pi ½}=0$ and $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}=0$.

Usage

Useful for constraining the number of big valleys of a sequence of domain variables, by ignoring too small peaks that artificially create small valleys wrt $\mathrm{\pi \pi Ύ\pi »\pi ΄\pi \pi °\pi ½\pi ²\pi ΄}$.

specialisation: $\mathrm{\pi \pi \pi \pi \pi \pi ’}$Β (the tolerance is set to 0 and removed).

Keywords
Automaton

FigureΒ 5.52.2 depicts the automaton associated with the $\mathrm{\pi \pi \pi }_\mathrm{\pi \pi \pi \pi \pi \pi ’}$ 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 signature variable ${S}_{i}$. The following signature constraint links ${\mathrm{\pi  \pi °\pi }}_{i}$, ${\mathrm{\pi  \pi °\pi }}_{i+1}$ and ${S}_{i}$: $\left({\mathrm{\pi  \pi °\pi }}_{i}<{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=0\right)\beta §\left({\mathrm{\pi  \pi °\pi }}_{i}={\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=1\right)\beta §\left({\mathrm{\pi  \pi °\pi }}_{i}>{\mathrm{\pi  \pi °\pi }}_{i+1}\beta {S}_{i}=2\right)$.