Global Constraint Catalog: Cno

5.231. no_valley

DESCRIPTION

LINKS

AUTOMATON

Origin

Derived from $𝚟𝚊𝚕𝚕𝚎𝚢$ .

Constraint

$𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢 (𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂)$

Argument

𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂

𝚌𝚘𝚕𝚕𝚎𝚌𝚝𝚒𝚘𝚗 (𝚟𝚊𝚛 - 𝚍𝚟𝚊𝚛)

Restrictions

| 𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 | > 0

𝚛𝚎𝚚𝚞𝚒𝚛𝚎𝚍

(𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂, 𝚟𝚊𝚛)

Purpose

A variable $V_{k}$ $(1 < k < m)$ of the sequence of variables $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂 = V_{1}, ..., V_{m}$ is a valley if and only if there exists an $i$ $(1 < i \leq k)$ such that $V_{i - 1} > V_{i}$ and $V_{i} = V_{i + 1} = ... = V_{k}$ and $V_{k} < V_{k + 1}$ . The total number of valleys of the sequence of variables $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ is equal to 0.

Example

(\begin{matrix} 〈\begin{matrix} 𝚟𝚊𝚛 - 1, \\ 𝚟𝚊𝚛 - 1, \\ 𝚟𝚊𝚛 - 4, \\ 𝚟𝚊𝚛 - 8, \\ 𝚟𝚊𝚛 - 8, \\ 𝚟𝚊𝚛 - 2 \end{matrix}〉 \end{matrix})

The $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint holds since the sequence $114882$ does not contain any valley.

Figure 5.231.1. A sequence without any valley

Symmetries

Items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ can be reversed.
One and the same constant can be added to the $𝚟𝚊𝚛$ attribute of all items of $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ .

See also

comparison swapped: $𝚗𝚘_𝚙𝚎𝚊𝚔$ .

generalisation: $𝚟𝚊𝚕𝚕𝚎𝚢$ (introduce a $𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎$ counting the number of valleys).

implied by: $𝚍𝚎𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐$ , $𝚐𝚕𝚘𝚋𝚊𝚕_𝚌𝚘𝚗𝚝𝚒𝚐𝚞𝚒𝚝𝚢$ , $𝚒𝚗𝚌𝚛𝚎𝚊𝚜𝚒𝚗𝚐$ .

related: $𝚙𝚎𝚊𝚔$ .

Keywords

characteristic of a constraint: automaton, automaton without counters, reified automaton constraint.

combinatorial object: sequence.

constraint network structure: sliding cyclic(1) constraint network(1).

Automaton: Figure 5.231.2 depicts the automaton associated with the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint. To each pair of consecutive variables $({𝚅𝙰𝚁}_{i}, {𝚅𝙰𝚁}_{i + 1})$ of the collection $𝚅𝙰𝚁𝙸𝙰𝙱𝙻𝙴𝚂$ corresponds a signature variable $𝚂_{i}$ . The following signature constraint links ${𝚅𝙰𝚁}_{i}$ , ${𝚅𝙰𝚁}_{i + 1}$ and $𝚂_{i}$ : $({𝚅𝙰𝚁}_{i} < {𝚅𝙰𝚁}_{i + 1} \Leftrightarrow 𝚂_{i} = 0) \land ({𝚅𝙰𝚁}_{i} = {𝚅𝙰𝚁}_{i + 1} \Leftrightarrow 𝚂_{i} = 1) \land ({𝚅𝙰𝚁}_{i} > {𝚅𝙰𝚁}_{i + 1} \Leftrightarrow 𝚂_{i} = 2)$ .

Figure 5.231.2. Automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint

Figure 5.231.3. Hypergraph of the reformulation corresponding to the automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint

Global Constraint Catalog

5. Global Constraint Catalogue

5.231. no_valley

Figure 5.231.1. A sequence without any valley

Figure 5.231.2. Automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint

Figure 5.231.3. Hypergraph of the reformulation corresponding to the automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint

W3C: XHTML - last update: 2010-12-6. SD.

Global Constraint Catalog

5. Global Constraint Catalogue

5.231. no_valley

Figure 5.231.1. A sequence without any valley

Figure 5.231.2. Automaton of the 𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢 constraint

Figure 5.231.3. Hypergraph of the reformulation corresponding to the automaton of the 𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢 constraint

W3C: XHTML - last update: 2010-12-6. SD.

Figure 5.231.2. Automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint

Figure 5.231.3. Hypergraph of the reformulation corresponding to the automaton of the $𝚗𝚘_𝚟𝚊𝚕𝚕𝚎𝚢$ constraint